Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = D3×D12 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | D3×#204 |
| Fitting | C32×C8 ⋊ C22 |
| Sylow | C32 ⋊ C2×SD16 |
146 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 44 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:
(1,2,3,5,4,6,7,8)(9,10,11)(12,13,14);(2,5)(3,7)(6,8)(13,14);(10,11)45 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#204, D3×#904, #10594677, … |
| maximal normal subgroup of | C2×D3×#204, D3×#904, #10594677, … |
| maximal quotient of | D3×#816, D3×#817, #10590674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 440);