889:
41:
923:
917:
911:
905:
1326:
or sporadic — it is almost but not strictly a group of Lie type — which is why in some sources the number of sporadic groups is given as 27, instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, or both. The Tits group is the
883:
in the 1860s and the other twenty-one were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list
1395:, p. 504) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received." (At the time, the other sporadic groups had not been discovered.)
4879:
3426:
2630:
2268:
5022:
4642:
3880:
1981:
4499:
3738:
2121:
3277:
3063:
5168:"The sporadic simple groups may be roughly sorted as the Mathieu groups, the Leech lattice groups, Fischer's 3-transposition groups, the further Monster centralizers, and the half-dozen oddments."
2487:
2840:
2737:
2376:
4740:
4367:
4073:
5261:
5329:
4269:
3519:
5129:
4171:
3157:
2944:
5381:
3973:
3608:
486:
461:
424:
1922:
5506:"The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if the Tits group
5465:"This completes the determination of matrix generators for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group)."
5183:
from standard generators of each sporadic group. Most sporadic groups have multiple presentations & semi-presentations; the more prominent examples are listed.
3751:
788:
5932:
4784:
3324:
2531:
2169:
6338:
4923:
4543:
3781:
6233:
6573:
848:
infinite families plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. The
1291:
1933:
4413:
3658:
2041:
4298:= 2·3·5·7·11·19 ≈ 2
802:
346:
6527:
6011:
5879:
3201:
2990:
296:
2420:
781:
291:
2780:
6615:
6132:
5731:
2680:
2319:
6051:
4686:
4313:
4019:
1300:≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in
4908:= 2·3·5·11 ≈ 8
4769:= 2·3·5·11 ≈ 1
4671:= 2·3·5·7·11 ≈ 4
4528:= 2·3·5·7·11·23 ≈ 1
4106:= 2·3·5·17·19 ≈ 5
5149:
The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups F
4398:= 2·3·5·7·11·23 ≈ 2
6675:
3744:
1105:
900:. A connecting line means the lower group is subquotient of the upper – and no sporadic subquotient in between.
707:
5995:
5941:
5195:
3439:
2750:
2003:
774:
5266:
4219:
3469:
6729:
5079:
4121:
3897:
3107:
2894:
2636:
1197:
841:
6347:
6334:"The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups"
391:
205:
5705:
4383:
4204:= 2·3·5·7 ≈ 6
3912:= 2·3·5·7·11 ≈ 4
3766:= 2·3·5·7·11 ≈ 9
3432:
3309:= 2·3·5·7·11·19·31 ≈ 5
2765:= 2·3·5·7·11·31·37·67 ≈ 5
1402:, p. 247). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.
1143:
880:
123:
6592:
5064:= 2·3·5·13 ≈ 2
4004:= 2·3·5·7·11·23·29·31·37·43 ≈ 9
3547:= 2·3·5·7·13·29 ≈ 1
3454:= 2·3·5·7·11·13 ≈ 4
3186:= 2·3·5·7·11·23 ≈ 5
3092:= 2·3·5·7·11·23 ≈ 4
2879:= 2·3·5·7·11·19 ≈ 3
2516:= 2·3·5·7·11·13 ≈ 6
6453:
6397:
5334:
5180:
2405:= 2·3·5·7·11·13·17·23 ≈ 4
3927:
3562:
6724:
589:
323:
200:
88:
5617:
Tabelle 2. Die
Entdeckung der sporadischen Gruppen (Table 2. The discovery of the sporadic groups)
469:
444:
407:
6519:
5866:
1428:
739:
529:
5153:(2)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
1886:
6181:
613:
6392:
5034:
1375:
6659:
6625:
6588:
6537:
6481:
6422:
6365:
6306:
6260:
6207:
6142:
6093:
6063:
6021:
5959:
5889:
5861:
5805:
5783:
5741:
1874:
553:
541:
159:
93:
6649:
6553:
6497:
6438:
6381:
6322:
6268:
6241:
Jahresber. Deutsch. Math.-Verein. (Annual Report of the German
Mathematicians Association)
6223:
6158:
6109:
5975:
5913:
5839:
3643:= 2·3·5·7·17 ≈ 4
2975:= 2·3·5·7·11·13·23 ≈ 4
867:, is the largest of the sporadic groups, and all but six of the other sporadic groups are
8:
6465:
6449:
6406:
2846:
2665:= 2·3·5·7·13·19·31 ≈ 9
2304:= 2·3·5·7·11·13·17·23·29 ≈ 1
2154:= 2·3·5·7·11·13·17·19·23·31·47 ≈ 4
1735:, if regarded as a sporadic group, would belong in this generation: there is a subgroup S
1345:
1312:
1167:
806:
128:
23:
6067:
5787:
6637:
6512:
6485:
6426:
6369:
6310:
6211:
6097:
6033:
5963:
5901:
5849:
5827:
5767:
5753:
2960:
2493:
2382:
2274:
2127:
1796:, and has no involvements in sporadic simple groups except the ones already mentioned.
1502:
1308:
1287:
1279:
1219:
1080:
1071:
1062:
113:
85:
5814:
5771:
1575:
is the group of automorphisms preserving a quaternionic structure (modulo its center).
6692:
6663:
6629:
6611:
6541:
6523:
6430:
6373:
6248:
6195:
6146:
6128:
6101:
6037:
6025:
6007:
5985:
5967:
5923:
5905:
5893:
5875:
5857:
5819:
5757:
5745:
5727:
4885:
4746:
4648:
4505:
4373:
3886:
1464:
1323:
1097:
970:
961:
952:
943:
934:
853:
518:
361:
255:
6641:
6489:
6314:
6215:
5950:
5927:
5831:
684:
6695:
6645:
6603:
6549:
6493:
6473:
6469:
6434:
6410:
6377:
6351:
6318:
6292:
6264:
6219:
6185:
6169:
6154:
6105:
6079:
6071:
5999:
5971:
5945:
5909:
5867:
ATLAS of Finite Groups: Maximal
Subgroups and Ordinary Characters for Simple Groups
5835:
5809:
5791:
5772:"A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups"
5717:
5709:
5695:
3893:
3294:
3163:
3069:
2950:
2650:
1275:
1271:
1050:
1041:
1032:
669:
661:
653:
645:
637:
625:
565:
505:
495:
337:
279:
154:
6621:
6607:
6578:
6533:
6477:
6418:
6361:
6302:
6256:
6203:
6138:
6124:
6089:
6017:
5955:
5928:"The classification of finite simple groups. I. Simple groups and local analysis"
5885:
5871:
5853:
5801:
5737:
3532:
2864:
2860:
1623:
1267:
1263:
833:
753:
746:
732:
689:
577:
500:
330:
244:
184:
64:
6709:
1554:
is the group of automorphisms preserving a complex structure (modulo its center)
4275:
4177:
4079:
3989:
3979:
3525:
3283:
1448:
1424:
1158:
1135:
1018:
1005:
992:
983:
760:
696:
386:
303:
268:
189:
179:
164:
149:
103:
80:
6356:
6333:
6029:
6003:
1579:
6718:
6633:
6545:
6252:
6199:
6047:
5722:
1989:
1510:
1492:
1488:
1436:
1416:
1242:
1059:
931:
860:
679:
601:
435:
308:
174:
6190:
6150:
5897:
5749:
5713:
888:
6297:
6280:
5823:
5049:
1844:
1805:
1028:
825:
534:
233:
222:
169:
144:
139:
98:
69:
32:
6414:
5796:
1618:′ has a triple cover which is the centralizer of an element of order 3 in
1290:
and semi-presentations. The degrees of minimal faithful representation or
6667:
6563:
6084:
3628:
2743:
1713:
Finally, the
Monster group itself is considered to be in this generation.
1603:
1506:
1498:
1476:
1189:
980:
893:
868:
1724:
and a group of order 11 is the centralizer of an element of order 11 in
6600:
The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249)
6507:
6393:"Smallest degrees of representations of exceptional groups of Lie type"
6075:
5994:. Mathematical Surveys and Monographs. Vol. 40. Providence, R.I.:
5989:
5699:
5028:
3614:
1850:
1732:
1316:
1113:
852:
is sometimes regarded as a sporadic group because it is not strictly a
849:
701:
429:
1706:
and a group of order 7 is the centralizer of an element of order 7 in
1689:
and a group of order 5 is the centralizer of an element of order 5 in
1671:
and a group of order 3 is the centralizer of an element of order 3 in
6700:
1584:
Consists of subgroups which are closely related to the
Monster group
845:
522:
16:
Finite simple group type not classified as Lie, cyclic or alternating
6514:
Symmetry and the
Monster: One of the Greatest Quests of Mathematics
4874:{\displaystyle o{\bigl (}{\bigr )}=o{\bigl (}ababab^{2}{\bigr )}=6}
1420:
59:
3421:{\displaystyle o{\bigl (}abab(b^{2}(b^{2})^{abab})^{5}{\bigr )}=5}
2625:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=12}
2263:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=23}
1482:
1442:
5017:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=4}
4637:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=8}
3875:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=7}
401:
315:
6602:. Cambridge, U.K: Cambridge University Press. pp. 261–273.
1364:
these finite simple groups coincide with the groups of Lie type
809:
which do not fit into any infinite family. These are called the
5516:
is counted also) which these infinite families do not include."
5165:, p. viii) organizes the 26 sporadic groups in likeness:
2007:
922:
916:
910:
904:
40:
1258:
Various constructions for these groups were first compiled in
840:
itself. The mentioned classification theorem states that the
1976:{\displaystyle \langle \langle a,b\mid o(z)\rangle \rangle }
1523:
is the quotient of the automorphism group by its center {±1}
1357:; thus in a strict sense not sporadic, nor of Lie type. For
4494:{\displaystyle o{\bigl (}ab(abab^{2})^{2}ab^{2}{\bigr )}=4}
1580:
Third generation (8 groups): other subgroups of the
Monster
6567:
3733:{\displaystyle o{\bigl (}ab^{2}abab^{2}ab^{2}{\bigr )}=10}
2116:{\displaystyle o{\bigl (}(ab)^{4}(ab^{2})^{2}{\bigr )}=50}
1792:(2)′ is also a subquotient of the (pariah) Rudvalis group
6658:
5666:
5527:
1283:
5991:
The classification of the finite simple groups, Number 3
5848:
5488:
5408:
5162:
1259:
6690:
5530:, Sporadic groups & Exceptional groups of Lie type)
6710:
6281:"Matrix generators for exceptional groups of Lie type"
6279:
Howlett, R. B.; Rylands, L. J.; Taylor, D. E. (2001).
5984:
5432:
3272:{\displaystyle o{\bigl (}(uvv)^{3}(uv)^{6}{\bigr )}=5}
3058:{\displaystyle o{\bigl (}ab(abab^{2})^{2}{\bigr )}=42}
6593:"Chapter: An Atlas of Sporadic Group Representations"
5337:
5269:
5198:
5082:
4926:
4787:
4689:
4546:
4416:
4316:
4222:
4124:
4022:
3930:
3784:
3661:
3565:
3472:
3327:
3204:
3110:
2993:
2897:
2783:
2683:
2534:
2423:
2322:
2172:
2044:
1936:
1889:
1532:
is the stabilizer of a type 2 (i.e., length 2) vector
1415:
Of the 26 sporadic groups, 20 can be seen inside the
472:
447:
410:
6278:
5459:
2482:{\displaystyle o{\bigl (}a^{bb}(ab)^{14}{\bigr )}=5}
1851:
Table of the sporadic group orders (with Tits group)
6454:"Semi-Presentations for the Sporadic Simple Groups"
6184:Department of Mathematics and Statistics: 106−118,
856:, in which case there would be 27 sporadic groups.
6511:
5375:
5323:
5255:
5123:
5016:
4873:
4734:
4636:
4493:
4361:
4263:
4165:
4067:
3967:
3874:
3732:
3602:
3513:
3420:
3271:
3151:
3057:
2938:
2834:
2731:
2624:
2481:
2370:
2262:
2115:
1975:
1916:
480:
455:
418:
6716:
6234:"Die Sporadischen Gruppen (The Sporadic Groups)"
2835:{\displaystyle o{\bigl (}ababab^{2}{\bigr )}=67}
1763:(2)′ is also a subquotient of the Fischer group
1463:= 11, 12, 22, 23 and 24 are multiply transitive
6447:
6170:"Polytopes Derived from Sporadic Simple Groups"
6168:Hartley, Michael I.; Hulpke, Alexander (2010),
5651:
2732:{\displaystyle o{\bigl (}(ab)^{3}b{\bigr )}=21}
2371:{\displaystyle o{\bigl (}(ab)^{3}b{\bigr )}=33}
1717:(This series continues further: the product of
1483:Second generation (7 groups): the Leech lattice
1443:First generation (5 groups): the Mathieu groups
1439:, and can be organized into three generations.
879:Five of the sporadic groups were discovered by
6123:. Springer Monographs in Mathematics. Berlin:
4735:{\displaystyle o{\bigl (}abab^{2}{\bigr )}=11}
4362:{\displaystyle o{\bigl (}abab^{2}{\bigr )}=19}
4068:{\displaystyle o{\bigl (}abab^{2}{\bigr )}=10}
6167:
5933:Bulletin of the American Mathematical Society
5662:
5660:
5500:
5110:
5088:
5003:
4932:
4860:
4828:
4815:
4793:
4721:
4695:
4623:
4552:
4480:
4422:
4348:
4322:
4250:
4228:
4152:
4130:
4054:
4028:
3861:
3790:
3719:
3667:
3500:
3478:
3407:
3333:
3258:
3210:
3138:
3116:
3044:
2999:
2925:
2903:
2821:
2789:
2718:
2689:
2611:
2540:
2468:
2429:
2357:
2328:
2249:
2178:
2152:4,154,781,481,226,426,191,177,580,544,000,000
2102:
2050:
782:
6568:"Orders of sporadic simple groups (A001228)"
2020:808,017,424,794,512,875,886,459,904,961,710,
1970:
1967:
1940:
1937:
1880:Degree of minimal faithful Brauer character
1541:is the stabilizer of a type 3 (i.e., length
6118:
6046:
5599:
5587:
5575:
5563:
5551:
5539:
5420:
6339:LMS Journal of Computation and Mathematics
5988:; Lyons, Richard; Solomon, Ronald (1998).
5922:
5657:
5476:
5057:
4901:
4762:
4664:
4521:
4391:
4291:
4197:
4099:
3997:
3905:
3759:
3636:
3540:
3447:
3302:
3179:
3085:
2968:
2872:
2758:
2658:
2509:
2398:
2297:
2147:
2015:
1651:has a double cover which is a subgroup of
1566:is the stabilizer of a type 2-3-3 triangle
1560:is the stabilizer of a type 2-2-3 triangle
789:
775:
6574:On-Line Encyclopedia of Integer Sequences
6355:
6296:
6189:
6083:
5949:
5813:
5795:
5721:
474:
449:
412:
5694:
5404:
5402:
5400:
1743:(2)′ normalising a 2C subgroup of
1392:
887:
5443:
5441:
5256:{\displaystyle u=(b^{2}(b^{2})abb)^{3}}
6717:
6666:; Nickerson, S.J.; Bray, J.N. (1999).
6587:
6562:
6390:
6331:
5766:
5677:
5639:
5628:
5482:
5324:{\displaystyle v=t(b^{2}(b^{2})t)^{2}}
4264:{\displaystyle o{\bigl (}{\bigr )}=12}
3514:{\displaystyle o{\bigl (}{\bigr )}=15}
1301:
803:classification of finite simple groups
347:Classification of finite simple groups
6691:
6672:ATLAS of Finite Group Representations
6506:
6174:Contributions to Discrete Mathematics
5447:
5433:Gorenstein, Lyons & Solomon (1998
5397:
5124:{\displaystyle o{\bigl (}{\bigr )}=5}
4166:{\displaystyle o{\bigl (}{\bigr )}=9}
3152:{\displaystyle o{\bigl (}{\bigr )}=4}
2939:{\displaystyle o{\bigl (}{\bigr )}=5}
1872:
1431:). These twenty have been called the
1399:
6231:
5611:
5438:
5460:Howlett, Rylands & Taylor (2001
2024:= 2·3·5·7·11·13·17·19·23·29·31
1471:points. They are all subgroups of M
1322:, which is sometimes considered of
13:
1475:, which is a permutation group on
1282:. These are also listed online at
902:The generations of Robert Griess:
14:
6741:
6684:
2302:1,255,205,709,190,661,721,292,800
1398:The diagram at right is based on
836:except for the trivial group and
6668:"ATLAS of Group Representations"
5701:Theory of groups of finite order
1602:has a double cover which is the
921:
915:
909:
903:
39:
6676:Queen Mary University of London
6285:Journal of Symbolic Computation
6119:Griess, Jr., Robert L. (1998).
5951:10.1090/S0273-0979-1979-14551-8
5671:
5645:
5633:
5622:
5605:
5593:
5581:
5569:
5557:
5545:
5533:
5521:
5376:{\displaystyle t=abab^{3}a^{2}}
5186:
5173:
5156:
1747:, giving rise to a subgroup 2·S
1410:
1405:
6474:10.1080/10586458.2005.10128927
5687:
5494:
5470:
5453:
5426:
5414:
5312:
5305:
5292:
5279:
5244:
5231:
5218:
5205:
5143:
5105:
5093:
4979:
4956:
4947:
4937:
4810:
4798:
4599:
4576:
4567:
4557:
4456:
4433:
4245:
4233:
4147:
4135:
3968:{\displaystyle o(abab^{2})=15}
3956:
3934:
3837:
3814:
3805:
3795:
3603:{\displaystyle o(abab^{2})=29}
3591:
3569:
3495:
3483:
3396:
3377:
3363:
3350:
3247:
3237:
3228:
3215:
3133:
3121:
3033:
3010:
2920:
2908:
2704:
2694:
2587:
2564:
2555:
2545:
2457:
2447:
2343:
2333:
2225:
2202:
2193:
2183:
2091:
2074:
2065:
2055:
1964:
1958:
1911:
1890:
1296:over fields of characteristic
708:Infinite dimensional Lie group
1:
5996:American Mathematical Society
5942:American Mathematical Society
5390:
1387:The earliest use of the term
6608:10.1017/CBO9780511565830.024
5776:Proc. Natl. Acad. Sci. U.S.A
5652:Nickerson & Wilson (2011
5070:
4914:
4775:
4677:
4534:
4404:
4304:
4210:
4112:
4010:
3918:
3772:
3649:
3553:
3460:
3315:
3192:
3098:
2981:
2885:
2771:
2671:
2522:
2411:
2310:
2160:
2032:
1755:(2)′ normalising a certain Q
1606:of an element of order 2 in
842:list of finite simple groups
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
7:
6348:London Mathematical Society
5704:(2nd ed.). Cambridge:
5061:
4905:
4766:
4668:
4525:
4395:
4295:
4201:
4103:
4001:
3909:
3763:
3640:
3544:
3451:
3306:
3183:
3089:
2972:
2876:
2762:
2662:
2513:
2402:
2301:
2151:
2022:757,005,754,368,000,000,000
2019:
1878:
1784:′, and of the Baby Monster
1286:, updated with their group
206:List of group theory topics
10:
6746:
6332:Jansen, Christoph (2005).
5874:. pp. xxxiii, 1–252.
5706:Cambridge University Press
5501:Hartley & Hulpke (2010
4002:86,775,571,046,077,562,880
1803:
1799:
1759:subgroup of the Monster. F
1486:
1446:
1344:of the infinite family of
6398:Communications in Algebra
6357:10.1112/S1461157000000930
6004:10.1112/S0024609398255457
2973:4,157,776,806,543,360,000
2403:4,089,470,473,293,004,800
1843:, sometimes known as the
1675:(in conjugacy class "3C")
1311:in the classification of
896:relations between the 26
6458:Experimental Mathematics
6056:Inventiones Mathematicae
5998:. pp. xiii, 1–362.
5708:. pp. xxiv, 1–512.
5602:, pp. 146, 150−152)
5136:
1917:{\displaystyle (a,b,ab)}
874:
805:, there are a number of
324:Elementary abelian group
201:Glossary of group theory
6522:. pp. vii, 1–255.
6520:Oxford University Press
6405:(5). Philadelphia, PA:
6191:10.11575/cdm.v5i2.61945
1810:The six exceptions are
832:that does not have any
6391:Lubeck, Frank (2001).
6298:10.1006/jsco.2000.0431
6232:Hiss, Gerhard (2003).
6121:Twelve Sporadic Groups
6048:Griess, Jr., Robert L.
5377:
5325:
5257:
5125:
5018:
4875:
4736:
4638:
4495:
4363:
4265:
4167:
4069:
3969:
3876:
3734:
3604:
3515:
3422:
3273:
3153:
3059:
2940:
2836:
2763:51,765,179,004,000,000
2733:
2663:90,745,943,887,872,000
2626:
2483:
2372:
2264:
2117:
1977:
1918:
1509:dimensions called the
927:
892:The diagram shows the
815:sporadic finite groups
811:sporadic simple groups
740:Linear algebraic group
482:
457:
420:
6415:10.1081/AGB-100002175
6182:University of Calgary
5797:10.1073/pnas.61.2.398
5714:10.1112/PLMS/S2-7.1.1
5378:
5326:
5258:
5126:
5019:
4876:
4737:
4639:
4496:
4364:
4266:
4168:
4070:
3970:
3877:
3735:
3605:
3516:
3423:
3274:
3154:
3060:
2941:
2837:
2734:
2627:
2484:
2373:
2265:
2118:
2026:·41·47·59·71 ≈ 8
1978:
1919:
891:
483:
458:
421:
6466:Taylor & Francis
6407:Taylor & Francis
6052:"The Friendly Giant"
5667:Wilson et al. (1999)
5335:
5267:
5196:
5080:
4924:
4785:
4687:
4544:
4414:
4314:
4220:
4122:
4020:
3928:
3782:
3659:
3563:
3470:
3325:
3202:
3108:
2991:
2895:
2781:
2681:
2532:
2421:
2320:
2170:
2042:
1934:
1887:
1313:finite simple groups
1284:Wilson et al. (1999)
1278:and orders of their
1260:Conway et al. (1985)
801:In the mathematical
470:
445:
408:
6730:Mathematical tables
6068:1982InMat..69....1G
5788:1968PNAS...61..398C
5642:, pp. 122–123)
5578:, pp. 146−150)
5566:, pp. 104–145)
5528:Wilson et al. (1999
5489:Conway et al. (1985
5450:, pp. 244–246)
5435:, pp. 262–302)
5409:Conway et al. (1985
5163:Conway et al. (1985
2877:273,030,912,000,000
1770:, and thus also of
1376:Ree groups of type
1280:outer automorphisms
1168:Harada-Norton group
114:Group homomorphisms
24:Algebraic structure
6693:Weisstein, Eric W.
6464:(3). Oxfordshire:
6180:(2), Alberta, CA:
6127:. pp. 1−169.
6076:10.1007/BF01389186
5373:
5321:
5253:
5181:semi-presentations
5121:
5014:
4871:
4732:
4634:
4491:
4359:
4261:
4163:
4065:
3965:
3872:
3730:
3600:
3511:
3418:
3269:
3149:
3090:42,305,421,312,000
3055:
2936:
2832:
2729:
2622:
2514:64,561,751,654,400
2479:
2368:
2260:
2113:
1984:Semi-presentation
1973:
1914:
1503:automorphism group
1465:permutation groups
1220:Baby Monster group
928:
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
6529:978-0-19-280722-9
6448:Nickerson, S.J.;
6013:978-0-8218-0391-2
5881:978-0-19-853199-9
5852:; Curtis, R. T.;
5723:2027/uc1.b4062919
5696:Burnside, William
5600:Griess, Jr. (1998
5590:, pp. 91−96)
5588:Griess, Jr. (1982
5576:Griess, Jr. (1998
5564:Griess, Jr. (1998
5554:, pp. 54–79)
5552:Griess, Jr. (1998
5540:Griess, Jr. (1982
5421:Griess, Jr. (1998
5134:
5133:
5045:
1635:is a subgroup of
1346:commutator groups
1293:Brauer characters
1276:Schur multipliers
1268:conjugacy classes
1098:Higman-Sims group
854:group of Lie type
799:
798:
374:
373:
256:Alternating group
213:
212:
6737:
6706:
6705:
6696:"Sporadic Group"
6679:
6653:
6597:
6582:
6564:Sloane, N. J. A.
6557:
6517:
6501:
6442:
6385:
6359:
6326:
6300:
6272:
6238:
6226:
6193:
6162:
6113:
6087:
6041:
5979:
5953:
5917:
5843:
5817:
5799:
5761:
5725:
5681:
5675:
5669:
5664:
5655:
5649:
5643:
5637:
5631:
5626:
5620:
5609:
5603:
5597:
5591:
5585:
5579:
5573:
5567:
5561:
5555:
5549:
5543:
5537:
5531:
5525:
5519:
5515:
5498:
5492:
5486:
5480:
5477:Gorenstein (1979
5474:
5468:
5457:
5451:
5445:
5436:
5430:
5424:
5418:
5412:
5406:
5384:
5382:
5380:
5379:
5374:
5372:
5371:
5362:
5361:
5330:
5328:
5327:
5322:
5320:
5319:
5304:
5303:
5291:
5290:
5262:
5260:
5259:
5254:
5252:
5251:
5230:
5229:
5217:
5216:
5190:
5184:
5179:Here listed are
5177:
5171:
5160:
5154:
5147:
5130:
5128:
5127:
5122:
5114:
5113:
5092:
5091:
5067:
5059:
5044:
5035:
5023:
5021:
5020:
5015:
5007:
5006:
5000:
4999:
4987:
4986:
4977:
4976:
4955:
4954:
4936:
4935:
4911:
4903:
4880:
4878:
4877:
4872:
4864:
4863:
4857:
4856:
4832:
4831:
4819:
4818:
4797:
4796:
4772:
4764:
4741:
4739:
4738:
4733:
4725:
4724:
4718:
4717:
4699:
4698:
4674:
4666:
4643:
4641:
4640:
4635:
4627:
4626:
4620:
4619:
4607:
4606:
4597:
4596:
4575:
4574:
4556:
4555:
4531:
4523:
4500:
4498:
4497:
4492:
4484:
4483:
4477:
4476:
4464:
4463:
4454:
4453:
4426:
4425:
4401:
4393:
4368:
4366:
4365:
4360:
4352:
4351:
4345:
4344:
4326:
4325:
4301:
4293:
4270:
4268:
4267:
4262:
4254:
4253:
4232:
4231:
4207:
4199:
4172:
4170:
4169:
4164:
4156:
4155:
4134:
4133:
4109:
4101:
4074:
4072:
4071:
4066:
4058:
4057:
4051:
4050:
4032:
4031:
4007:
3999:
3974:
3972:
3971:
3966:
3955:
3954:
3915:
3907:
3881:
3879:
3878:
3873:
3865:
3864:
3858:
3857:
3845:
3844:
3835:
3834:
3813:
3812:
3794:
3793:
3769:
3761:
3739:
3737:
3736:
3731:
3723:
3722:
3716:
3715:
3703:
3702:
3684:
3683:
3671:
3670:
3646:
3638:
3609:
3607:
3606:
3601:
3590:
3589:
3550:
3542:
3520:
3518:
3517:
3512:
3504:
3503:
3482:
3481:
3457:
3449:
3427:
3425:
3424:
3419:
3411:
3410:
3404:
3403:
3394:
3393:
3375:
3374:
3362:
3361:
3337:
3336:
3312:
3304:
3278:
3276:
3275:
3270:
3262:
3261:
3255:
3254:
3236:
3235:
3214:
3213:
3189:
3181:
3158:
3156:
3155:
3150:
3142:
3141:
3120:
3119:
3095:
3087:
3064:
3062:
3061:
3056:
3048:
3047:
3041:
3040:
3031:
3030:
3003:
3002:
2978:
2970:
2945:
2943:
2942:
2937:
2929:
2928:
2907:
2906:
2882:
2874:
2841:
2839:
2838:
2833:
2825:
2824:
2818:
2817:
2793:
2792:
2768:
2760:
2738:
2736:
2735:
2730:
2722:
2721:
2712:
2711:
2693:
2692:
2668:
2660:
2631:
2629:
2628:
2623:
2615:
2614:
2608:
2607:
2595:
2594:
2585:
2584:
2563:
2562:
2544:
2543:
2519:
2511:
2488:
2486:
2485:
2480:
2472:
2471:
2465:
2464:
2446:
2445:
2433:
2432:
2408:
2400:
2377:
2375:
2374:
2369:
2361:
2360:
2351:
2350:
2332:
2331:
2307:
2299:
2269:
2267:
2266:
2261:
2253:
2252:
2246:
2245:
2233:
2232:
2223:
2222:
2201:
2200:
2182:
2181:
2157:
2149:
2122:
2120:
2119:
2114:
2106:
2105:
2099:
2098:
2089:
2088:
2073:
2072:
2054:
2053:
2029:
2025:
2021:
2017:
1982:
1980:
1979:
1974:
1927:
1923:
1921:
1920:
1915:
1870:
1855:
1854:
1547:
1546:
1505:of a lattice in
1373:
1363:
1356:
1343:
1334:
1272:maximal subgroup
1264:character tables
1106:McLaughlin group
925:
919:
913:
907:
834:normal subgroups
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
485:
484:
479:
477:
462:
460:
459:
454:
452:
425:
423:
422:
417:
415:
338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
228:
227:
54:
53:
43:
30:
19:
18:
6745:
6744:
6740:
6739:
6738:
6736:
6735:
6734:
6725:Sporadic groups
6715:
6714:
6687:
6682:
6618:
6595:
6579:OEIS Foundation
6530:
6236:
6135:
6125:Springer-Verlag
6014:
5882:
5872:Clarendon Press
5734:
5690:
5685:
5684:
5680:, p. 2151)
5676:
5672:
5665:
5658:
5650:
5646:
5638:
5634:
5627:
5623:
5610:
5606:
5598:
5594:
5586:
5582:
5574:
5570:
5562:
5558:
5550:
5546:
5538:
5534:
5526:
5522:
5513:
5507:
5499:
5495:
5487:
5483:
5475:
5471:
5458:
5454:
5446:
5439:
5431:
5427:
5419:
5415:
5407:
5398:
5393:
5388:
5387:
5367:
5363:
5357:
5353:
5336:
5333:
5332:
5315:
5311:
5299:
5295:
5286:
5282:
5268:
5265:
5264:
5247:
5243:
5225:
5221:
5212:
5208:
5197:
5194:
5193:
5191:
5187:
5178:
5174:
5161:
5157:
5152:
5148:
5144:
5139:
5109:
5108:
5087:
5086:
5081:
5078:
5077:
5065:
5063:
5042:
5036:
5002:
5001:
4995:
4991:
4982:
4978:
4972:
4968:
4950:
4946:
4931:
4930:
4925:
4922:
4921:
4909:
4907:
4891:
4859:
4858:
4852:
4848:
4827:
4826:
4814:
4813:
4792:
4791:
4786:
4783:
4782:
4770:
4768:
4752:
4720:
4719:
4713:
4709:
4694:
4693:
4688:
4685:
4684:
4672:
4670:
4654:
4622:
4621:
4615:
4611:
4602:
4598:
4592:
4588:
4570:
4566:
4551:
4550:
4545:
4542:
4541:
4529:
4527:
4511:
4479:
4478:
4472:
4468:
4459:
4455:
4449:
4445:
4421:
4420:
4415:
4412:
4411:
4399:
4397:
4379:
4347:
4346:
4340:
4336:
4321:
4320:
4315:
4312:
4311:
4299:
4297:
4281:
4249:
4248:
4227:
4226:
4221:
4218:
4217:
4205:
4203:
4183:
4151:
4150:
4129:
4128:
4123:
4120:
4119:
4107:
4105:
4085:
4053:
4052:
4046:
4042:
4027:
4026:
4021:
4018:
4017:
4005:
4003:
3985:
3950:
3946:
3929:
3926:
3925:
3913:
3911:
3860:
3859:
3853:
3849:
3840:
3836:
3830:
3826:
3808:
3804:
3789:
3788:
3783:
3780:
3779:
3767:
3765:
3718:
3717:
3711:
3707:
3698:
3694:
3679:
3675:
3666:
3665:
3660:
3657:
3656:
3644:
3642:
3625:
3585:
3581:
3564:
3561:
3560:
3548:
3546:
3545:145,926,144,000
3499:
3498:
3477:
3476:
3471:
3468:
3467:
3455:
3453:
3452:448,345,497,600
3406:
3405:
3399:
3395:
3380:
3376:
3370:
3366:
3357:
3353:
3332:
3331:
3326:
3323:
3322:
3310:
3308:
3307:460,815,505,920
3257:
3256:
3250:
3246:
3231:
3227:
3209:
3208:
3203:
3200:
3199:
3187:
3185:
3184:495,766,656,000
3169:
3137:
3136:
3115:
3114:
3109:
3106:
3105:
3093:
3091:
3075:
3043:
3042:
3036:
3032:
3026:
3022:
2998:
2997:
2992:
2989:
2988:
2976:
2974:
2956:
2924:
2923:
2902:
2901:
2896:
2893:
2892:
2880:
2878:
2857:
2820:
2819:
2813:
2809:
2788:
2787:
2782:
2779:
2778:
2766:
2764:
2717:
2716:
2707:
2703:
2688:
2687:
2682:
2679:
2678:
2666:
2664:
2647:
2610:
2609:
2603:
2599:
2590:
2586:
2580:
2576:
2558:
2554:
2539:
2538:
2533:
2530:
2529:
2517:
2515:
2499:
2467:
2466:
2460:
2456:
2438:
2434:
2428:
2427:
2422:
2419:
2418:
2406:
2404:
2388:
2356:
2355:
2346:
2342:
2327:
2326:
2321:
2318:
2317:
2305:
2303:
2288:
2280:
2248:
2247:
2241:
2237:
2228:
2224:
2218:
2214:
2196:
2192:
2177:
2176:
2171:
2168:
2167:
2155:
2153:
2138:
2101:
2100:
2094:
2090:
2084:
2080:
2068:
2064:
2049:
2048:
2043:
2040:
2039:
2027:
2023:
2000:
1983:
1935:
1932:
1931:
1930:
1925:
1924:
1888:
1885:
1884:
1883:
1879:
1873:
1868:
1864:
1853:
1830:
1823:
1816:
1808:
1802:
1791:
1783:
1776:
1769:
1762:
1758:
1754:
1750:
1742:
1738:
1723:
1705:
1695:The product of
1688:
1678:The product of
1670:
1660:The product of
1657:
1650:
1641:
1634:
1624:conjugacy class
1617:
1601:
1582:
1574:
1544:
1542:
1540:
1531:
1522:
1495:
1485:
1474:
1458:
1451:
1445:
1413:
1408:
1382:
1371:
1365:
1358:
1354:
1348:
1341:
1335:
1328:
1254:
1241:Fischer-Griess
1238:
1231:
1216:
1209:
1186:
1179:
1155:
1132:
1125:
1093:
1086:
1077:
1068:
1054:
1045:
1036:
1024:
1011:
998:
989:
976:
967:
958:
949:
940:
901:
898:sporadic groups
885:
877:
844:consists of 18
819:sporadic groups
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
723:
722:
673:
665:
657:
649:
641:
614:Special unitary
525:
511:
510:
492:
491:
473:
471:
468:
467:
448:
446:
443:
442:
411:
409:
406:
405:
397:
396:
387:Discrete groups
376:
375:
331:Frobenius group
276:
263:
252:
245:Symmetric group
241:
225:
215:
214:
65:Normal subgroup
51:
31:
22:
17:
12:
11:
5:
6743:
6733:
6732:
6727:
6713:
6712:
6707:
6686:
6685:External links
6683:
6681:
6680:
6655:
6654:
6616:
6584:
6583:
6566:, ed. (1996).
6559:
6558:
6528:
6503:
6502:
6444:
6443:
6387:
6386:
6328:
6327:
6291:(4): 429–445.
6275:
6274:
6247:(4): 169−193.
6228:
6227:
6164:
6163:
6133:
6115:
6114:
6043:
6042:
6012:
5986:Gorenstein, D.
5981:
5980:
5936:. New Series.
5924:Gorenstein, D.
5919:
5918:
5880:
5845:
5844:
5782:(2): 398–400.
5763:
5762:
5732:
5691:
5689:
5686:
5683:
5682:
5670:
5656:
5644:
5632:
5621:
5619:
5618:
5604:
5592:
5580:
5568:
5556:
5544:
5532:
5520:
5518:
5517:
5511:
5493:
5481:
5469:
5467:
5466:
5452:
5437:
5425:
5423:, p. 146)
5413:
5395:
5394:
5392:
5389:
5386:
5385:
5370:
5366:
5360:
5356:
5352:
5349:
5346:
5343:
5340:
5318:
5314:
5310:
5307:
5302:
5298:
5294:
5289:
5285:
5281:
5278:
5275:
5272:
5250:
5246:
5242:
5239:
5236:
5233:
5228:
5224:
5220:
5215:
5211:
5207:
5204:
5201:
5185:
5172:
5170:
5169:
5155:
5150:
5141:
5140:
5138:
5135:
5132:
5131:
5120:
5117:
5112:
5107:
5104:
5101:
5098:
5095:
5090:
5085:
5075:
5072:
5069:
5060:
5055:
5052:
5047:
5040:
5025:
5024:
5013:
5010:
5005:
4998:
4994:
4990:
4985:
4981:
4975:
4971:
4967:
4964:
4961:
4958:
4953:
4949:
4945:
4942:
4939:
4934:
4929:
4919:
4916:
4913:
4904:
4899:
4896:
4893:
4889:
4882:
4881:
4870:
4867:
4862:
4855:
4851:
4847:
4844:
4841:
4838:
4835:
4830:
4825:
4822:
4817:
4812:
4809:
4806:
4803:
4800:
4795:
4790:
4780:
4777:
4774:
4765:
4760:
4757:
4754:
4750:
4743:
4742:
4731:
4728:
4723:
4716:
4712:
4708:
4705:
4702:
4697:
4692:
4682:
4679:
4676:
4667:
4662:
4659:
4656:
4652:
4645:
4644:
4633:
4630:
4625:
4618:
4614:
4610:
4605:
4601:
4595:
4591:
4587:
4584:
4581:
4578:
4573:
4569:
4565:
4562:
4559:
4554:
4549:
4539:
4536:
4533:
4524:
4519:
4516:
4513:
4509:
4502:
4501:
4490:
4487:
4482:
4475:
4471:
4467:
4462:
4458:
4452:
4448:
4444:
4441:
4438:
4435:
4432:
4429:
4424:
4419:
4409:
4406:
4403:
4394:
4389:
4386:
4381:
4377:
4370:
4369:
4358:
4355:
4350:
4343:
4339:
4335:
4332:
4329:
4324:
4319:
4309:
4306:
4303:
4294:
4289:
4286:
4283:
4279:
4272:
4271:
4260:
4257:
4252:
4247:
4244:
4241:
4238:
4235:
4230:
4225:
4215:
4212:
4209:
4200:
4195:
4192:
4189:
4181:
4174:
4173:
4162:
4159:
4154:
4149:
4146:
4143:
4140:
4137:
4132:
4127:
4117:
4114:
4111:
4102:
4097:
4094:
4091:
4083:
4076:
4075:
4064:
4061:
4056:
4049:
4045:
4041:
4038:
4035:
4030:
4025:
4015:
4012:
4009:
4000:
3995:
3992:
3987:
3983:
3976:
3975:
3964:
3961:
3958:
3953:
3949:
3945:
3942:
3939:
3936:
3933:
3923:
3920:
3917:
3908:
3903:
3900:
3891:
3883:
3882:
3871:
3868:
3863:
3856:
3852:
3848:
3843:
3839:
3833:
3829:
3825:
3822:
3819:
3816:
3811:
3807:
3803:
3800:
3797:
3792:
3787:
3777:
3774:
3771:
3762:
3757:
3754:
3749:
3741:
3740:
3729:
3726:
3721:
3714:
3710:
3706:
3701:
3697:
3693:
3690:
3687:
3682:
3678:
3674:
3669:
3664:
3654:
3651:
3648:
3639:
3634:
3631:
3626:
3623:
3611:
3610:
3599:
3596:
3593:
3588:
3584:
3580:
3577:
3574:
3571:
3568:
3558:
3555:
3552:
3543:
3538:
3535:
3530:
3522:
3521:
3510:
3507:
3502:
3497:
3494:
3491:
3488:
3485:
3480:
3475:
3465:
3462:
3459:
3450:
3445:
3442:
3437:
3429:
3428:
3417:
3414:
3409:
3402:
3398:
3392:
3389:
3386:
3383:
3379:
3373:
3369:
3365:
3360:
3356:
3352:
3349:
3346:
3343:
3340:
3335:
3330:
3320:
3317:
3314:
3305:
3300:
3297:
3292:
3280:
3279:
3268:
3265:
3260:
3253:
3249:
3245:
3242:
3239:
3234:
3230:
3226:
3223:
3220:
3217:
3212:
3207:
3197:
3194:
3191:
3182:
3177:
3174:
3171:
3167:
3160:
3159:
3148:
3145:
3140:
3135:
3132:
3129:
3126:
3123:
3118:
3113:
3103:
3100:
3097:
3088:
3083:
3080:
3077:
3073:
3066:
3065:
3054:
3051:
3046:
3039:
3035:
3029:
3025:
3021:
3018:
3015:
3012:
3009:
3006:
3001:
2996:
2986:
2983:
2980:
2971:
2966:
2963:
2958:
2954:
2947:
2946:
2935:
2932:
2927:
2922:
2919:
2916:
2913:
2910:
2905:
2900:
2890:
2887:
2884:
2875:
2870:
2867:
2858:
2855:
2843:
2842:
2831:
2828:
2823:
2816:
2812:
2808:
2805:
2802:
2799:
2796:
2791:
2786:
2776:
2773:
2770:
2761:
2756:
2753:
2748:
2740:
2739:
2728:
2725:
2720:
2715:
2710:
2706:
2702:
2699:
2696:
2691:
2686:
2676:
2673:
2670:
2661:
2656:
2653:
2648:
2645:
2633:
2632:
2621:
2618:
2613:
2606:
2602:
2598:
2593:
2589:
2583:
2579:
2575:
2572:
2569:
2566:
2561:
2557:
2553:
2550:
2547:
2542:
2537:
2527:
2524:
2521:
2512:
2507:
2504:
2501:
2497:
2490:
2489:
2478:
2475:
2470:
2463:
2459:
2455:
2452:
2449:
2444:
2441:
2437:
2431:
2426:
2416:
2413:
2410:
2401:
2396:
2393:
2390:
2386:
2379:
2378:
2367:
2364:
2359:
2354:
2349:
2345:
2341:
2338:
2335:
2330:
2325:
2315:
2312:
2309:
2300:
2295:
2292:
2289:
2286:
2278:
2271:
2270:
2259:
2256:
2251:
2244:
2240:
2236:
2231:
2227:
2221:
2217:
2213:
2210:
2207:
2204:
2199:
2195:
2191:
2188:
2185:
2180:
2175:
2165:
2162:
2159:
2150:
2145:
2142:
2139:
2136:
2124:
2123:
2112:
2109:
2104:
2097:
2093:
2087:
2083:
2079:
2076:
2071:
2067:
2063:
2060:
2057:
2052:
2047:
2037:
2034:
2031:
2018:
2013:
2010:
2001:
1998:
1986:
1985:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1928:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1881:
1877:
1871:
1866:
1862:
1859:
1852:
1849:
1828:
1821:
1814:
1804:Main article:
1801:
1798:
1789:
1781:
1774:
1767:
1760:
1756:
1752:
1748:
1740:
1736:
1721:
1715:
1714:
1711:
1703:
1693:
1686:
1676:
1668:
1658:
1655:
1648:
1643:
1639:
1632:
1627:
1615:
1610:
1599:
1581:
1578:
1577:
1576:
1572:
1567:
1561:
1555:
1549:
1538:
1533:
1529:
1524:
1520:
1484:
1481:
1472:
1454:
1449:Mathieu groups
1447:Main article:
1444:
1441:
1427:of subgroups (
1412:
1409:
1407:
1404:
1393:Burnside (1911
1389:sporadic group
1380:
1374:also known as
1369:
1352:
1339:
1256:
1255:
1252:
1239:
1236:
1229:
1217:
1214:
1207:
1198:Thompson group
1195:
1187:
1184:
1177:
1165:
1156:
1153:
1141:
1136:Rudvalis group
1133:
1130:
1123:
1111:
1103:
1095:
1091:
1084:
1075:
1066:
1060:Fischer groups
1057:
1052:
1043:
1034:
1026:
1022:
1009:
996:
987:
978:
974:
965:
956:
947:
938:
932:Mathieu groups
876:
873:
865:friendly giant
817:, or just the
797:
796:
794:
793:
786:
779:
771:
768:
767:
764:
763:
761:Elliptic curve
757:
756:
750:
749:
743:
742:
736:
731:
730:
727:
726:
721:
720:
717:
714:
710:
706:
705:
704:
699:
697:Diffeomorphism
693:
692:
687:
682:
676:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
633:
622:
621:
610:
609:
598:
597:
586:
585:
574:
573:
562:
561:
554:Special linear
550:
549:
542:General linear
538:
537:
532:
526:
517:
516:
513:
512:
509:
508:
503:
498:
490:
489:
476:
464:
451:
438:
436:Modular groups
434:
433:
432:
427:
414:
398:
395:
394:
389:
383:
382:
381:
378:
377:
372:
371:
370:
369:
364:
359:
356:
350:
349:
343:
342:
341:
340:
334:
333:
327:
326:
321:
312:
311:
309:Hall's theorem
306:
304:Sylow theorems
300:
299:
294:
286:
285:
284:
283:
277:
272:
269:Dihedral group
265:
264:
259:
253:
248:
242:
237:
226:
221:
220:
217:
216:
211:
210:
209:
208:
203:
195:
194:
193:
192:
187:
182:
177:
172:
167:
162:
160:multiplicative
157:
152:
147:
142:
134:
133:
132:
131:
126:
118:
117:
109:
108:
107:
106:
104:Wreath product
101:
96:
91:
89:direct product
83:
81:Quotient group
75:
74:
73:
72:
67:
62:
52:
49:
48:
45:
44:
36:
35:
15:
9:
6:
4:
3:
2:
6742:
6731:
6728:
6726:
6723:
6722:
6720:
6711:
6708:
6703:
6702:
6697:
6694:
6689:
6688:
6677:
6673:
6669:
6665:
6661:
6657:
6656:
6651:
6647:
6643:
6639:
6635:
6631:
6627:
6623:
6619:
6617:9780511565830
6613:
6609:
6605:
6601:
6594:
6590:
6586:
6585:
6580:
6576:
6575:
6569:
6565:
6561:
6560:
6555:
6551:
6547:
6543:
6539:
6535:
6531:
6525:
6521:
6516:
6515:
6509:
6505:
6504:
6499:
6495:
6491:
6487:
6483:
6479:
6475:
6471:
6467:
6463:
6459:
6455:
6451:
6446:
6445:
6440:
6436:
6432:
6428:
6424:
6420:
6416:
6412:
6409:: 2147−2169.
6408:
6404:
6400:
6399:
6394:
6389:
6388:
6383:
6379:
6375:
6371:
6367:
6363:
6358:
6353:
6349:
6345:
6341:
6340:
6335:
6330:
6329:
6324:
6320:
6316:
6312:
6308:
6304:
6299:
6294:
6290:
6286:
6282:
6277:
6276:
6270:
6266:
6262:
6258:
6254:
6250:
6246:
6242:
6235:
6230:
6229:
6225:
6221:
6217:
6213:
6209:
6205:
6201:
6197:
6192:
6187:
6183:
6179:
6175:
6171:
6166:
6165:
6160:
6156:
6152:
6148:
6144:
6140:
6136:
6134:9783540627784
6130:
6126:
6122:
6117:
6116:
6111:
6107:
6103:
6099:
6095:
6091:
6086:
6085:2027.42/46608
6081:
6077:
6073:
6069:
6065:
6061:
6057:
6053:
6049:
6045:
6044:
6039:
6035:
6031:
6027:
6023:
6019:
6015:
6009:
6005:
6001:
5997:
5993:
5992:
5987:
5983:
5982:
5977:
5973:
5969:
5965:
5961:
5957:
5952:
5947:
5943:
5939:
5935:
5934:
5929:
5925:
5921:
5920:
5915:
5911:
5907:
5903:
5899:
5895:
5891:
5887:
5883:
5877:
5873:
5869:
5868:
5863:
5862:Wilson, R. A.
5859:
5858:Parker, R. A.
5855:
5854:Norton, S. P.
5851:
5850:Conway, J. H.
5847:
5846:
5841:
5837:
5833:
5829:
5825:
5821:
5816:
5811:
5807:
5803:
5798:
5793:
5789:
5785:
5781:
5777:
5773:
5769:
5768:Conway, J. H.
5765:
5764:
5759:
5755:
5751:
5747:
5743:
5739:
5735:
5733:0-486-49575-2
5729:
5724:
5719:
5715:
5711:
5707:
5703:
5702:
5697:
5693:
5692:
5679:
5674:
5668:
5663:
5661:
5653:
5648:
5641:
5636:
5630:
5629:Sloane (1996)
5625:
5616:
5615:
5613:
5608:
5601:
5596:
5589:
5584:
5577:
5572:
5565:
5560:
5553:
5548:
5542:, p. 91)
5541:
5536:
5529:
5524:
5510:
5505:
5504:
5502:
5497:
5490:
5485:
5478:
5473:
5464:
5463:
5461:
5456:
5449:
5444:
5442:
5434:
5429:
5422:
5417:
5410:
5405:
5403:
5401:
5396:
5368:
5364:
5358:
5354:
5350:
5347:
5344:
5341:
5338:
5316:
5308:
5300:
5296:
5287:
5283:
5276:
5273:
5270:
5248:
5240:
5237:
5234:
5226:
5222:
5213:
5209:
5202:
5199:
5189:
5182:
5176:
5167:
5166:
5164:
5159:
5146:
5142:
5118:
5115:
5102:
5099:
5096:
5083:
5076:
5073:
5056:
5053:
5051:
5048:
5046:
5039:
5032:
5031:
5027:
5026:
5011:
5008:
4996:
4992:
4988:
4983:
4973:
4969:
4965:
4962:
4959:
4951:
4943:
4940:
4927:
4920:
4917:
4900:
4897:
4894:
4892:
4888:
4884:
4883:
4868:
4865:
4853:
4849:
4845:
4842:
4839:
4836:
4833:
4823:
4820:
4807:
4804:
4801:
4788:
4781:
4778:
4761:
4758:
4755:
4753:
4749:
4745:
4744:
4729:
4726:
4714:
4710:
4706:
4703:
4700:
4690:
4683:
4680:
4663:
4660:
4657:
4655:
4651:
4647:
4646:
4631:
4628:
4616:
4612:
4608:
4603:
4593:
4589:
4585:
4582:
4579:
4571:
4563:
4560:
4547:
4540:
4537:
4520:
4517:
4514:
4512:
4508:
4504:
4503:
4488:
4485:
4473:
4469:
4465:
4460:
4450:
4446:
4442:
4439:
4436:
4430:
4427:
4417:
4410:
4407:
4390:
4387:
4385:
4382:
4380:
4376:
4372:
4371:
4356:
4353:
4341:
4337:
4333:
4330:
4327:
4317:
4310:
4307:
4290:
4287:
4284:
4282:
4278:
4274:
4273:
4258:
4255:
4242:
4239:
4236:
4223:
4216:
4213:
4196:
4193:
4190:
4188:
4184:
4180:
4176:
4175:
4160:
4157:
4144:
4141:
4138:
4125:
4118:
4115:
4098:
4095:
4092:
4090:
4086:
4082:
4078:
4077:
4062:
4059:
4047:
4043:
4039:
4036:
4033:
4023:
4016:
4013:
3996:
3993:
3991:
3988:
3986:
3982:
3978:
3977:
3962:
3959:
3951:
3947:
3943:
3940:
3937:
3931:
3924:
3921:
3904:
3901:
3899:
3895:
3892:
3890:
3889:
3885:
3884:
3869:
3866:
3854:
3850:
3846:
3841:
3831:
3827:
3823:
3820:
3817:
3809:
3801:
3798:
3785:
3778:
3775:
3758:
3755:
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3750:
3748:
3747:
3743:
3742:
3727:
3724:
3712:
3708:
3704:
3699:
3695:
3691:
3688:
3685:
3680:
3676:
3672:
3662:
3655:
3652:
3641:4,030,387,200
3635:
3632:
3630:
3627:
3622:
3618:
3617:
3613:
3612:
3597:
3594:
3586:
3582:
3578:
3575:
3572:
3566:
3559:
3556:
3539:
3536:
3534:
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3528:
3524:
3523:
3508:
3505:
3492:
3489:
3486:
3473:
3466:
3463:
3446:
3443:
3441:
3438:
3436:
3435:
3431:
3430:
3415:
3412:
3400:
3390:
3387:
3384:
3381:
3371:
3367:
3358:
3354:
3347:
3344:
3341:
3338:
3328:
3321:
3318:
3301:
3298:
3296:
3293:
3291:
3287:
3286:
3282:
3281:
3266:
3263:
3251:
3243:
3240:
3232:
3224:
3221:
3218:
3205:
3198:
3195:
3178:
3175:
3172:
3170:
3166:
3162:
3161:
3146:
3143:
3130:
3127:
3124:
3111:
3104:
3101:
3084:
3081:
3078:
3076:
3072:
3068:
3067:
3052:
3049:
3037:
3027:
3023:
3019:
3016:
3013:
3007:
3004:
2994:
2987:
2984:
2967:
2964:
2962:
2959:
2957:
2953:
2949:
2948:
2933:
2930:
2917:
2914:
2911:
2898:
2891:
2888:
2871:
2868:
2866:
2862:
2859:
2854:
2850:
2849:
2845:
2844:
2829:
2826:
2814:
2810:
2806:
2803:
2800:
2797:
2794:
2784:
2777:
2774:
2757:
2754:
2752:
2749:
2747:
2746:
2742:
2741:
2726:
2723:
2713:
2708:
2700:
2697:
2684:
2677:
2674:
2657:
2654:
2652:
2649:
2644:
2640:
2639:
2635:
2634:
2619:
2616:
2604:
2600:
2596:
2591:
2581:
2577:
2573:
2570:
2567:
2559:
2551:
2548:
2535:
2528:
2525:
2508:
2505:
2502:
2500:
2496:
2492:
2491:
2476:
2473:
2461:
2453:
2450:
2442:
2439:
2435:
2424:
2417:
2414:
2397:
2394:
2391:
2389:
2385:
2381:
2380:
2365:
2362:
2352:
2347:
2339:
2336:
2323:
2316:
2313:
2296:
2293:
2290:
2285:
2281:
2277:
2273:
2272:
2257:
2254:
2242:
2238:
2234:
2229:
2219:
2215:
2211:
2208:
2205:
2197:
2189:
2186:
2173:
2166:
2163:
2146:
2143:
2140:
2135:
2131:
2130:
2126:
2125:
2110:
2107:
2095:
2085:
2081:
2077:
2069:
2061:
2058:
2045:
2038:
2035:
2014:
2011:
2009:
2005:
2002:
1997:
1993:
1992:
1988:
1987:
1961:
1955:
1952:
1949:
1946:
1943:
1929:
1908:
1905:
1902:
1899:
1896:
1893:
1882:
1876:
1867:
1863:
1860:
1857:
1856:
1848:
1846:
1842:
1838:
1834:
1827:
1820:
1813:
1807:
1797:
1795:
1787:
1780:
1773:
1766:
1746:
1734:
1729:
1727:
1720:
1712:
1709:
1702:
1698:
1694:
1692:
1685:
1681:
1677:
1674:
1667:
1663:
1659:
1654:
1647:
1644:
1638:
1631:
1628:
1625:
1621:
1614:
1611:
1609:
1605:
1598:
1594:
1591:
1590:
1589:
1587:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1537:
1534:
1528:
1525:
1519:
1516:
1515:
1514:
1512:
1511:Leech lattice
1508:
1504:
1500:
1494:
1493:Conway groups
1490:
1489:Leech lattice
1480:
1478:
1470:
1466:
1462:
1457:
1450:
1440:
1438:
1437:Robert Griess
1434:
1430:
1426:
1422:
1418:
1417:monster group
1403:
1401:
1396:
1394:
1390:
1385:
1383:
1379:
1368:
1361:
1351:
1347:
1338:
1332:
1325:
1321:
1318:
1314:
1310:
1305:
1303:
1302:Jansen (2005)
1299:
1295:
1294:
1289:
1288:presentations
1285:
1281:
1277:
1274:, as well as
1273:
1270:and lists of
1269:
1266:, individual
1265:
1261:
1251:
1247:
1244:
1243:Monster group
1240:
1235:
1228:
1224:
1221:
1218:
1213:
1206:
1202:
1199:
1196:
1194:
1191:
1188:
1183:
1176:
1172:
1169:
1166:
1163:
1160:
1157:
1152:
1148:
1145:
1142:
1140:
1137:
1134:
1129:
1122:
1118:
1115:
1112:
1110:
1107:
1104:
1102:
1099:
1096:
1094:
1090:
1083:
1078:
1074:
1069:
1065:
1061:
1058:
1056:
1055:
1047:
1046:
1038:
1037:
1030:
1029:Conway groups
1027:
1025:
1021:
1016:
1015:
1008:
1003:
1002:
995:
990:
986:
982:
979:
977:
973:
968:
964:
959:
955:
950:
946:
941:
937:
933:
930:
929:
924:
918:
912:
906:
899:
895:
890:
886:
882:
881:Émile Mathieu
872:
870:
866:
862:
861:monster group
857:
855:
851:
847:
843:
839:
835:
831:
827:
822:
820:
816:
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
345:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
111:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
21:
20:
6699:
6671:
6664:Parker, R.A.
6660:Wilson, R.A.
6599:
6571:
6518:. New York:
6513:
6461:
6457:
6450:Wilson, R.A.
6402:
6396:
6343:
6337:
6288:
6284:
6244:
6240:
6177:
6173:
6120:
6059:
6055:
5990:
5937:
5931:
5865:
5779:
5775:
5700:
5678:Lubeck (2001
5673:
5647:
5640:Jansen (2005
5635:
5624:
5607:
5595:
5583:
5571:
5559:
5547:
5535:
5523:
5508:
5496:
5484:
5472:
5455:
5428:
5416:
5188:
5175:
5158:
5145:
5037:
5029:
4886:
4747:
4649:
4506:
4374:
4276:
4186:
4178:
4088:
4080:
3980:
3887:
3745:
3620:
3615:
3526:
3433:
3289:
3284:
3164:
3070:
2951:
2852:
2847:
2744:
2642:
2637:
2494:
2383:
2283:
2275:
2133:
2128:
1995:
1990:
1840:
1836:
1832:
1825:
1818:
1811:
1809:
1806:Pariah group
1793:
1785:
1778:
1771:
1764:
1744:
1730:
1725:
1718:
1716:
1707:
1700:
1696:
1690:
1683:
1679:
1672:
1665:
1661:
1652:
1645:
1636:
1629:
1619:
1612:
1607:
1596:
1592:
1585:
1583:
1569:
1563:
1557:
1551:
1535:
1526:
1517:
1499:subquotients
1496:
1468:
1460:
1455:
1452:
1433:happy family
1432:
1414:
1411:Happy Family
1406:Organization
1397:
1388:
1386:
1377:
1366:
1359:
1349:
1336:
1330:
1319:
1306:
1297:
1292:
1262:, including
1257:
1249:
1245:
1233:
1226:
1222:
1211:
1204:
1200:
1192:
1181:
1174:
1170:
1161:
1150:
1146:
1144:Suzuki group
1138:
1127:
1120:
1116:
1108:
1100:
1088:
1081:
1072:
1063:
1049:
1040:
1031:
1019:
1013:
1006:
1000:
993:
984:
981:Janko groups
971:
962:
953:
944:
935:
897:
878:
869:subquotients
864:
858:
837:
829:
826:simple group
823:
818:
814:
810:
800:
629:
617:
605:
593:
581:
569:
557:
545:
366:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
6589:Wilson, R.A
6508:Ronan, Mark
6468:: 359−371.
6350:: 122−144.
5688:Works cited
5448:Ronan (2006
4396:244,823,040
3764:898,128,000
1861:Discoverer
1604:centralizer
1400:Ronan (2006
1333:= 0)-member
1190:Lyons group
1159:O'Nan group
1087:′ or
894:subquotient
828:is a group
519:Topological
358:alternating
6719:Categories
6650:0914.20016
6554:1113.00002
6498:1087.20025
6439:1004.20003
6382:1089.20006
6323:0987.20003
6269:1042.20007
6224:1320.51021
6159:0908.20007
6110:0498.20013
6030:6907721813
5976:0414.20009
5944:: 43–199.
5914:0568.20001
5870:. Oxford:
5840:0186.32401
5614:, p. 172)
5612:Hiss (2003
5411:, p. viii)
5391:References
5062:17,971,200
4779:2B, 3B, 11
4681:2A, 4A, 11
4526:10,200,960
4408:2B, 3A, 23
4116:2A, 3A, 19
4104:50,232,960
4014:2A, 4A, 37
3922:2A, 5A, 11
3910:44,352,000
3776:2A, 5A, 11
3752:McLaughlin
3653:2A, 7C, 17
3557:2B, 4A, 13
3464:2B, 3B, 13
3319:2A, 4A, 11
3196:2A, 7C, 17
3102:2A, 5A, 28
2985:2B, 3C, 40
2889:2A, 3B, 22
2526:2A, 13, 11
2415:2B, 3D, 28
2314:2A, 3E, 29
2164:2C, 3A, 55
2036:2A, 3B, 29
1926:Generators
1869:Generation
1733:Tits group
1487:See also:
1317:Tits group
1307:A further
1114:Held group
850:Tits group
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
6701:MathWorld
6634:726827806
6546:180766312
6431:122060727
6374:121362819
6253:0012-0456
6200:1715-0868
6102:123597150
6062:: 1−102.
6038:209854856
5968:121953006
5906:117473588
5758:117347785
5654:, p. 365)
5503:, p.106)
5491:, p.viii)
5462:, p.429)
5074:2A, 3, 13
4214:2B, 3B, 7
2775:2, 5A, 14
2675:2, 3A, 19
1971:⟩
1968:⟩
1953:∣
1941:⟨
1938:⟨
1425:quotients
1421:subgroups
1309:exception
846:countably
813:, or the
690:Conformal
578:Euclidean
185:nilpotent
6642:59394831
6591:(1998).
6510:(2006).
6490:13100616
6452:(2011).
6315:14682147
6273:(German)
6216:40845205
6151:38910263
6050:(1982).
5926:(1979).
5898:12106933
5864:(1985).
5832:29358882
5824:16591697
5770:(1968).
5750:54407807
5698:(1911).
5479:, p.111)
4918:2, 4, 11
4538:2, 4, 23
3533:Rudvalis
2651:Thompson
1548:) vector
1497:All the
1479:points.
1429:sections
1324:Lie type
1154:3−
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
6626:1647427
6538:2215662
6482:2172713
6423:1837968
6366:2153793
6307:1823074
6261:2033760
6208:2791293
6143:1707296
6094:0671653
6064:Bibcode
6022:1490581
5960:0513750
5890:0827219
5806:0237634
5784:Bibcode
5742:0069818
4895:Mathieu
4756:Mathieu
4669:443,520
4658:Mathieu
4515:Mathieu
4384:Mathieu
4308:2, 3, 7
4296:175,560
4202:604,800
2503:Fischer
2392:Fischer
2291:Fischer
2141:Fischer
2004:Fischer
1845:pariahs
1800:Pariahs
1751:×F
1739:×F
1543:√
1501:of the
1391:may be
1315:is the
871:of it.
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
6648:
6640:
6632:
6624:
6614:
6552:
6544:
6536:
6526:
6496:
6488:
6480:
6437:
6429:
6421:
6380:
6372:
6364:
6321:
6313:
6305:
6267:
6259:
6251:
6222:
6214:
6206:
6198:
6157:
6149:
6141:
6131:
6108:
6100:
6092:
6036:
6028:
6020:
6010:
5974:
5966:
5958:
5912:
5904:
5896:
5888:
5878:
5838:
5830:
5822:
5815:225171
5812:
5804:
5756:
5748:
5740:
5730:
5192:Where
4767:95,040
4292:Pariah
4100:Pariah
3998:Pariah
3894:Higman
3541:Pariah
3440:Suzuki
3303:Pariah
3173:Conway
3079:Conway
2961:Conway
2865:Norton
2861:Harada
2759:Pariah
2033:196883
2008:Griess
1858:Group
1839:, and
1362:> 0
926:Pariah
807:groups
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
6638:S2CID
6596:(PDF)
6486:S2CID
6427:S2CID
6370:S2CID
6311:S2CID
6237:(PDF)
6212:S2CID
6098:S2CID
6034:S2CID
5964:S2CID
5940:(1).
5902:S2CID
5828:S2CID
5754:S2CID
5331:with
5137:Notes
5054:1964
4906:7,920
4898:1861
4759:1861
4661:1861
4518:1861
4388:1861
4288:1965
4285:Janko
4194:1968
4191:Janko
4096:1968
4093:Janko
3994:1976
3990:Janko
3902:1967
3756:1969
3633:1969
3537:1972
3444:1969
3316:10944
3299:1976
3295:O'Nan
3176:1969
3082:1969
2965:1969
2869:1976
2755:1972
2751:Lyons
2655:1976
2506:1971
2395:1971
2294:1971
2144:1973
2012:1973
1875:Order
1865:Year
1626:"3A")
920:3rd,
914:2nd,
908:1st,
875:Names
863:, or
719:Sp(∞)
716:SU(∞)
129:image
6630:OCLC
6612:ISBN
6572:The
6542:OCLC
6524:ISBN
6249:ISSN
6196:ISSN
6147:OCLC
6129:ISBN
6026:OCLC
6008:ISBN
5894:OCLC
5876:ISBN
5820:PMID
5746:OCLC
5728:ISBN
5514:(2)′
5263:and
5050:Tits
5043:(2)′
4011:1333
3898:Sims
3629:Held
2772:2480
2311:8671
2161:4371
1777:and
1731:The
1622:(in
1491:and
1459:for
1372:(2),
1355:(2)′
1342:(2)′
1164:(ON)
859:The
713:O(∞)
702:Loop
521:and
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