Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C22×C5×D9 |
| unambiguous semidirect product | C9⋊(C23×C5) |
| unambiguous semidirect product | (C2×C9)⋊(C22×C5) |
| Fitting | C22×C5×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C23×C5 |
| Hall | C5×C9 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | D9 |
| commutator, cosocle, hypocenter | C9 | C23×C5 |
| Frattini | C3 | C22×C5×D3 |
| Fitting | C22×C5×C9 | C2 |
| socle | C22×C3×C5 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C22×C5 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4)(5,6,7,8,9)(11,13)(12,16)(14,17)(15,18);(3,4)(10,15,13,12,17,14,16,11,18);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D9)⋊Dic5, C5×(C2×D9)⋊C4, C2×D9×Dic5, … |
| maximal normal subgroup of | (C2×D9)⋊Dic5, C5×(C2×D9)⋊C4, C2×D9×Dic5, … |
| maximal quotient of | C2×C5×Dic18, C2×C4×C5×D9, C2×C5×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 48);