Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×D3×Dic3 |
| unambiguous semidirect product | (C3×D3)⋊C42 |
| derived, Sylow | C32 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D32 |
| commutator, hypocenter | C32 | C2×C42 |
| Frattini | C22 | C2×D32 |
| Fitting | C2×C32×C4 | C22 |
| socle, cosocle | C22×C32 | C23 |
| prime radical | C22×C3×D3 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C4 | |
231 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 96 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,3)(5,6)(11,12,13,14);(2,3)(4,5,6);(7,8,9,10)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×Dic3, D3×Dic3⋊C8, #10590655, … |
| maximal normal subgroup of | D3×C8×Dic3, D3×Dic3⋊C8, #10590655, … |
| maximal quotient of | C4×D3×C3⋊C8, D3×#803, #10590318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 523);