Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×D4×D9 |
| Fitting | C2×D4×C9 ⋊ C2 |
| Sylow | C9 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D9 |
| commutator, cosocle | C2×C9 | C24 |
| Frattini | C2×C3 | C23×D3 |
| Fitting | C2×D4×C9 | C2 |
| socle | C22×C3 | C22×D3 |
| hypercenter | C2×D4 | D9 |
| hypocenter | C9 | C22×D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
354 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(8,10)(9,13)(11,14)(12,15);(3,4)(5,6)(7,14,15,9,8,10,13,12,11);(4,6);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#99, D9×#100, D9×#102, … |
| maximal normal subgroup of | D9×#99, D9×#100, D9×#102, … |
| maximal quotient of | C2×#63440, C2×C22⋊C4×D9, C2×#611⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 356);