Finite Group DatabasebetaFinite Group Databasebeta| order | 90 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C4×AGL(1,ℤ/9) (order 216) · Inn(G) = D9 · Out(G) = C3×C4 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×#92265⋊C2 |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×D9 |
| unambiguous semidirect product | C9⋊(C2×C5) |
| Fitting, Hall | C5×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D9 |
| commutator, cosocle, hypocenter | C9 | C2×C5 |
| Frattini | C3 | C5×D3 |
| Fitting | C5×C9 | C2 |
| socle | C3×C5 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C5 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,3,4,5)(7,9)(8,12)(10,13)(11,14);(6,11,9,8,13,10,12,7,14)30 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D9, C2×C5×D9, C5×D27, … |
| maximal normal subgroup of | D5×D9, C2×C5×D9, C3×C5×D9, … |
| maximal quotient of | C5×Dic9, C2×C5×D9, C5×D27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(90, 1);