Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = C4×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = C2×D9 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D36 |
| unambiguous semidirect product | C4⋊(C5×D9) |
| Fitting | C4×C5×C9 ⋊ C2 |
| Hall | C5×C9 ⋊ D4 |
| Sylow | C9 ⋊ C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×D9 |
| commutator, cosocle | C2×C9 | C22×C5 |
| Frattini | C2×C3 | C2×C5×D3 |
| Fitting | C4×C5×C9 | C2 |
| socle | C2×C3×C5 | C2×D3 |
| hypercenter | C4×C5 | D9 |
| hypocenter | C9 | C5×D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5)(6,7,8,9)(10,11,12,13,14,15,16,17,18);(7,9)(11,18)(12,17)(13,16)(14,15)105 conjugacy classes — view the full character table.
| maximal subgroup of | #10623542, #10623544, #10623548, … |
| maximal normal subgroup of | #10623542, #10623544, #10623548, … |
| maximal quotient of | C5×Dic36, C5×#3664, C5×D72, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 22);