Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of (C22⋊C9)⋊C2 |
| automorphism | Aut(G) = C22×((C22⋊C9)⋊C2)⋊C3 (order 864) · Inn(G) = (C22⋊C9)⋊C2 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×(C22⋊C9)⋊C2 |
| derived | C22⋊C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | C22⋊C9 | C2×C4 |
| Frattini | C2×C3 | C2×S4 |
| Fitting | C22×C3×C4 | D3 |
| socle | C23×C3 | C2×D3 |
| prime radical | C2×(C22⋊C9)⋊C2 | C2 |
| cosocle | C2×C22⋊C9 | C22 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 ◁ C4 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 17 points:
(5,6,8,7)(10,12)(11,15)(13,14)(16,17);(1,2,3,4)(6,7,8)(9,17,12,15,14,13,11,10,16)36 conjugacy classes — view the full character table.
| maximal subgroup of | C8×(C22⋊C9)⋊C2, #10589502, C2×C4×(C22⋊C9)⋊C2, … |
| maximal normal subgroup of | C8×(C22⋊C9)⋊C2, #10589502, C2×C4×(C22⋊C9)⋊C2, … |
| maximal quotient of | C4×#3688, #10589490, C4×#3689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 333);