Finite Group DatabasebetaFinite Group Databasebeta| order | 58920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic14730 |
| Hall | C3×C5×C491 ⋊ Q8 |
| Hall | C5×C491 ⋊ Dic6 |
| Hall | C3×C491 ⋊ Dic10 |
| Sylow | C491 ⋊ Dic30 |
| Hall | C3×C5 ⋊ Dic982 |
| Sylow | C5 ⋊ Dic2946 |
| Sylow | C3 ⋊ Dic4910 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 507 |
| transitive degree | μt(G) = 58920 |
A minimal faithful permutation representation — 2 generators on 507 points:
(1,2,3,4)(5,8,7,6)(9,10,11)(12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,507)(19,506)(20,505)(21,504)(22,503)(23,502)(24,501)(25,500)(26,499)(27,498)(28,497)(29,496)(30,495)(31,494)(32,493)(33,492)(34,491)(35,490)(36,489)(37,488)(38,487)(39,486)(40,485)(41,484)(42,483)(43,482)(44,481)(45,480)(46,479)(47,478)(48,477)(49,476)(50,475)(51,474)(52,473)(53,472)(54,471)(55,470)(56,469)(57,468)(58,467)(59,466)(60,465)(61,464)(62,463)(63,462)(64,461)(65,460)(66,459)(67,458)(68,457)(69,456)(70,455)(71,454)(72,453)(73,452)(74,451)(75,450)(76,449)(77,448)(78,447)(79,446)(80,445)(81,444)(82,443)(83,442)(84,441)(85,440)(86,439)(87,438)(88,437)(89,436)(90,435)(91,434)(92,433)(93,432)(94,431)(95,430)(96,429)(97,428)(98,427)(99,426)(100,425)(101,424)(102,423)(103,422)(104,421)(105,420)(106,419)(107,418)(108,417)(109,416)(110,415)(111,414)(112,413)(113,412)(114,411)(115,410)(116,409)(117,408)(118,407)(119,406)(120,405)(121,404)(122,403)(123,402)(124,401)(125,400)(126,399)(127,398)(128,397)(129,396)(130,395)(131,394)(132,393)(133,392)(134,391)(135,390)(136,389)(137,388)(138,387)(139,386)(140,385)(141,384)(142,383)(143,382)(144,381)(145,380)(146,379)(147,378)(148,377)(149,376)(150,375)(151,374)(152,373)(153,372)(154,371)(155,370)(156,369)(157,368)(158,367)(159,366)(160,365)(161,364)(162,363)(163,362)(164,361)(165,360)(166,359)(167,358)(168,357)(169,356)(170,355)(171,354)(172,353)(173,352)(174,351)(175,350)(176,349)(177,348)(178,347)(179,346)(180,345)(181,344)(182,343)(183,342)(184,341)(185,340)(186,339)(187,338)(188,337)(189,336)(190,335)(191,334)(192,333)(193,332)(194,331)(195,330)(196,329)(197,328)(198,327)(199,326)(200,325)(201,324)(202,323)(203,322)(204,321)(205,320)(206,319)(207,318)(208,317)(209,316)(210,315)(211,314)(212,313)(213,312)(214,311)(215,310)(216,309)(217,308)(218,307)(219,306)(220,305)(221,304)(222,303)(223,302)(224,301)(225,300)(226,299)(227,298)(228,297)(229,296)(230,295)(231,294)(232,293)(233,292)(234,291)(235,290)(236,289)(237,288)(238,287)(239,286)(240,285)(241,284)(242,283)(243,282)(244,281)(245,280)(246,279)(247,278)(248,277)(249,276)(250,275)(251,274)(252,273)(253,272)(254,271)(255,270)(256,269)(257,268)(258,267)(259,266)(260,265)(261,264)(262,263)Polycyclic presentation:
2^3,3,5,491|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^49014733 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 491]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^490);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);