Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×C4×D3 (order 192) · Inn(G) = D3 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C3×D3×C16 |
| Fitting | C32×C16 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C16 |
| Sylow | C32 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C16 | D3 |
| commutator, hypocenter | C3 | C2×C3×C16 |
| Frattini | C8 | C2×C3×D3 |
| Fitting | C32×C16 | C2 |
| socle | C2×C32 | C2×C8 |
| prime radical | C2×C3×D3 | C8 |
| cosocle | C3×C8 | C22×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C16 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(5,6);(4,5,6)(7,12,15,14,17,22,9,13,11,19,21,8,10,18,16,20)144 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊C32, #10589068, C3×D3×C32, … |
| maximal normal subgroup of | D3×C3⋊C32, #10589068, C3×D3×C32, … |
| maximal quotient of | C3×D3×C32, C3×#4447, C3×Dic3×C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 231);