Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = D3×#67 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D3×#213 |
| Fitting | C32×D4 ⋊ C22 |
| Sylow | C32 ⋊ C2×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#67 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C2×D32 |
| Fitting | C32×D4 | C22 |
| socle | C2×C32 | C2×D4 |
| hypercenter | D4 | D32 |
| hypocenter | C32 | C2×D8 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
163 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 44 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);(10,11)36 conjugacy classes — view the full character table.
| maximal subgroup of | D32×D8, D3×#911, #10594732, … |
| maximal normal subgroup of | D32×D8, D3×#911, #10594732, … |
| maximal quotient of | #10590411, #10590422, #10590438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 572);