Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×C4⋊(A4⋊S4) |
| Fitting | C25×C4 ⋊ C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×A4⋊S4 |
| commutator, cosocle | C2×A42 | C23 |
| Frattini | C2 | C22×A4⋊S4 |
| Fitting | C25×C4 | C3⋊D3 |
| socle | C26 | C2×C3⋊D3 |
| hypercenter | C2×C4 | A4⋊S4 |
| hypocenter | A42 | C2×D4 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
3105 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 3105 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(3,4,5,6)(7,9,8)(11,14,12);(1,2)(12,13,14);(4,6)(9,10)(13,14)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×A4⋊S4, #72226315, C2×#72228078, … |
| maximal normal subgroup of | C2×D4×A4⋊S4, #72226315, C2×#72228078, … |
| maximal quotient of | #72226345, C2×#72228078, #72258936, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5,6)(7,9,8)(11,14,12), (1,2)(12,13,14), (4,6)(9,10)(13,14));