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×#918 |
| Fitting | C32×Q16 ⋊ C2 |
| Sylow | C32 ⋊ #137 |
129 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6)(7,15,11,9)(8,21,14,12)(10,17,20,19)(13,16,18,22);(7,14,11,8)(9,22,15,16)(10,13,20,18)(12,17,21,19);(5,6)(9,15)(12,21)(16,22)(17,19)54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#918, #10594760, #10594762, … |
| maximal normal subgroup of | D3×#918, #10594760, #10594762, … |
| maximal quotient of | C3×#4789, C3×#4791, C3×#4794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 689);