Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = C2×D3×A4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×#67 |
| unambiguous semidirect product | (C22×#67)⋊C3 |
| Fitting | C24×C3 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×A4 |
| commutator, socle, cosocle | C23×C3 | C22×C3 |
| Fitting | C24×C3 | C2×C3 |
| hypercenter | C22 | D3×A4 |
| hypocenter | C22×C3 | C3×D4 |
| LCS | G ▷ C23×C3 ▷ C22×C3 | |
| UCS | C1 ◁ C2 ◁ C22 | |
142 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2,3)(5,6,7)(9,11);(2,3)(4,7)(5,6)(8,9)(10,11)36 conjugacy classes — view the full character table.
| maximal subgroup of | #10596467, #10596468, #67×S4, … |
| maximal normal subgroup of | #10596467, #10596468, #67×S4, … |
| maximal quotient of | A4×Dic3⋊C4, A4×(C2×D3)⋊C4, A4×#213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 928);