Finite Group DatabasebetaFinite Group Databasebeta| order | 90 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C4×D3 (order 48) · Inn(G) = D3 · Out(G) = C2×C4 |
| holomorph | G ⋊ Aut(G) = D33×AGL(1,𝔽5) |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×D3 |
| unambiguous semidirect product | C3⋊(C2×C3×C5) |
| Fitting, socle, Hall | C32×C5 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C5 |
| Sylow | C32 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C5 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2,3)(5,6)(7,8,9,10,11);(4,5,6)45 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D5, C5×D32, D3×D15, … |
| maximal normal subgroup of | C3×D3×D5, C5×D32, D3×D15, … |
| maximal quotient of | C3×C5×Dic3, C2×C3×C5×D3, C3×C5×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(90, 6);