Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×D3×D12 |
| Fitting | C2×C32×C4 ⋊ C22 |
| Sylow | C32 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D32 |
| commutator, cosocle | C2×C32 | C24 |
| Frattini | C2 | C22×D32 |
| Fitting | C2×C32×C4 | C22 |
| socle | C22×C32 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C22×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
499 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 48 |
A minimal faithful permutation representation — 4 generators on 12 points:
(3,4,5)(6,7,8)(9,10,11,12);(1,2)(4,5)(10,12);(7,8);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#809, #10590592, #10590595, … |
| maximal normal subgroup of | D3×#809, #10590592, #10590595, … |
| maximal quotient of | D3×#869, #10593893, #10593894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 951);