Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C3×C4×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C3×C42) |
| derived | C3 ⋊ C2×C3×C42 |
| Sylow | C32 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C4 | D3 |
| commutator, hypocenter | C3 | C2×C3×C42 |
| Frattini | C22 | C22×C3×D3 |
| Fitting | C22×C32×C4 | C2 |
| socle, prime radical | C23×C32 | C22 |
| cosocle | C22×C3 | C23×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C3×C4 | |
243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(4,5)(6,7,8)(9,10,11,12);(3,4,5)(13,14,15,16);(1,2)144 conjugacy classes — view the full character table.
| maximal subgroup of | #10589155, #10589161, #10589162, … |
| maximal normal subgroup of | #10589155, #10589161, #10589162, … |
| maximal quotient of | C2×C3×C4×C3⋊C8, C2×C3×#803, C3×C4×#208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 693);