Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×D3×S4 |
| unambiguous semidirect product | (C3×S4)⋊C22 |
| unambiguous semidirect product | (C3⋊S4)⋊C22 |
| unambiguous semidirect product | (D3×A4)⋊C22 |
| Fitting, socle | C23×C3 ⋊ C2×D3 |
| derived | C3×A4 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×S4 |
| commutator, cosocle, hypocenter | C3×A4 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3 | C2×D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
272 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 9 points:
(3,4,5)(6,7,9,8);(1,2)(4,5)(7,8,9);(1,2)30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1028);