Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D3×#239 |
| Fitting | C32×Q8 ⋊ C22 |
| Sylow | C32 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×Q8 | C22 |
| hypercenter | Q8 | D32 |
| hypocenter | C32 | C2×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
348 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,3,4,7)(2,5,6,8)(9,10,11)(12,13,14);(1,2,4,6)(3,8,7,5);(3,7)(5,8)(13,14);(10,11)45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#914, D3×#916, #10594744, … |
| maximal normal subgroup of | D3×#914, D3×#916, #10594744, … |
| maximal quotient of | #10594320, #10594321, #10594360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 966);