Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #72209715 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | C25×2−1+4 | A5 |
| LCS | G | |
| UCS | C1 ◁ C22 | |
At least 83716 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has at least 83716 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) = 332 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 332 points:
(1,10,2,9,11)(3,12,6,8,5)(13,241,78,277,313)(14,94,125,257,260)(15,248,243,311,215)(16,242,84,282,197)(17,206,133,217,303)(18,105,279,154,261)(19,111,266,239,317)(20,183,332,119,230)(21,224,128,222,323)(22,88,113,269,187)(23,137,203,258,115)(24,169,177,320,305)(25,245,223,290,147)(26,295,273,172,65)(27,171,259,211,145)(28,40,58,121,330)(29,236,86,264,214)(30,307,252,299,195)(31,81,85,288,293)(32,170,199,219,198)(33,104,129,240,164)(34,231,116,189,318)(35,141,122,253,71)(36,47,61,310,163)(37,204,235,300,156)(38,296,276,43,249)(39,244,110,124,327)(41,101,218,112,329)(42,275,331,213,175)(44,73,127,190,285)(45,83,268,272,130)(46,146,138,315,297)(48,136,325,97,139)(49,178,93,165,120)(50,247,225,326,134)(51,271,152,274,221)(52,267,54,306,283)(53,59,250,292,207)(55,99,168,166,312)(56,149,210,234,157)(57,314,64,153,180)(60,233,227,118,79)(62,192,287,232,324)(63,103,286,308,80)(66,226,182,188,256)(67,309,238,160,96)(68,228,298,254,90)(69,278,301,76,150)(70,155,185,212,174)(72,132,184,289,220)(74,148,100,161,294)(75,251,316,280,107)(77,194,328,291,173)(82,193,196,201,321)(87,131,98,229,319)(89,202,302,270,92)(91,108,162,255,263)(95,159,143,216,322)(102,208,205,135,114)(106,265,200,158,142)(109,237,167,284,186)(117,181,191,140,262)(123,304,209,144,126)(151,179,176,281,246);(1,2,4,3,6,7)(5,9,12,11,8,10)(13,14,18,31,39,55)(15,22,43,49,29,46)(16,25,20,37,32,68)(17,28,62,44,97,134)(19,34,74,82,95,64)(21,41,76,87,117,52)(23,47,65,104,96,45)(24,51)(26,57,118,83,161,188)(27,60,126,30,66,63)(33,71,144,115,205,80)(35,77,151,135,132,75)(36,79,90,160,256,230)(38,84,119,138,235,254)(40,88,81,139,236,241)(42,92,70,142,109,56)(48,102,120,58,122,215)(50,105,193,192,99,111)(53,108)(54,112,163,69,140,238)(59,124,219,162,257,290)(61,121,212,67,136,234)(72,130,198,173,172,147)(73,148,249,206,153,146)(78,154,164,85,166,258)(86,167,213,113,202,200)(89,165,260,237,311,327)(91,174,158,207,157,175)(93,176,220,243,316,291)(94,178,231,244,248,159)(98,182,214,128,227,187)(100,185,276,180,210,297)(101,181)(103,190,284,123,217,302)(106,189,281,275,216,280)(107,197,145,246,156,195)(110,199,221,125,223,305)(114,203,261,141,240,312)(116,201,272,143,239,273)(127,225,263,133,232,292)(129,229,253,137,222,208)(131,169,267,224,271,150)(149,250,317,155,255,321)(152,252,308,177,259,209)(168,265,313,279,331,293)(170,268,329,245,295,262)(171,269,330,307,264,325)(179,274,323,251,320,319)(183,278,300,228,306,282)(184,277,299,194,288,211)(186,270)(191,286,285,218,304,303)(196,289,283,266,328,301)(204,296,294,242,315,314)(226,309,326,233,310,324)(247,298,318,287,332,322)66 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((1,10,2,9,11)(3,12,6,8,5)(13,241,78,277,313)(14,94,125,257,260)(15,248,243,311,215)(16,242,84,282,197)(17,206,133,217,303)(18,105,279,154,261)(19,111,266,239,317)(20,183,332,119,230)(21,224,128,222,323)(22,88,113,269,187)(23,137,203,258,115)(24,169,177,320,305)(25,245,223,290,147)(26,295,273,172,65)(27,171,259,211,145)(28,40,58,121,330)(29,236,86,264,214)(30,307,252,299,195)(31,81,85,288,293)(32,170,199,219,198)(33,104,129,240,164)(34,231,116,189,318)(35,141,122,253,71)(36,47,61,310,163)(37,204,235,300,156)(38,296,276,43,249)(39,244,110,124,327)(41,101,218,112,329)(42,275,331,213,175)(44,73,127,190,285)(45,83,268,272,130)(46,146,138,315,297)(48,136,325,97,139)(49,178,93,165,120)(50,247,225,326,134)(51,271,152,274,221)(52,267,54,306,283)(53,59,250,292,207)(55,99,168,166,312)(56,149,210,234,157)(57,314,64,153,180)(60,233,227,118,79)(62,192,287,232,324)(63,103,286,308,80)(66,226,182,188,256)(67,309,238,160,96)(68,228,298,254,90)(69,278,301,76,150)(70,155,185,212,174)(72,132,184,289,220)(74,148,100,161,294)(75,251,316,280,107)(77,194,328,291,173)(82,193,196,201,321)(87,131,98,229,319)(89,202,302,270,92)(91,108,162,255,263)(95,159,143,216,322)(102,208,205,135,114)(106,265,200,158,142)(109,237,167,284,186)(117,181,191,140,262)(123,304,209,144,126)(151,179,176,281,246), (1,2,4,3,6,7)(5,9,12,11,8,10)(13,14,18,31,39,55)(15,22,43,49,29,46)(16,25,20,37,32,68)(17,28,62,44,97,134)(19,34,74,82,95,64)(21,41,76,87,117,52)(23,47,65,104,96,45)(24,51)(26,57,118,83,161,188)(27,60,126,30,66,63)(33,71,144,115,205,80)(35,77,151,135,132,75)(36,79,90,160,256,230)(38,84,119,138,235,254)(40,88,81,139,236,241)(42,92,70,142,109,56)(48,102,120,58,122,215)(50,105,193,192,99,111)(53,108)(54,112,163,69,140,238)(59,124,219,162,257,290)(61,121,212,67,136,234)(72,130,198,173,172,147)(73,148,249,206,153,146)(78,154,164,85,166,258)(86,167,213,113,202,200)(89,165,260,237,311,327)(91,174,158,207,157,175)(93,176,220,243,316,291)(94,178,231,244,248,159)(98,182,214,128,227,187)(100,185,276,180,210,297)(101,181)(103,190,284,123,217,302)(106,189,281,275,216,280)(107,197,145,246,156,195)(110,199,221,125,223,305)(114,203,261,141,240,312)(116,201,272,143,239,273)(127,225,263,133,232,292)(129,229,253,137,222,208)(131,169,267,224,271,150)(149,250,317,155,255,321)(152,252,308,177,259,209)(168,265,313,279,331,293)(170,268,329,245,295,262)(171,269,330,307,264,325)(179,274,323,251,320,319)(183,278,300,228,306,282)(184,277,299,194,288,211)(186,270)(191,286,285,218,304,303)(196,289,283,266,328,301)(204,296,294,242,315,314)(226,309,326,233,310,324)(247,298,318,287,332,322));