Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#237 |
| Fitting | C2×C32×D4 ⋊ C2 |
| Sylow | C32 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C22×D3 |
| commutator, cosocle | C2×C3 | C24×C3 |
| Frattini | C2 | C23×C3×D3 |
| Fitting | C2×C32×D4 | C2 |
| socle | C22×C32 | C23 |
| hypercenter | C2×C3×D4 | D3 |
| hypocenter | C3 | C2×C3×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×D4 | |
355 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4,5)(6,7,8)(10,14)(11,15);(9,11,12,15)(10,13,14,16);(4,5)(9,10)(11,13)(12,14)(15,16);(1,2)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10591414, #10591420, #10591450, … |
| maximal normal subgroup of | #10591414, #10591420, #10591450, … |
| maximal quotient of | C2×C3×#877, C2×C3×#878, C2×C3×#879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 993);