Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2×C44 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #72209703 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | H(5,𝔽4) | A5 |
| LCS | G | |
| UCS | C1 ◁ C22 | |
15715 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 15715 conjugacy classes of subgroups — too many to draw. Only its 12 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) = 768 |
A minimal faithful permutation representation — 3 generators on 416 points:
(2,4)(3,8)(6,14)(7,16)(10,19)(11,23)(17,18)(20,24)(21,27)(26,32)(28,29)(30,31)(33,35,43,45)(34,39,56,108)(36,48,70,83)(37,51,71,140)(38,53,80,129)(40,59,60,119)(41,63,109,47)(42,66,68,131)(44,73,75,149)(46,78,76,151)(49,85,139,257)(50,88,90,178)(52,95,96,189)(54,102,160,202)(55,105,205,65)(57,111,112,216)(58,115,209,106)(61,121,120,153)(62,124,125,163)(64,127,210,99)(67,133,134,248)(69,137,135,250)(72,141,142,116)(74,147,150,268)(77,154,145,264)(79,159,152,274)(81,161,126,226)(82,164,166,228)(84,167,238,359)(86,171,123,117)(87,173,162,287)(89,177,179,305)(91,180,122,236)(92,182,258,309)(93,185,132,246)(94,158,259,273)(97,191,190,322)(98,192,194,104)(100,198,242,329)(101,200,172,293)(103,156,279,271)(107,207,339,343)(110,211,215,244)(113,219,217,312)(114,222,224,199)(118,232,233,356)(128,241,144,263)(130,243,196,327)(136,253,235,355)(138,256,251,204)(143,262,234,357)(146,266,237,300)(148,270,252,362)(155,277,265,282)(157,195,272,326)(165,260,240,212)(168,193,325,183)(169,291,285,229)(170,249,363,366)(174,296,303,181)(175,299,304,254)(176,301,247,223)(184,311,245,364)(186,315,375,401)(187,317,320,314)(188,319,321,295)(197,328,239,360)(201,284,333,384)(203,330,334,404)(206,341,342,409)(208,307,398,281)(213,344,288,221)(214,278,275,381)(218,346,345,231)(220,280,340,382)(225,294,286,267)(227,348,261,316)(230,352,292,387)(255,373,369,338)(269,353,379,276)(283,385,383,386)(289,350,349,391)(290,337,408,347)(297,393,394,415)(298,365,390,313)(302,358,380,306)(308,368,374,397)(310,351,331,323)(318,400,332,407)(324,403,335,392)(336,377,389,361)(354,411,410,405)(367,371,378,412)(370,396,388,372)(376,402,413,399)(395,416,414,406);(1,2,7,15,6,21)(3,10)(4,12,18,14,25,29)(5,11,22,13,26,9)(8,28,16,19,17,27)(20,23,31,30,32,24)(33,173,200)(34,377,132)(35,407,184)(36,105,333)(37,214,252)(38,315,180)(39,404,69)(40,309,412)(41,89,391)(42,406,141)(43,287,293)(44,383,298)(45,400,245)(46,270,237)(47,266,269)(48,49,197)(50,355,166)(51,273,203)(52,119,396)(53,230,215)(54,145,279)(55,264,382)(56,361,93)(57,268,415)(58,227,349)(59,372,96)(60,182,371)(61,150,226)(62,314,220)(63,300,379)(64,131,395)(65,201,70)(66,414,210)(67,342,128)(68,416,116)(71,275,148)(72,171,123)(73,281,262)(74,161,120)(75,283,390)(76,362,146)(77,103,160)(78,316,410)(79,290,225)(80,401,236)(81,113,92)(82,90,253)(83,139,239)(84,376,351)(85,190,365)(86,142,117)(87,159,178)(88,162,274)(91,114,320)(94,411,402)(95,248,153)(97,313,257)(98,370,235)(99,288,221)(100,196,101)(102,346,397)(104,294,341)(106,303,331)(107,375,398)(108,330,135)(109,179,350)(110,129,292)(111,318,373)(112,147,393)(115,174,310)(118,363,256)(121,189,133)(122,224,187)(124,374,387)(125,317,340)(126,217,258)(127,213,344)(130,172,242)(134,206,144)(136,194,388)(137,167,232)(138,282,336)(140,158,334)(143,244,260)(149,307,357)(151,348,354)(152,408,286)(154,280,205)(155,403,302)(156,328,325)(157,305,324)(163,308,352)(164,378,329)(165,345,254)(168,271,360)(169,347,311)(170,204,233)(175,183,385)(176,278,185)(177,335,272)(181,195,366)(186,208,339)(188,384,199)(191,319,343)(192,267,409)(193,386,304)(198,228,367)(202,231,368)(207,322,295)(209,261,289)(211,212,234)(216,332,338)(218,299,240)(219,369,327)(222,321,284)(223,306,353)(229,297,263)(238,413,323)(241,291,394)(243,312,255)(246,247,381)(249,296,326)(250,359,356)(251,277,389)(259,405,399)(265,392,380)(276,301,358)(285,337,364);(2,3,7)(4,11,18)(5,12,9)(6,10,21)(8,17,24)(13,25,22)(14,26,29)(16,20,23)(19,28,31)(27,30,32)(33,34,38)(35,42,65)(36,47,40)(37,50,87)(39,55,104)(41,62,123)(43,56,80)(44,72,67)(45,68,105)(46,77,153)(48,82,163)(49,84,64)(51,92,181)(52,81,89)(53,98,131)(54,101,91)(57,110,106)(58,114,221)(59,117,228)(60,70,63)(61,97,103)(66,129,194)(69,136,252)(71,90,162)(73,144,165)(74,146,138)(75,142,134)(76,145,121)(78,156,278)(79,143,155)(83,166,124)(85,168,100)(86,109,125)(88,174,295)(93,184,170)(94,159,239)(95,187,316)(96,126,179)(99,196,193)(102,201,276)(107,206,340)(108,205,192)(111,213,229)(112,215,115)(113,218,208)(116,212,223)(118,231,315)(119,171,164)(120,190,279)(122,160,172)(127,130,183)(128,240,149)(132,245,363)(133,247,241)(135,235,148)(137,254,372)(139,238,210)(140,258,296)(141,260,301)(147,267,308)(150,237,251)(151,271,381)(152,234,265)(154,275,322)(157,281,232)(158,283,262)(161,227,338)(167,198,327)(169,199,211)(173,188,309)(175,270,297)(176,263,248)(177,255,314)(178,303,319)(180,306,384)(182,287,321)(185,313,318)(186,312,195)(189,320,348)(191,264,214)(197,259,274)(200,269,380)(202,333,353)(203,335,391)(204,337,225)(207,230,351)(209,224,344)(216,288,291)(217,345,398)(219,326,375)(220,339,342)(222,244,285)(226,261,373)(233,346,401)(236,358,284)(242,257,325)(243,359,329)(246,365,332)(249,367,390)(250,299,396)(253,370,393)(256,347,286)(266,368,408)(268,294,374)(272,307,356)(273,383,357)(277,385,328)(280,376,352)(282,386,360)(289,389,410)(290,300,397)(292,323,343)(293,379,302)(298,366,378)(304,362,394)(305,369,317)(310,399,409)(311,400,412)(324,350,334)(330,405,395)(331,402,341)(336,354,349)(355,388,415)(361,403,406)(364,407,371)(377,392,416)(382,413,387)(404,411,414)41 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((2,4)(3,8)(6,14)(7,16)(10,19)(11,23)(17,18)(20,24)(21,27)(26,32)(28,29)(30,31)(33,35,43,45)(34,39,56,108)(36,48,70,83)(37,51,71,140)(38,53,80,129)(40,59,60,119)(41,63,109,47)(42,66,68,131)(44,73,75,149)(46,78,76,151)(49,85,139,257)(50,88,90,178)(52,95,96,189)(54,102,160,202)(55,105,205,65)(57,111,112,216)(58,115,209,106)(61,121,120,153)(62,124,125,163)(64,127,210,99)(67,133,134,248)(69,137,135,250)(72,141,142,116)(74,147,150,268)(77,154,145,264)(79,159,152,274)(81,161,126,226)(82,164,166,228)(84,167,238,359)(86,171,123,117)(87,173,162,287)(89,177,179,305)(91,180,122,236)(92,182,258,309)(93,185,132,246)(94,158,259,273)(97,191,190,322)(98,192,194,104)(100,198,242,329)(101,200,172,293)(103,156,279,271)(107,207,339,343)(110,211,215,244)(113,219,217,312)(114,222,224,199)(118,232,233,356)(128,241,144,263)(130,243,196,327)(136,253,235,355)(138,256,251,204)(143,262,234,357)(146,266,237,300)(148,270,252,362)(155,277,265,282)(157,195,272,326)(165,260,240,212)(168,193,325,183)(169,291,285,229)(170,249,363,366)(174,296,303,181)(175,299,304,254)(176,301,247,223)(184,311,245,364)(186,315,375,401)(187,317,320,314)(188,319,321,295)(197,328,239,360)(201,284,333,384)(203,330,334,404)(206,341,342,409)(208,307,398,281)(213,344,288,221)(214,278,275,381)(218,346,345,231)(220,280,340,382)(225,294,286,267)(227,348,261,316)(230,352,292,387)(255,373,369,338)(269,353,379,276)(283,385,383,386)(289,350,349,391)(290,337,408,347)(297,393,394,415)(298,365,390,313)(302,358,380,306)(308,368,374,397)(310,351,331,323)(318,400,332,407)(324,403,335,392)(336,377,389,361)(354,411,410,405)(367,371,378,412)(370,396,388,372)(376,402,413,399)(395,416,414,406), (1,2,7,15,6,21)(3,10)(4,12,18,14,25,29)(5,11,22,13,26,9)(8,28,16,19,17,27)(20,23,31,30,32,24)(33,173,200)(34,377,132)(35,407,184)(36,105,333)(37,214,252)(38,315,180)(39,404,69)(40,309,412)(41,89,391)(42,406,141)(43,287,293)(44,383,298)(45,400,245)(46,270,237)(47,266,269)(48,49,197)(50,355,166)(51,273,203)(52,119,396)(53,230,215)(54,145,279)(55,264,382)(56,361,93)(57,268,415)(58,227,349)(59,372,96)(60,182,371)(61,150,226)(62,314,220)(63,300,379)(64,131,395)(65,201,70)(66,414,210)(67,342,128)(68,416,116)(71,275,148)(72,171,123)(73,281,262)(74,161,120)(75,283,390)(76,362,146)(77,103,160)(78,316,410)(79,290,225)(80,401,236)(81,113,92)(82,90,253)(83,139,239)(84,376,351)(85,190,365)(86,142,117)(87,159,178)(88,162,274)(91,114,320)(94,411,402)(95,248,153)(97,313,257)(98,370,235)(99,288,221)(100,196,101)(102,346,397)(104,294,341)(106,303,331)(107,375,398)(108,330,135)(109,179,350)(110,129,292)(111,318,373)(112,147,393)(115,174,310)(118,363,256)(121,189,133)(122,224,187)(124,374,387)(125,317,340)(126,217,258)(127,213,344)(130,172,242)(134,206,144)(136,194,388)(137,167,232)(138,282,336)(140,158,334)(143,244,260)(149,307,357)(151,348,354)(152,408,286)(154,280,205)(155,403,302)(156,328,325)(157,305,324)(163,308,352)(164,378,329)(165,345,254)(168,271,360)(169,347,311)(170,204,233)(175,183,385)(176,278,185)(177,335,272)(181,195,366)(186,208,339)(188,384,199)(191,319,343)(192,267,409)(193,386,304)(198,228,367)(202,231,368)(207,322,295)(209,261,289)(211,212,234)(216,332,338)(218,299,240)(219,369,327)(222,321,284)(223,306,353)(229,297,263)(238,413,323)(241,291,394)(243,312,255)(246,247,381)(249,296,326)(250,359,356)(251,277,389)(259,405,399)(265,392,380)(276,301,358)(285,337,364), (2,3,7)(4,11,18)(5,12,9)(6,10,21)(8,17,24)(13,25,22)(14,26,29)(16,20,23)(19,28,31)(27,30,32)(33,34,38)(35,42,65)(36,47,40)(37,50,87)(39,55,104)(41,62,123)(43,56,80)(44,72,67)(45,68,105)(46,77,153)(48,82,163)(49,84,64)(51,92,181)(52,81,89)(53,98,131)(54,101,91)(57,110,106)(58,114,221)(59,117,228)(60,70,63)(61,97,103)(66,129,194)(69,136,252)(71,90,162)(73,144,165)(74,146,138)(75,142,134)(76,145,121)(78,156,278)(79,143,155)(83,166,124)(85,168,100)(86,109,125)(88,174,295)(93,184,170)(94,159,239)(95,187,316)(96,126,179)(99,196,193)(102,201,276)(107,206,340)(108,205,192)(111,213,229)(112,215,115)(113,218,208)(116,212,223)(118,231,315)(119,171,164)(120,190,279)(122,160,172)(127,130,183)(128,240,149)(132,245,363)(133,247,241)(135,235,148)(137,254,372)(139,238,210)(140,258,296)(141,260,301)(147,267,308)(150,237,251)(151,271,381)(152,234,265)(154,275,322)(157,281,232)(158,283,262)(161,227,338)(167,198,327)(169,199,211)(173,188,309)(175,270,297)(176,263,248)(177,255,314)(178,303,319)(180,306,384)(182,287,321)(185,313,318)(186,312,195)(189,320,348)(191,264,214)(197,259,274)(200,269,380)(202,333,353)(203,335,391)(204,337,225)(207,230,351)(209,224,344)(216,288,291)(217,345,398)(219,326,375)(220,339,342)(222,244,285)(226,261,373)(233,346,401)(236,358,284)(242,257,325)(243,359,329)(246,365,332)(249,367,390)(250,299,396)(253,370,393)(256,347,286)(266,368,408)(268,294,374)(272,307,356)(273,383,357)(277,385,328)(280,376,352)(282,386,360)(289,389,410)(290,300,397)(292,323,343)(293,379,302)(298,366,378)(304,362,394)(305,369,317)(310,399,409)(311,400,412)(324,350,334)(330,405,395)(331,402,341)(336,354,349)(355,388,415)(361,403,406)(364,407,371)(377,392,416)(382,413,387)(404,411,414));