Finite Group DatabasebetaFinite Group Databasebeta| order | 40320 |
| isoclinism | isoclinic class of #72227518 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | A8 |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups; only the 3 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 3 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 240 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 240 points:
(1,2,4,5)(3,6,7,9)(8,10,12,14)(11,15,17,22)(13,18,20,26)(16,23,24,34)(19,27,29,39)(21,30,32,43)(25,35,37,50)(28,40,41,57)(31,44,46,61)(33,47,48,65)(36,51,53,70)(38,54,55,74)(42,58,60,78)(45,62,63,83)(49,66,68,86)(52,71,72,93)(56,75,76,98)(59,79,81,103)(64,84,85,109)(67,87,89,111)(69,90,91,115)(73,94,96,119)(77,99,101,127)(80,104,105,133)(82,106,107,114)(88,112,113,141)(92,116,117,144)(95,120,122,149)(97,123,125,152)(100,128,129,157)(102,130,131,126)(108,135,137,165)(110,138,139,170)(118,145,147,180)(121,150,151,185)(124,153,155,187)(132,159,161,148)(134,162,142,174)(136,166,168,199)(140,171,173,203)(143,176,178,182)(146,181,183,211)(154,188,189,164)(156,190,158,192)(160,194,196,209)(163,198,175,206)(167,200,201,169)(172,186,204,215)(177,191,207,193)(179,208,210,226)(184,212,213,227)(195,216,217,230)(197,218,205,223)(202,221,222,233)(214,228,229,237)(219,231,224,234)(220,232,225,235)(236,238,240,239);(1,3,4,7)(6,8,11,16)(9,12,17,24)(10,13,19,28)(14,20,29,41)(15,21,31,45)(18,25,36,52)(22,32,46,63)(23,33,39,55)(26,37,53,72)(27,38,34,48)(30,42,59,80)(35,49,67,88)(40,56)(43,60,81,105)(47,64)(50,68,89,113)(51,69,70,91)(54,73,95,121)(57,76)(58,77,100,86)(62,82,103,131)(65,85)(66,78,101,129)(71,92,87,110)(74,96,122,151)(75,97,124,154)(79,102,83,107)(84,108,136,167)(90,114,142,175)(93,117,111,139)(94,118,146,182)(98,125,155,189)(99,126,156,191)(104,132,160,195)(106,134,163,115)(109,137,168,201)(112,140,172,166)(116,143,177,128)(119,147,183,176)(120,148,149,159)(123,145,179,209)(127,130,158,193)(133,161,196,217)(135,164,165,188)(138,169,198,220)(141,173,204,199)(144,178,207,157)(150,184,181,190)(152,180,210,194)(153,186,214,227)(162,197,219,216)(170,200,206,225)(171,202,203,222)(174,205,224,230)(185,213,211,192)(187,215,229,212)(208,226)(218,221)(223,233)(228,236,237,240)(231,238)(232,235)(234,239)23 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((1,2,4,5)(3,6,7,9)(8,10,12,14)(11,15,17,22)(13,18,20,26)(16,23,24,34)(19,27,29,39)(21,30,32,43)(25,35,37,50)(28,40,41,57)(31,44,46,61)(33,47,48,65)(36,51,53,70)(38,54,55,74)(42,58,60,78)(45,62,63,83)(49,66,68,86)(52,71,72,93)(56,75,76,98)(59,79,81,103)(64,84,85,109)(67,87,89,111)(69,90,91,115)(73,94,96,119)(77,99,101,127)(80,104,105,133)(82,106,107,114)(88,112,113,141)(92,116,117,144)(95,120,122,149)(97,123,125,152)(100,128,129,157)(102,130,131,126)(108,135,137,165)(110,138,139,170)(118,145,147,180)(121,150,151,185)(124,153,155,187)(132,159,161,148)(134,162,142,174)(136,166,168,199)(140,171,173,203)(143,176,178,182)(146,181,183,211)(154,188,189,164)(156,190,158,192)(160,194,196,209)(163,198,175,206)(167,200,201,169)(172,186,204,215)(177,191,207,193)(179,208,210,226)(184,212,213,227)(195,216,217,230)(197,218,205,223)(202,221,222,233)(214,228,229,237)(219,231,224,234)(220,232,225,235)(236,238,240,239), (1,3,4,7)(6,8,11,16)(9,12,17,24)(10,13,19,28)(14,20,29,41)(15,21,31,45)(18,25,36,52)(22,32,46,63)(23,33,39,55)(26,37,53,72)(27,38,34,48)(30,42,59,80)(35,49,67,88)(40,56)(43,60,81,105)(47,64)(50,68,89,113)(51,69,70,91)(54,73,95,121)(57,76)(58,77,100,86)(62,82,103,131)(65,85)(66,78,101,129)(71,92,87,110)(74,96,122,151)(75,97,124,154)(79,102,83,107)(84,108,136,167)(90,114,142,175)(93,117,111,139)(94,118,146,182)(98,125,155,189)(99,126,156,191)(104,132,160,195)(106,134,163,115)(109,137,168,201)(112,140,172,166)(116,143,177,128)(119,147,183,176)(120,148,149,159)(123,145,179,209)(127,130,158,193)(133,161,196,217)(135,164,165,188)(138,169,198,220)(141,173,204,199)(144,178,207,157)(150,184,181,190)(152,180,210,194)(153,186,214,227)(162,197,219,216)(170,200,206,225)(171,202,203,222)(174,205,224,230)(185,213,211,192)(187,215,229,212)(208,226)(218,221)(223,233)(228,236,237,240)(231,238)(232,235)(234,239));