Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C4×D32 |
| Fitting | C2×C32×C4 ⋊ C22 |
| derived, Sylow | C32 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D32 |
| commutator, hypocenter | C32 | C23×C4 |
| Frattini | C2 | C22×D32 |
| Fitting | C2×C32×C4 | C22 |
| socle | C22×C32 | C23 |
| prime radical | C22×D32 | C2 |
| cosocle | C2×C32 | C24 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C4 | |
499 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 180 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:
(1,2)(4,5)(6,7,8);(3,4,5)(7,8);(9,10,11,12);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C2×D3)⋊C8, #10590935, #10590936, … |
| maximal normal subgroup of | D3×(C2×D3)⋊C8, #10590935, #10590936, … |
| maximal quotient of | #10593848, C4×D3×Dic6, #10593885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 950);