Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #126⋊C3 |
| automorphism | Aut(G) = D3×#560⋊C3 (order 1152) · Inn(G) = #126⋊C3 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#126⋊C3 |
| unambiguous semidirect product | #126⋊C32 |
| unambiguous semidirect product | (C3×#126)⋊C3 |
| Fitting | C3×#126 ⋊ C3 |
| derived | C42 ⋊ C2×C32 |
| Sylow | #126 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #126⋊C3 |
| commutator, cosocle, hypocenter | C42 | C2×C32 |
| Frattini | C22 | C2×C3×A4 |
| Fitting | C3×#126 | C3 |
| socle | C22×C3 | C2×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C3 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,10,11,6,3,13)(4,5,7)(8,9,12,16,14,15);(1,13,5,16)(2,10,9,3)(4,8,7,11)(6,15,14,12)(17,18,19)30 conjugacy classes — view the full character table.
| maximal subgroup of | (#126⋊C3)⋊D3, D3×#126⋊C3, C2×C3×#126⋊C3, … |
| maximal normal subgroup of | (#126⋊C3)⋊D3, D3×#126⋊C3, C2×C3×#126⋊C3, … |
| maximal quotient of | C3×#383⋊C3, C3×#396⋊C3, C3×#401⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 635);