Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| Fitting | #72258201 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #72209703 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | #72258201 | A5 |
| socle | C2 | #72219284 |
| LCS | G | |
| UCS | C1 ◁ C4 | |
At least 25628 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has at least 25628 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 384 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 384 points:
(1,35,23,19,100,66)(2,93,68,9,34,59)(3,7,20,44,29,61)(4,120,115,16,95,227)(5,39,54,58,105,97)(6,65,11,21,24,37)(8,116,124,27,164,47)(10,188,180,30,291,286)(12,55,98,40,18,36)(13,48,237,114,111,109)(14,56,73,49,131,60)(15,123,75,50,45,26)(17,127,171,51,182,152)(22,356,332,62,262,224)(25,223,74,79,278,168)(28,166,158,76,282,279)(31,248,295,80,133,336)(32,84,228,81,174,179)(33,307,195,82,351,134)(38,129,322,101,241,376)(41,160,281,107,78,243)(42,222,112,226,122,43)(46,302,375,119,273,330)(52,126,285,125,69,275)(53,276,277,113,202,155)(57,183,178,132,288,91)(63,335,247,139,110,317)(64,254,352,140,142,257)(67,305,216,147,349,106)(70,92,242,156,159,170)(71,99,169,172,154,77)(72,220,128,162,108,240)(83,181,177,173,287,87)(85,136,193,175,249,294)(86,290,165,176,185,138)(88,196,292,163,274,189)(89,221,161,153,197,167)(90,94,251,135,194,296)(96,201,347,203,130,380)(102,377,337,204,324,339)(103,208,348,205,309,245)(104,321,340,206,213,345)(117,284,250,235,304,150)(118,231,270,146,342,338)(121,298,283,239,368,366)(137,192,308,151,191,364)(141,315,144,253,200,256)(143,199,331,255,318,238)(145,280,365,229,333,360)(148,218,244,266,313,187)(149,269,301,267,329,369)(157,233,209,214,327,319)(184,346,325,289,236,297)(186,303,258,252,300,353)(190,370,312,293,310,371)(198,260,355,314,328,261)(207,378,320,264,379,211)(210,323,374,311,361,384)(212,344,350,316,373,382)(215,259,359,246,354,381)(217,363,268,306,271,362)(219,265,232,272,234,230)(225,357,383,334,263,358)(299,326,341,367,372,343);(1,61)(2,132,9,57)(3,13)(4,25,16,79)(5,45)(6,96,21,203)(7,51,29,17)(8,221,27,197)(10,55,30,18)(11,239,37,121)(12,74,40,168)(14,216)(15,48)(19,20)(22,35,62,100)(23,138)(24,99,65,154)(26,107,75,41)(28,120,76,95)(31,313,80,218)(32,89)(33,127)(34,133,93,248)(36,126,98,69)(38,124,101,47)(39,119)(42,331,226,238)(43,278,112,223)(44,114)(46,105)(49,106)(50,111)(52,103)(53,109,113,237)(54,229,97,145)(56,194,131,94)(58,123)(59,228,68,179)(60,67)(63,314)(64,305,140,349)(66,165)(70,366)(72,183)(73,147)(77,178)(78,160)(81,153)(82,182)(83,308)(84,159)(85,319)(86,196,176,274)(87,371,177,312)(88,243)(91,169)(92,174)(102,310,204,370)(104,356)(108,241)(110,222,335,122)(115,151,227,137)(116,201,164,130)(117,211,235,320)(118,347,146,380)(125,205)(128,284,240,304)(129,220)(134,330,195,375)(136,188)(139,198)(141,338,253,270)(143,225)(144,264)(148,367,266,299)(149,185)(150,336,250,295)(152,171)(155,332)(156,283)(157,282)(158,318)(162,288)(163,281)(166,214)(170,187,242,244)(173,364)(175,209)(180,286)(181,208)(184,307)(186,327,252,233)(189,325)(190,294,293,193)(191,192)(199,279)(200,298)(202,333)(206,262)(207,256)(210,236,311,346)(212,365)(213,321)(215,322)(217,245,306,348)(219,302)(224,277)(230,345,232,340)(231,354,342,259)(234,323)(246,376)(247,285)(249,291)(255,334)(258,261,353,355)(260,328)(263,271,357,363)(265,361)(267,290)(268,337,362,339)(269,344,329,373)(272,273)(275,317)(276,280)(287,309)(289,351)(292,297)(300,324,303,377)(301,374)(315,368)(316,360)(341,359)(343,381)(350,382)(358,383)(369,384);(1,2,6)(3,11,34)(4,14,8)(5,18,54)(7,23,65)(9,21,19)(10,31,83)(12,39,36)(13,42,109)(15,47,56)(16,49,27)(17,52,126)(20,59,35)(22,63,141)(24,29,66)(25,70,28)(26,73,120)(30,80,173)(32,87,133)(33,90,85)(37,93,44)(38,102,207)(40,105,98)(41,108,197)(43,111,222)(45,115,164)(46,117,230)(48,122,112)(50,124,131)(51,125,69)(53,89,128)(55,97,58)(57,134,136)(60,95,75)(61,68,100)(62,139,253)(64,144,110)(67,148,268)(71,158,92)(72,78,155)(74,166,154)(76,79,156)(77,170,278)(81,177,248)(82,135,175)(84,180,181)(86,185,258)(88,191,189)(91,193,94)(96,198,259)(99,168,282)(101,204,264)(103,211,324)(104,213,209)(106,217,329)(107,220,221)(113,153,240)(114,226,237)(116,123,227)(118,234,284)(119,235,232)(121,236,343)(127,152,275)(129,245,337)(130,143,261)(132,195,249)(137,196,308)(138,252,300)(140,256,335)(142,224,200)(145,225,263)(146,265,304)(147,266,362)(149,271,313)(150,273,270)(151,274,364)(157,233,340)(159,172,279)(160,277,162)(161,202,243)(163,192,292)(165,186,303)(167,276,281)(169,242,223)(171,285,182)(174,286,287)(176,290,353)(178,294,194)(179,295,188)(183,296,307)(184,298,350)(187,305,301)(190,310,374)(199,246,260)(201,255,355)(203,314,354)(205,320,377)(206,321,319)(208,322,378)(210,323,312)(212,326,325)(214,327,345)(215,328,318)(216,306,269)(218,267,363)(219,231,330)(228,336,291)(229,334,357)(238,347,359)(239,346,341)(241,348,339)(244,349,369)(247,262,257)(250,302,338)(251,351,288)(254,332,315)(272,342,375)(280,365,358)(283,367,344)(289,368,382)(293,370,384)(297,316,372)(299,373,366)(309,376,379)(311,361,371)(317,356,352)(331,380,381)(333,360,383)56 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((1,35,23,19,100,66)(2,93,68,9,34,59)(3,7,20,44,29,61)(4,120,115,16,95,227)(5,39,54,58,105,97)(6,65,11,21,24,37)(8,116,124,27,164,47)(10,188,180,30,291,286)(12,55,98,40,18,36)(13,48,237,114,111,109)(14,56,73,49,131,60)(15,123,75,50,45,26)(17,127,171,51,182,152)(22,356,332,62,262,224)(25,223,74,79,278,168)(28,166,158,76,282,279)(31,248,295,80,133,336)(32,84,228,81,174,179)(33,307,195,82,351,134)(38,129,322,101,241,376)(41,160,281,107,78,243)(42,222,112,226,122,43)(46,302,375,119,273,330)(52,126,285,125,69,275)(53,276,277,113,202,155)(57,183,178,132,288,91)(63,335,247,139,110,317)(64,254,352,140,142,257)(67,305,216,147,349,106)(70,92,242,156,159,170)(71,99,169,172,154,77)(72,220,128,162,108,240)(83,181,177,173,287,87)(85,136,193,175,249,294)(86,290,165,176,185,138)(88,196,292,163,274,189)(89,221,161,153,197,167)(90,94,251,135,194,296)(96,201,347,203,130,380)(102,377,337,204,324,339)(103,208,348,205,309,245)(104,321,340,206,213,345)(117,284,250,235,304,150)(118,231,270,146,342,338)(121,298,283,239,368,366)(137,192,308,151,191,364)(141,315,144,253,200,256)(143,199,331,255,318,238)(145,280,365,229,333,360)(148,218,244,266,313,187)(149,269,301,267,329,369)(157,233,209,214,327,319)(184,346,325,289,236,297)(186,303,258,252,300,353)(190,370,312,293,310,371)(198,260,355,314,328,261)(207,378,320,264,379,211)(210,323,374,311,361,384)(212,344,350,316,373,382)(215,259,359,246,354,381)(217,363,268,306,271,362)(219,265,232,272,234,230)(225,357,383,334,263,358)(299,326,341,367,372,343), (1,61)(2,132,9,57)(3,13)(4,25,16,79)(5,45)(6,96,21,203)(7,51,29,17)(8,221,27,197)(10,55,30,18)(11,239,37,121)(12,74,40,168)(14,216)(15,48)(19,20)(22,35,62,100)(23,138)(24,99,65,154)(26,107,75,41)(28,120,76,95)(31,313,80,218)(32,89)(33,127)(34,133,93,248)(36,126,98,69)(38,124,101,47)(39,119)(42,331,226,238)(43,278,112,223)(44,114)(46,105)(49,106)(50,111)(52,103)(53,109,113,237)(54,229,97,145)(56,194,131,94)(58,123)(59,228,68,179)(60,67)(63,314)(64,305,140,349)(66,165)(70,366)(72,183)(73,147)(77,178)(78,160)(81,153)(82,182)(83,308)(84,159)(85,319)(86,196,176,274)(87,371,177,312)(88,243)(91,169)(92,174)(102,310,204,370)(104,356)(108,241)(110,222,335,122)(115,151,227,137)(116,201,164,130)(117,211,235,320)(118,347,146,380)(125,205)(128,284,240,304)(129,220)(134,330,195,375)(136,188)(139,198)(141,338,253,270)(143,225)(144,264)(148,367,266,299)(149,185)(150,336,250,295)(152,171)(155,332)(156,283)(157,282)(158,318)(162,288)(163,281)(166,214)(170,187,242,244)(173,364)(175,209)(180,286)(181,208)(184,307)(186,327,252,233)(189,325)(190,294,293,193)(191,192)(199,279)(200,298)(202,333)(206,262)(207,256)(210,236,311,346)(212,365)(213,321)(215,322)(217,245,306,348)(219,302)(224,277)(230,345,232,340)(231,354,342,259)(234,323)(246,376)(247,285)(249,291)(255,334)(258,261,353,355)(260,328)(263,271,357,363)(265,361)(267,290)(268,337,362,339)(269,344,329,373)(272,273)(275,317)(276,280)(287,309)(289,351)(292,297)(300,324,303,377)(301,374)(315,368)(316,360)(341,359)(343,381)(350,382)(358,383)(369,384), (1,2,6)(3,11,34)(4,14,8)(5,18,54)(7,23,65)(9,21,19)(10,31,83)(12,39,36)(13,42,109)(15,47,56)(16,49,27)(17,52,126)(20,59,35)(22,63,141)(24,29,66)(25,70,28)(26,73,120)(30,80,173)(32,87,133)(33,90,85)(37,93,44)(38,102,207)(40,105,98)(41,108,197)(43,111,222)(45,115,164)(46,117,230)(48,122,112)(50,124,131)(51,125,69)(53,89,128)(55,97,58)(57,134,136)(60,95,75)(61,68,100)(62,139,253)(64,144,110)(67,148,268)(71,158,92)(72,78,155)(74,166,154)(76,79,156)(77,170,278)(81,177,248)(82,135,175)(84,180,181)(86,185,258)(88,191,189)(91,193,94)(96,198,259)(99,168,282)(101,204,264)(103,211,324)(104,213,209)(106,217,329)(107,220,221)(113,153,240)(114,226,237)(116,123,227)(118,234,284)(119,235,232)(121,236,343)(127,152,275)(129,245,337)(130,143,261)(132,195,249)(137,196,308)(138,252,300)(140,256,335)(142,224,200)(145,225,263)(146,265,304)(147,266,362)(149,271,313)(150,273,270)(151,274,364)(157,233,340)(159,172,279)(160,277,162)(161,202,243)(163,192,292)(165,186,303)(167,276,281)(169,242,223)(171,285,182)(174,286,287)(176,290,353)(178,294,194)(179,295,188)(183,296,307)(184,298,350)(187,305,301)(190,310,374)(199,246,260)(201,255,355)(203,314,354)(205,320,377)(206,321,319)(208,322,378)(210,323,312)(212,326,325)(214,327,345)(215,328,318)(216,306,269)(218,267,363)(219,231,330)(228,336,291)(229,334,357)(238,347,359)(239,346,341)(241,348,339)(244,349,369)(247,262,257)(250,302,338)(251,351,288)(254,332,315)(272,342,375)(280,365,358)(283,367,344)(289,368,382)(293,370,384)(297,316,372)(299,373,366)(309,376,379)(311,361,371)(317,356,352)(331,380,381)(333,360,383));