Finite Group DatabasebetaFinite Group Databasebeta| order | 8640 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
| direct product | S4×A6 |
| Fitting | C22 ⋊ D3×A6 |
| socle | C22×A6 ⋊ D3 |
| derived | A4×A6 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | A4×A6 | C2 |
| Fitting | C22 | D3×A6 |
| socle | C22×A6 | D3 |
| cosocle | A4 | C2×A6 |
| LCS | G ▷ A4×A6 | |
| UCS | C1 | |
444 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,4,2)(5,6,9,8,10);(2,3)(5,10,6,9,7)35 conjugacy classes — view the full character table.
| maximal subgroup of | GL(3,𝔽2)×A6 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2)(5,6,9,8,10), (2,3)(5,10,6,9,7));