Finite Group DatabasebetaFinite Group Databasebeta| order | 8640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | D3×A4⋊S5 |
| Fitting | C22×C3 ⋊ C2×ΓL(2,𝔽4) |
| socle | C22×C3×A5 ⋊ C2×D3 |
| derived | C3×A4×A5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A4×A5 | C22 |
| Fitting | C22×C3 | C2×ΓL(2,𝔽4) |
| socle | C22×C3×A5 | C2×D3 |
| LCS | G ▷ C3×A4×A5 | |
| UCS | C1 | |
1152 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,3)(4,7,6)(8,9,10);(2,3)(4,7,5,6)(8,12,11,9)57 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72219548 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3)(4,7,6)(8,9,10), (2,3)(4,7,5,6)(8,12,11,9));