Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C3 |
| direct product | C2×D4×C3⋊D3 |
| Fitting | C2×C32×D4 ⋊ C2 |
| Sylow | C32 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C3⋊D3 |
| commutator, cosocle | C2×C32 | C24 |
| Frattini | C2 | C23×C3⋊D3 |
| Fitting | C2×C32×D4 | C2 |
| socle | C22×C32 | C23 |
| hypercenter | C2×D4 | C3⋊D3 |
| hypocenter | C32 | C22×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
708 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 173 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) = 72 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(4,5)(7,8);(3,4,5)(9,10)(11,12);(6,7,8)(10,12);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10591416, #10591426, #10591456, … |
| maximal normal subgroup of | #10591416, #10591426, #10591456, … |
| maximal quotient of | C2×#64087, C2×C22⋊C4×C3⋊D3, C2×#64090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1007);