Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C42 |
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 79795 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has at least 79795 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) = 216 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 216 points:
(1,19,4,9,17,6)(2,7,8,14,22,21)(3,12,20,18,24,5)(10,23,11,15,16,13)(25,88,148,58,67,185,47,199,140,84,62,175)(26,105,193,194,201,96,65,167,107,68,90,181)(27,60,162,50,118,79,81,152,171,205,44,177)(28,142,191,95,102,214,98,203,145,46,135,128)(29,150,59,123,86,146,51,204,180,115,61,154)(30,141,209,53,147,164,112,132,139,93,63,134)(31,178,187,153,210,111,69,169,42,160,215,196)(32,211,48,64,138,100,116,172,52,156,213,126)(33,35,202,87,190,163,113,166,195,109,168,197)(34,200,108,192,208,206,151,73,165,124,106,71)(36,137,143,80,125,117,174,38,149,119,121,49)(37,186,91,120,89,77,184,133,182,40,198,179)(39,78,85,55,183,92,104,83,82,144,75,176)(41,159,74,43,103,216,189,57,129,97,207,127)(45,131,161,114,66,72,122,155,173,76,70,188)(54,101,212,136,110,56,157,99,130,170,94,158);(1,18,9,3)(2,4,14,6)(5,24,20,12)(7,10,22,15)(8,19,21,17)(11,23,13,16)(25,82,196,76,47,85,111,114)(26,70,186,60,65,66,133,152)(27,52,197,41,81,48,163,189)(28,121,123,193,98,125,115,107)(29,202,46,50,51,195,95,205)(30,176,110,122,112,92,94,45)(31,169,64,209,69,178,156,139)(32,138,160,128,116,213,153,214)(33,198,55,68,113,89,144,194)(34,155,149,118,151,131,143,44)(35,167,80,162,166,105,119,171)(36,91,168,72,174,182,190,188)(37,187,71,58,184,42,206,84)(38,100,96,43,137,126,181,97)(39,192,109,117,104,124,87,49)(40,101,108,93,120,99,165,53)(54,135,140,132,157,102,148,141)(56,185,150,170,158,175,204,136)(57,215,199,134,159,210,88,164)(59,63,74,142,180,147,129,203)(61,127,212,154,86,216,130,146)(62,172,207,145,67,211,103,191)(73,201,75,79,200,90,183,177)(77,208,161,78,179,106,173,83)62 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((1,19,4,9,17,6)(2,7,8,14,22,21)(3,12,20,18,24,5)(10,23,11,15,16,13)(25,88,148,58,67,185,47,199,140,84,62,175)(26,105,193,194,201,96,65,167,107,68,90,181)(27,60,162,50,118,79,81,152,171,205,44,177)(28,142,191,95,102,214,98,203,145,46,135,128)(29,150,59,123,86,146,51,204,180,115,61,154)(30,141,209,53,147,164,112,132,139,93,63,134)(31,178,187,153,210,111,69,169,42,160,215,196)(32,211,48,64,138,100,116,172,52,156,213,126)(33,35,202,87,190,163,113,166,195,109,168,197)(34,200,108,192,208,206,151,73,165,124,106,71)(36,137,143,80,125,117,174,38,149,119,121,49)(37,186,91,120,89,77,184,133,182,40,198,179)(39,78,85,55,183,92,104,83,82,144,75,176)(41,159,74,43,103,216,189,57,129,97,207,127)(45,131,161,114,66,72,122,155,173,76,70,188)(54,101,212,136,110,56,157,99,130,170,94,158), (1,18,9,3)(2,4,14,6)(5,24,20,12)(7,10,22,15)(8,19,21,17)(11,23,13,16)(25,82,196,76,47,85,111,114)(26,70,186,60,65,66,133,152)(27,52,197,41,81,48,163,189)(28,121,123,193,98,125,115,107)(29,202,46,50,51,195,95,205)(30,176,110,122,112,92,94,45)(31,169,64,209,69,178,156,139)(32,138,160,128,116,213,153,214)(33,198,55,68,113,89,144,194)(34,155,149,118,151,131,143,44)(35,167,80,162,166,105,119,171)(36,91,168,72,174,182,190,188)(37,187,71,58,184,42,206,84)(38,100,96,43,137,126,181,97)(39,192,109,117,104,124,87,49)(40,101,108,93,120,99,165,53)(54,135,140,132,157,102,148,141)(56,185,150,170,158,175,204,136)(57,215,199,134,159,210,88,164)(59,63,74,142,180,147,129,203)(61,127,212,154,86,216,130,146)(62,172,207,145,67,211,103,191)(73,201,75,79,200,90,183,177)(77,208,161,78,179,106,173,83));