Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#940 |
| Fitting | C2×C32×D4 ⋊ C2 |
| Sylow | C32 ⋊ #127 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C22×D3 |
| commutator, cosocle | C22×C3 | C23×C3 |
| Frattini | C22 | C22×C3×D3 |
| Fitting | C2×C32×D4 | C2 |
| socle | C22×C32 | C23 |
| hypercenter | C2×C3×D4 | D3 |
| hypocenter | C3 | C3×#127 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×D4 | |
243 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 243 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)(12,14);(11,14,13,12);(5,6)(7,8)(9,10)(12,14)72 conjugacy classes — view the full character table.
| maximal subgroup of | #10589178, C3×#4474, #10591428, … |
| maximal normal subgroup of | #10589178, C3×#4474, #10591428, … |
| maximal quotient of | C3×#4949, C3×#4957, C3×#4960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 711);