Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#3811 |
| Fitting | C23×C32 ⋊ C22 |
| Sylow | C32 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D32 |
| commutator, cosocle | C2×C32 | C24 |
| Frattini | C2 | C22×D32 |
| Fitting | C23×C32 | C22 |
| socle | C22×C32 | C23 |
| hypercenter | C23 | D32 |
| hypocenter | C32 | C22×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C23 | |
539 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 12 points:
(3,4,5)(6,7,8)(9,10)(11,12);(1,2)(7,8)(10,12);(4,5)(10,12);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #10590404, C2×#63738, #10593490, … |
| maximal normal subgroup of | #10590404, C2×#63738, #10593490, … |
| maximal quotient of | #10594805, C2×#63840, C2×#63853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 977);