Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #634 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C5×C4⋊(C3⋊D3) |
| unambiguous semidirect product | C4⋊(C5×C3⋊D3) |
| Fitting | C32×C4×C5 ⋊ C2 |
| Hall | C32×C5 ⋊ D4 |
| Sylow | C32 ⋊ C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×C3⋊D3 |
| commutator, cosocle | C2×C32 | C22×C5 |
| Frattini | C2 | C2×C5×C3⋊D3 |
| Fitting | C32×C4×C5 | C2 |
| socle | C2×C32×C5 | C22 |
| hypercenter | C4×C5 | C3⋊D3 |
| hypocenter | C32 | C5×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
96 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)(8,9,10)(11,12,13,14,15);(2,4)(6,7)(9,10);(5,6,7)105 conjugacy classes — view the full character table.
| maximal subgroup of | #10623757, #10623759, #10623761, … |
| maximal normal subgroup of | #10623757, #10623759, #10623761, … |
| maximal quotient of | C5×#3744, C5×#3745, C5×#3746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 107);