Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = D3×D4×S4 (order 1152) · Inn(G) = C22×A4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×2−1+4⋊C3 |
| unambiguous semidirect product | 2−1+4⋊C32 |
| unambiguous semidirect product | (C3×2−1+4)⋊C3 |
| Fitting | C3×2−1+4 ⋊ C3 |
| Sylow | 2−1+4 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C32 |
| Frattini | C2 | C22×C3×A4 |
| Fitting | C3×2−1+4 | C3 |
| hypercenter | C3×D4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,6)(3,4,9)(7,14,15,13,8,16)(10,12,11);(1,15,5,16)(2,9,6,11)(3,14,10,8)(4,7,12,13)(17,18,19)57 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#5430, C3×#5431, #10593664, … |
| maximal normal subgroup of | C3×#5430, C3×#5431, #10593664, … |
| maximal quotient of | C3×#519⋊C3, C3×#520⋊C3, C3×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 985);