Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C23×D3×D4 (order 384) · Inn(G) = C22×D3 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×D3×M16 |
| Fitting | C32×M16 ⋊ C2 |
| Sylow | C32 ⋊ C2×M16 |
146 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,5)(6,8)(9,10,11)(12,13,14);(1,6,7,2,4,8,3,5);(13,14)90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#822, #10590973, D3×#823, … |
| maximal normal subgroup of | D3×#822, #10590973, D3×#823, … |
| maximal quotient of | C3×#4701, C3×D3×#97, C3×#4705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 677);