Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C28 |
| direct product | C23×B(4,𝔽2)⋊C3 |
| Fitting, Sylow | C23×B(4,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C24×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C24×C3 |
| Frattini | C23 | C24×A4 |
| Fitting | C23×B(4,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C24 | |
12520 conjugacy classes of subgroups; only the 268 normal subgroups are shown.
This group has 12520 conjugacy classes of subgroups — too many to draw. Only its 268 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(5,6)(7,8,10)(11,12,14);(3,4)(7,8)(9,14,13,10)(11,12);(9,13);(1,2)128 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11714171, C23×#73026, C22×#60399⋊C3, … |
| maximal normal subgroup of | C22×#11714171, C23×#73026, C22×#60399⋊C3, … |
| maximal quotient of | C2×D4×B(4,𝔽2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,10)(11,12,14), (3,4)(7,8)(9,14,13,10)(11,12), (9,13), (1,2));