Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72227399 |
| commutator, hypocenter | G | C1 |
| Frattini | C26 | #11757163 |
| Fitting, cosocle | C25×2−1+4 | A5 |
| socle | C25 | #14515304 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
At least 78846 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has at least 78846 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,33,9,12,38,5,34,28,2,40)(3,20,21,7,39,11,35,37,15,31)(4,24,25,29,22,17,30,14,23,27)(6,32,19,26,13,16,36,8,10,18)(41,54,70,72,50)(42,44,64,68,47)(43,53,65,52,49)(45,56,69,67,51)(46,57,60,62,63)(48,61,55,58,59);(2,4,8)(3,5,11)(6,13,9)(7,10,19)(12,17,27)(14,18,28)(15,26,22)(16,24,21)(20,31,36,33,38,32)(23,35,40,29,34,39)(25,30,37)(41,42,45,43)(44,50,54,49,61,46,53,51,56,47,57,48)(52,66,65,63,62,60,68,71,64,59,58,55)(67,70,72,69)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((1,33,9,12,38,5,34,28,2,40)(3,20,21,7,39,11,35,37,15,31)(4,24,25,29,22,17,30,14,23,27)(6,32,19,26,13,16,36,8,10,18)(41,54,70,72,50)(42,44,64,68,47)(43,53,65,52,49)(45,56,69,67,51)(46,57,60,62,63)(48,61,55,58,59), (2,4,8)(3,5,11)(6,13,9)(7,10,19)(12,17,27)(14,18,28)(15,26,22)(16,24,21)(20,31,36,33,38,32)(23,35,40,29,34,39)(25,30,37)(41,42,45,43)(44,50,54,49,61,46,53,51,56,47,57,48)(52,66,65,63,62,60,68,71,64,59,58,55)(67,70,72,69));