Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×C32×D5 |
| unambiguous semidirect product | C5⋊(C23×C32) |
| unambiguous semidirect product | (C2×C5)⋊(C22×C32) |
| Fitting, socle | C22×C32×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C23×C32 |
| Hall | C32×C5 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | D5 |
| commutator, cosocle, hypocenter | C5 | C23×C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C22×C32 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(3,4)(5,6,7)(12,13)(14,15);(3,4)(8,9,10)(11,12,14,15,13);(1,2)144 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#6665, #10623770, C32×#685, … |
| maximal normal subgroup of | C3×#6665, #10623770, C32×#685, … |
| maximal quotient of | C2×C32×Dic10, C2×C32×C4×D5, C2×C32×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 157);