Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C23×D3×D4 (order 384) · Inn(G) = D3×D4 · Out(G) = C23 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×D3×SD16 |
| Fitting | C32×SD16 ⋊ C2 |
| Sylow | C32 ⋊ C2×SD16 |
146 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,8)(6,7)(9,10,11)(12,13,14);(1,7,4,2)(3,6,8,5);(13,14)63 conjugacy classes — view the full character table.
| maximal subgroup of | D32×SD16, D3×#914, D3×#915, … |
| maximal normal subgroup of | D32×SD16, D3×#914, D3×#915, … |
| maximal quotient of | C3×#4758, C3×#4760, C3×#4761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 684);