Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| isoclinism | isoclinic class of #72237324 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini, Fitting, cosocle | C45 | A5 |
| socle | C24 | #72209605 |
| LCS | G | |
| UCS | C1 | |
2030 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 2030 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 120 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 120 points:
(1,27,89,72,80,32,7,53,97,59,5,48,91,104,75,14,24,99,117,98)(2,77,55,46,110,9,43,25,28,119,16,106,64,61,103,22,96,65,70,116)(3,100,71,78,90)(4,60,41,88,83,15,63,73,114,52,18,29,95,113,57,36,62,105,107,23)(6,120,35,92,56)(8,111,31,50,74,26,108,40,54,58,39,85,30,87,44,33,115,10,94,19)(11,34,79,86,76)(12,67,45,49,101)(13,84,81,38,109)(17,69,102,93,66)(20,112,37,82,21)(42,118,68,47,51);(1,2,6)(3,10,28)(4,13,33)(5,16,38)(7,21,45)(8,15,37)(9,20,32)(11,30,61)(12,31,46)(14,22,42)(17,40,70)(18,35,26)(19,41,60)(23,49,84)(24,51,79)(25,53,67)(27,56,71)(29,58,95)(34,65,99)(36,68,39)(43,75,72)(44,73,62)(47,57,93)(48,81,102)(50,76,107)(52,86,92)(54,90,113)(55,91,69)(59,97,77)(63,74,105)(64,89,100)(66,88,94)(78,82,83)(80,104,96)(85,109,119)(87,101,114)(98,117,106)(103,108,118)(110,115,112)(111,120,116)80 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((1,27,89,72,80,32,7,53,97,59,5,48,91,104,75,14,24,99,117,98)(2,77,55,46,110,9,43,25,28,119,16,106,64,61,103,22,96,65,70,116)(3,100,71,78,90)(4,60,41,88,83,15,63,73,114,52,18,29,95,113,57,36,62,105,107,23)(6,120,35,92,56)(8,111,31,50,74,26,108,40,54,58,39,85,30,87,44,33,115,10,94,19)(11,34,79,86,76)(12,67,45,49,101)(13,84,81,38,109)(17,69,102,93,66)(20,112,37,82,21)(42,118,68,47,51), (1,2,6)(3,10,28)(4,13,33)(5,16,38)(7,21,45)(8,15,37)(9,20,32)(11,30,61)(12,31,46)(14,22,42)(17,40,70)(18,35,26)(19,41,60)(23,49,84)(24,51,79)(25,53,67)(27,56,71)(29,58,95)(34,65,99)(36,68,39)(43,75,72)(44,73,62)(47,57,93)(48,81,102)(50,76,107)(52,86,92)(54,90,113)(55,91,69)(59,97,77)(63,74,105)(64,89,100)(66,88,94)(78,82,83)(80,104,96)(85,109,119)(87,101,114)(98,117,106)(103,108,118)(110,115,112)(111,120,116));