Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | S4×AΓL(1,𝔽8) |
| Fitting, socle | C25 ⋊ D3×ΓL(1,𝔽8) |
| derived | A4×AGL(1,𝔽8) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×AGL(1,𝔽8) | C2×C3 |
| Fitting, socle | C25 | D3×ΓL(1,𝔽8) |
| LCS | G ▷ A4×AGL(1,𝔽8) | |
| UCS | C1 | |
209 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 16 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) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,4,3)(5,9,7,11,12,6)(8,10);(2,3,4)(6,10,8,9,7,12,11)40 conjugacy classes — view the full character table.
| maximal subgroup of | S4×AGL(3,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,3)(5,9,7,11,12,6)(8,10), (2,3,4)(6,10,8,9,7,12,11));