Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C2×#41×AGL(1,ℤ/9) (order 1728) · Inn(G) = D9 · Out(G) = C2×C3×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| direct product | C2×C9⋊C16 |
| unambiguous semidirect product | (C2×C9)⋊C16 |
| derived, Sylow | C9 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | D9 |
| commutator, hypocenter | C9 | C2×C16 |
| Frattini | C3×C8 | C2×D3 |
| Fitting | C2×C8×C9 | C2 |
| socle, prime radical | C22×C3 | C3⋊C8 |
| cosocle | C8×C9 | C22 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C8 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2)(4,6)(5,9)(7,8)(10,11)(12,13,14,17,15,18,20,23,16,19,21,24,22,25,26,27);(1,2)(3,11,6,9,8,7,5,4,10)96 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C9⋊C16, #10588138, C4⋊(C9⋊C16), … |
| maximal normal subgroup of | C4×C9⋊C16, #10588138, C4⋊(C9⋊C16), … |
| maximal quotient of | C4×C9⋊C16, C4⋊(C9⋊C16), C2×C9⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 18);