Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×A4×C4⋊S4 |
| Fitting | C25×C4 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×A4×S4 |
| commutator, cosocle | C23×A4 | C23×C3 |
| Frattini | C2 | C22×A4×S4 |
| Fitting | C25×C4 | C3×D3 |
| socle | C26 | C2×C3×D3 |
| hypercenter | C2×C4 | A4×S4 |
| hypocenter | C22×A4 | C2×C3×D4 |
| LCS | G ▷ C23×A4 ▷ C22×A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
4049 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 4049 conjugacy classes of subgroups — too many to draw. Only its 105 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 — 3 generators on 14 points:
(4,6,5)(7,8,10)(11,12,13,14);(1,2)(3,6)(4,5);(8,9)(12,14)112 conjugacy classes — view the full character table.
| maximal quotient of | A4×#85440 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6,5)(7,8,10)(11,12,13,14), (1,2)(3,6)(4,5), (8,9)(12,14));