Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×D3×D8 |
| Fitting | C32×D8 ⋊ C2 |
| Sylow | C32 ⋊ C2×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | D3×D4 |
| commutator, cosocle | C3×C4 | C23×C3 |
| Frattini | C4 | C22×C3×D3 |
| Fitting | C32×D8 | C2 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C3×D8 | D3 |
| hypocenter | C3 | C2×C3×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8 | |
163 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(2,5)(3,8)(6,7)(9,10,11)(12,13,14);(1,2)(3,5)(4,7)(6,8);(13,14)63 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#826, D3×#827, #10591031, … |
| maximal normal subgroup of | D3×#826, D3×#827, #10591031, … |
| maximal quotient of | C3×#4756, C3×#4759, C3×#4760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 681);