Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#882 |
| Fitting | C32×C22⋊C4 ⋊ C2 |
| Sylow | C32 ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C22×D3 |
| commutator, cosocle | C22×C3 | C23×C3 |
| Frattini | C22 | C22×C3×D3 |
| Fitting | C32×C22⋊C4 | C2 |
| socle | C22×C32 | C23 |
| hypercenter | C3×C22⋊C4 | D3 |
| hypocenter | C3 | C3×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C22⋊C4 | |
277 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 277 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 14 points:
(1,2,3)(4,5,6)(8,10)(11,14,13,12);(7,8)(9,10);(5,6)(12,14)72 conjugacy classes — view the full character table.
| maximal subgroup of | #10590407, C3×#4741, #10594054, … |
| maximal normal subgroup of | #10590407, C3×#4741, #10594054, … |
| maximal quotient of | C3×#4656, C3×#4665, C3×#4668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 653);