Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×A4⋊S5 |
| Fitting | C22×C3 ⋊ ΓL(2,𝔽4) |
| socle | C22×C3×A5 ⋊ D3 |
| derived | A4×A5 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4⋊S5 |
| commutator, cosocle, hypocenter | A4×A5 | C2×C3 |
| Frattini | C1 | G |
| Fitting | C22×C3 | ΓL(2,𝔽4) |
| socle | C22×C3×A5 | D3 |
| LCS | G ▷ A4×A5 | |
| UCS | C1 ◁ C3 | |
334 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 14 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:
(1,2,3)(4,6,5,7)(8,9,11,10);(4,5,6,7)(9,12)57 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72219548, D3×A4⋊S5 |
| maximal normal subgroup of | D3×A4⋊S5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,6,5,7)(8,9,11,10), (4,5,6,7)(9,12));