Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C28 |
| direct product | C23×A4×S4 |
| Fitting, socle | C27 ⋊ C3×D3 |
| derived | C22×A4 ⋊ C24×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | A4×S4 |
| commutator, cosocle, hypocenter | C22×A4 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C27 | C3×D3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C23 | |
19681 conjugacy classes of subgroups; only the 297 normal subgroups are shown.
This group has 19681 conjugacy classes of subgroups — too many to draw. Only its 297 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) = 72 |
A minimal faithful permutation representation — 4 generators on 14 points:
(5,6)(8,10,9)(11,14,12);(3,4)(7,10)(8,9);(13,14);(1,2)160 conjugacy classes — view the full character table.
| maximal subgroup of | C23×S42 |
| maximal normal subgroup of | C23×S42 |
| maximal quotient of | A4×#87477 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(8,10,9)(11,14,12), (3,4)(7,10)(8,9), (13,14), (1,2));