Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | A4×B(4,𝔽2)⋊C3 |
| Fitting, Sylow | C22×B(4,𝔽2) ⋊ C32 |
| derived | C22×H(5,𝔽2) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×#120⋊C3 |
| commutator, cosocle, hypocenter | C22×H(5,𝔽2) | C2×C32 |
| Frattini | C23 | C2×A42 |
| Fitting | C22×B(4,𝔽2) | C32 |
| socle | C23 | C3×#120⋊C3 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
778 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,4,3)(6,11,10,9,7,12);(2,3,4)(5,10)(6,7)(8,12)(9,11)64 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#11757163, A5×B(4,𝔽2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,11,10,9,7,12), (2,3,4)(5,10)(6,7)(8,12)(9,11));