Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #72209721 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | H(5,𝔽4) | A5 |
| LCS | G | |
| UCS | C1 ◁ C22 | |
11613 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 11613 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 416 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 416 points:
(1,2,7,3,22,11,21,12)(4,13,30,31,17,10,29,14)(5,9,24,19,6,20,32,23)(8,18,15,26,16,27,28,25)(33,105)(34,340)(35,363)(36,153)(37,306,43,309)(38,383)(39,408)(40,308)(41,124,56,130)(42,375,93,326)(44,416,94,365)(45,237,70,284)(46,90,155,228)(47,183,163,368)(48,371,169,399)(49,345,65,352)(50,398,69,405)(51,158)(52,300)(53,194)(54,83,91,199)(55,391,140,380)(57,310,156,189)(58,62,102,103)(59,165,145,403)(60,388,146,394)(61,325,147,409)(63,258,101,252)(64,411)(66,268)(67,279,185,104)(68,173)(71,342,186,181)(72,81)(73,139,188,318)(74,129)(75,312)(76,209)(77,407,114,261)(78,107)(79,412,217,410)(80,269,180,219)(82,190)(84,374)(85,125)(86,278)(87,372,132,413)(88,174)(89,113)(92,119,287,201)(95,343,179,220)(96,182)(97,270,254,293)(98,195)(99,246,255,171)(100,234)(106,393,280,164)(108,197)(109,402)(110,151)(111,134)(112,339,141,331)(115,305)(116,211,259,283)(117,349)(118,235)(120,187)(121,198)(122,236,276,392)(123,154,290,150)(126,142)(127,336)(128,364)(131,271)(133,205)(135,289)(136,304)(137,362)(138,192)(143,218,320,301)(144,357)(148,253,384,350)(149,406,334,265)(152,302)(157,206)(159,378)(160,281)(161,251)(162,200)(166,367,168,370)(167,373)(170,256)(172,415,247,315)(175,291,359,260)(176,327)(177,321,266,346)(178,322)(184,307)(191,347)(193,351,239,355)(196,245)(202,204)(203,207)(208,216)(210,377)(212,397,230,214)(213,227)(215,297)(221,223,344,222)(224,395)(225,275)(226,323)(229,250,231,244)(232,360)(233,294,333,414)(238,240)(241,267,249,324)(242,382)(243,401)(248,274)(257,381)(262,354)(263,285)(264,292)(272,313)(273,329)(277,389)(282,396)(286,338,316,328)(288,298)(295,390)(296,341)(299,385)(303,366)(311,404)(314,386)(317,319)(330,356)(332,361)(335,348)(337,358)(353,379)(369,400)(376,387);(2,4,3,6,8)(5,16,11,17,12)(7,15,25,10,32)(9,23,14,29,27)(13,24,21,28,26)(18,20,19,31,30)(33,63,266,201,254)(34,69,185,147,291)(35,36,289,108,311)(37,165,216,281,354)(38,229,255,399,293)(39,52,323,386,400)(41,397,277,347,117)(42,142,88,361,164)(43,403,336,366,285)(44,58,225,217,186)(45,271,350,171,156)(46,167,349,303,318)(47,228,248,129,191)(48,352,341,112,342)(49,205,236,333,189)(50,67,61,260,64)(51,101,177,119,97)(53,85,364,203,159)(54,375,245,374,121)(55,304,78,390,60)(56,214,272,348,206)(57,70,278,253,246)(59,66,256,107,315)(62,292,324,346,413)(65,113,392,233,310)(68,376,393,93,319)(71,94,102,337,79)(72,343,379,153,280)(73,383,398,240,363)(74,211,380,176,262)(76,295,174,344,218)(77,87,80,235,95)(81,351,294,416,175)(82,258,389,251,172)(83,340,339,274,378)(84,91,326,238,198)(86,241,227,314,150)(89,148,307,128,394)(90,144,111,335,163)(92,284,381,316,104)(96,231,99,371,270)(98,157,160,139,155)(100,204,415,145,184)(103,305,267,321,372)(105,412,195,308,368)(106,187,220,327,125)(109,161,192,396,356)(110,360,322,149,212)(114,132,180,162,179)(115,122,209,138,406)(116,296,244,127,408)(118,328,317,207,306)(120,355,414,365,359)(123,143,382,387,170)(124,358,370,152,369)(126,197,309,200,338)(130,275,367,136,330)(131,249,151,282,154)(133,384,268,135,388)(134,283,391,353,263)(137,243,222,257,239)(140,302,202,385,146)(141,181,169,345,288)(158,410,373,312,183)(166,261,325,219,215)(168,407,409,269,190)(173,221,301,273,299)(178,232,334,230,213)(182,405,196,194,188)(193,242,210,223,224)(199,411,331,357,404)(208,402,259,298,250)(226,265,264,276,329)(234,290,320,362,332)(237,395,286,279,287)(247,297,252,313,300)55 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,7,3,22,11,21,12)(4,13,30,31,17,10,29,14)(5,9,24,19,6,20,32,23)(8,18,15,26,16,27,28,25)(33,105)(34,340)(35,363)(36,153)(37,306,43,309)(38,383)(39,408)(40,308)(41,124,56,130)(42,375,93,326)(44,416,94,365)(45,237,70,284)(46,90,155,228)(47,183,163,368)(48,371,169,399)(49,345,65,352)(50,398,69,405)(51,158)(52,300)(53,194)(54,83,91,199)(55,391,140,380)(57,310,156,189)(58,62,102,103)(59,165,145,403)(60,388,146,394)(61,325,147,409)(63,258,101,252)(64,411)(66,268)(67,279,185,104)(68,173)(71,342,186,181)(72,81)(73,139,188,318)(74,129)(75,312)(76,209)(77,407,114,261)(78,107)(79,412,217,410)(80,269,180,219)(82,190)(84,374)(85,125)(86,278)(87,372,132,413)(88,174)(89,113)(92,119,287,201)(95,343,179,220)(96,182)(97,270,254,293)(98,195)(99,246,255,171)(100,234)(106,393,280,164)(108,197)(109,402)(110,151)(111,134)(112,339,141,331)(115,305)(116,211,259,283)(117,349)(118,235)(120,187)(121,198)(122,236,276,392)(123,154,290,150)(126,142)(127,336)(128,364)(131,271)(133,205)(135,289)(136,304)(137,362)(138,192)(143,218,320,301)(144,357)(148,253,384,350)(149,406,334,265)(152,302)(157,206)(159,378)(160,281)(161,251)(162,200)(166,367,168,370)(167,373)(170,256)(172,415,247,315)(175,291,359,260)(176,327)(177,321,266,346)(178,322)(184,307)(191,347)(193,351,239,355)(196,245)(202,204)(203,207)(208,216)(210,377)(212,397,230,214)(213,227)(215,297)(221,223,344,222)(224,395)(225,275)(226,323)(229,250,231,244)(232,360)(233,294,333,414)(238,240)(241,267,249,324)(242,382)(243,401)(248,274)(257,381)(262,354)(263,285)(264,292)(272,313)(273,329)(277,389)(282,396)(286,338,316,328)(288,298)(295,390)(296,341)(299,385)(303,366)(311,404)(314,386)(317,319)(330,356)(332,361)(335,348)(337,358)(353,379)(369,400)(376,387), (2,4,3,6,8)(5,16,11,17,12)(7,15,25,10,32)(9,23,14,29,27)(13,24,21,28,26)(18,20,19,31,30)(33,63,266,201,254)(34,69,185,147,291)(35,36,289,108,311)(37,165,216,281,354)(38,229,255,399,293)(39,52,323,386,400)(41,397,277,347,117)(42,142,88,361,164)(43,403,336,366,285)(44,58,225,217,186)(45,271,350,171,156)(46,167,349,303,318)(47,228,248,129,191)(48,352,341,112,342)(49,205,236,333,189)(50,67,61,260,64)(51,101,177,119,97)(53,85,364,203,159)(54,375,245,374,121)(55,304,78,390,60)(56,214,272,348,206)(57,70,278,253,246)(59,66,256,107,315)(62,292,324,346,413)(65,113,392,233,310)(68,376,393,93,319)(71,94,102,337,79)(72,343,379,153,280)(73,383,398,240,363)(74,211,380,176,262)(76,295,174,344,218)(77,87,80,235,95)(81,351,294,416,175)(82,258,389,251,172)(83,340,339,274,378)(84,91,326,238,198)(86,241,227,314,150)(89,148,307,128,394)(90,144,111,335,163)(92,284,381,316,104)(96,231,99,371,270)(98,157,160,139,155)(100,204,415,145,184)(103,305,267,321,372)(105,412,195,308,368)(106,187,220,327,125)(109,161,192,396,356)(110,360,322,149,212)(114,132,180,162,179)(115,122,209,138,406)(116,296,244,127,408)(118,328,317,207,306)(120,355,414,365,359)(123,143,382,387,170)(124,358,370,152,369)(126,197,309,200,338)(130,275,367,136,330)(131,249,151,282,154)(133,384,268,135,388)(134,283,391,353,263)(137,243,222,257,239)(140,302,202,385,146)(141,181,169,345,288)(158,410,373,312,183)(166,261,325,219,215)(168,407,409,269,190)(173,221,301,273,299)(178,232,334,230,213)(182,405,196,194,188)(193,242,210,223,224)(199,411,331,357,404)(208,402,259,298,250)(226,265,264,276,329)(234,290,320,362,332)(237,395,286,279,287)(247,297,252,313,300));