Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| isoclinism | isoclinic class of #72235472 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C26 | #11757162 |
| Fitting, cosocle | #72235019 | A5 |
| socle | C24 | #72209603 |
| LCS | G | |
| UCS | C1 | |
1206 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 320 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 320 points:
(1,4,15,54)(2,8,31,96)(3,11,42,90)(5,19,65,153)(6,23,33,74)(7,27,59,146)(9,34,103,161)(10,38,21,50)(12,44,120,121)(13,18,62,111)(14,51)(16,20,67,140)(17,32,35,105)(22,72,100,98)(24,76,58,112)(25,78,164,251)(26,81,70,126)(28,88,166,177)(29,60,139,75)(30,93,184,274)(36,108,106,149)(37,43,69,117)(39,114,61,101)(40,53,46,82)(41,113,201,138)(45,63,95,56)(47,64,152,233)(48,118,128,71)(49,131,143,203)(52,73,159,127)(55,66,110,151)(57,144,91,125)(68,141,109,147)(77,85,80,167)(79,162,89,170)(83,157,247,156)(84,173,256,196)(86,165,137,231)(87,123,97,104)(92,182,270,259)(94,187,199,185)(99,191,188,171)(102,195)(107,198,168,254)(115,163)(116,205,134,222)(119,154,148,189)(122,209,250,176)(124,213,206,227)(129,215,266,234)(130,219,292,297)(132,224,200,290)(133,178)(135,229,238,211)(136,142)(145,207,193,217)(150,242,230,221)(155,160)(158,248,300,240)(169,175,197,252)(172,228,183,180)(174,210)(179,267,225,181)(186,276,293,312)(190,281,255,313)(192,264)(194,287,310,316)(202,272,208,296)(204,216,294,244)(212,237,235,226)(214,298,239,218)(220,282,309,262)(223,303,241,277)(232,286,246,304)(236,257,305,319)(243,295,249,299)(245,280,284,275)(253,285,263,307)(258,268,317,260)(261,315,269,302)(265,308,291,289)(271,283,301,288)(273,318,278,320)(279,311,306,314);(1,110,140)(2,69,107)(3,25,68)(4,94,132)(5,75,20)(6,89,31)(7,109,12)(8,51,255)(9,111,86)(10,21,22)(11,193,215)(13,155,214)(14,300,74)(15,60,61)(16,197,179)(17,18,156)(19,115,150)(23,159,158)(24,137,26)(27,171,282)(28,87,118)(29,169,143)(30,232,38)(32,148,260)(33,164,106)(34,201,283)(35,45,166)(36,58,77)(37,79,97)(40,144,293)(41,194,149)(42,59,103)(43,113,265)(44,189,206)(46,47,208)(48,154,216)(49,139,122)(50,250,246)(52,239,62)(53,95,157)(54,184,200)(55,65,100)(56,91,226)(57,134,90)(63,233,291)(64,116,76)(66,173,230)(67,199,225)(70,71,80)(72,192,238)(73,278,147)(78,176,253)(81,142,92)(82,120,168)(83,191,314)(84,172,167)(85,195,311)(88,252,269)(93,297,251)(96,127,270)(98,188,245)(99,135,114)(101,102,284)(104,217,186)(105,178,273)(108,207,288)(112,138,234)(117,145,237)(119,190,126)(121,136,124)(123,125,202)(128,133,244)(129,307,295)(130,242,308)(131,266,236)(141,160,268)(146,174,319)(151,210,243)(152,310,161)(153,196,249)(162,264,262)(163,183,247)(165,274,263)(170,256,257)(175,277,254)(177,185,303)(180,304,235)(181,205,220)(182,287,271)(187,219,231)(198,209,261)(203,298,241)(204,285,286)(211,212,223)(213,315,290)(218,309,229)(221,248,305)(222,302,275)(224,289,306)(227,228,280)(240,292,267)(258,320,259)(272,296,317)(276,318,301)(279,299,294)(312,316,313)31 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((1,4,15,54)(2,8,31,96)(3,11,42,90)(5,19,65,153)(6,23,33,74)(7,27,59,146)(9,34,103,161)(10,38,21,50)(12,44,120,121)(13,18,62,111)(14,51)(16,20,67,140)(17,32,35,105)(22,72,100,98)(24,76,58,112)(25,78,164,251)(26,81,70,126)(28,88,166,177)(29,60,139,75)(30,93,184,274)(36,108,106,149)(37,43,69,117)(39,114,61,101)(40,53,46,82)(41,113,201,138)(45,63,95,56)(47,64,152,233)(48,118,128,71)(49,131,143,203)(52,73,159,127)(55,66,110,151)(57,144,91,125)(68,141,109,147)(77,85,80,167)(79,162,89,170)(83,157,247,156)(84,173,256,196)(86,165,137,231)(87,123,97,104)(92,182,270,259)(94,187,199,185)(99,191,188,171)(102,195)(107,198,168,254)(115,163)(116,205,134,222)(119,154,148,189)(122,209,250,176)(124,213,206,227)(129,215,266,234)(130,219,292,297)(132,224,200,290)(133,178)(135,229,238,211)(136,142)(145,207,193,217)(150,242,230,221)(155,160)(158,248,300,240)(169,175,197,252)(172,228,183,180)(174,210)(179,267,225,181)(186,276,293,312)(190,281,255,313)(192,264)(194,287,310,316)(202,272,208,296)(204,216,294,244)(212,237,235,226)(214,298,239,218)(220,282,309,262)(223,303,241,277)(232,286,246,304)(236,257,305,319)(243,295,249,299)(245,280,284,275)(253,285,263,307)(258,268,317,260)(261,315,269,302)(265,308,291,289)(271,283,301,288)(273,318,278,320)(279,311,306,314), (1,110,140)(2,69,107)(3,25,68)(4,94,132)(5,75,20)(6,89,31)(7,109,12)(8,51,255)(9,111,86)(10,21,22)(11,193,215)(13,155,214)(14,300,74)(15,60,61)(16,197,179)(17,18,156)(19,115,150)(23,159,158)(24,137,26)(27,171,282)(28,87,118)(29,169,143)(30,232,38)(32,148,260)(33,164,106)(34,201,283)(35,45,166)(36,58,77)(37,79,97)(40,144,293)(41,194,149)(42,59,103)(43,113,265)(44,189,206)(46,47,208)(48,154,216)(49,139,122)(50,250,246)(52,239,62)(53,95,157)(54,184,200)(55,65,100)(56,91,226)(57,134,90)(63,233,291)(64,116,76)(66,173,230)(67,199,225)(70,71,80)(72,192,238)(73,278,147)(78,176,253)(81,142,92)(82,120,168)(83,191,314)(84,172,167)(85,195,311)(88,252,269)(93,297,251)(96,127,270)(98,188,245)(99,135,114)(101,102,284)(104,217,186)(105,178,273)(108,207,288)(112,138,234)(117,145,237)(119,190,126)(121,136,124)(123,125,202)(128,133,244)(129,307,295)(130,242,308)(131,266,236)(141,160,268)(146,174,319)(151,210,243)(152,310,161)(153,196,249)(162,264,262)(163,183,247)(165,274,263)(170,256,257)(175,277,254)(177,185,303)(180,304,235)(181,205,220)(182,287,271)(187,219,231)(198,209,261)(203,298,241)(204,285,286)(211,212,223)(213,315,290)(218,309,229)(221,248,305)(222,302,275)(224,289,306)(227,228,280)(240,292,267)(258,320,259)(272,296,317)(276,318,301)(279,299,294)(312,316,313));