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×#915 |
| Fitting | C32×SD16 ⋊ C2 |
| Sylow | C32 ⋊ #137 |
130 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 130 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) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6)(8,13)(9,15)(10,18)(12,19)(16,17)(20,22);(7,17,11,9)(8,20,13,12)(10,16,18,15)(14,19,21,22);(5,6)(7,8)(9,12)(10,14)(11,13)(15,22)(16,19)(17,20)(18,21)54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#915, #10594742, #10594744, … |
| maximal normal subgroup of | D3×#915, #10594742, #10594744, … |
| maximal quotient of | C3×#4757, C3×#4759, C3×#4765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 686);