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×#907 |
| Fitting | C32×M16 ⋊ C2 |
| Sylow | C32 ⋊ #131 |
134 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 74 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)(7,16,17,9,11,21,10,15)(8,20,19,13,12,22,14,18);(8,12)(9,15)(14,19)(16,21);(5,6)(7,8)(9,13)(10,14)(11,12)(15,18)(16,20)(17,19)(21,22)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10590975, #10590976, #10590994, … |
| maximal normal subgroup of | #10590975, #10590976, #10590994, … |
| maximal quotient of | C3×#4701, C3×#4706, C3×#4708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 678);