Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = C2×D32 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | D3×#208 |
| Sylow | C32 ⋊ C2×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D32 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting | C2×C32×C4 | C22 |
| prime radical | C22×C3×D3 | C4 |
| cosocle | C32×C4 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C2×M16 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
146 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 64 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)(6,8)(9,10,11)(12,13,14);(1,2,3,6,4,5,7,8)(13,14);(10,11)54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#805, #10590392, D3×#819, … |
| maximal normal subgroup of | D3×#805, #10590392, D3×#819, … |
| maximal quotient of | #10590248, #10590251, #10590252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 461);