Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of S4 |
| automorphism | Aut(G) = C23×S4 (order 192) · Inn(G) = S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×C4×S4 |
| Fitting | C22×C3×C4 ⋊ D3 |
| derived | A4 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | S4 |
| commutator, hypocenter | A4 | C2×C3×C4 |
| Frattini | C2 | C2×C3×S4 |
| Fitting | C22×C3×C4 | D3 |
| socle | C23×C3 | C2×D3 |
| prime radical | C2×C3×S4 | C2 |
| cosocle | C2×A4 | C22×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C3×C4 | |
118 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 11 points:
(5,6,8,7)(9,10,11);(1,4,3,2)(6,7,8)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 897);