Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of Dic30 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D60 |
| unambiguous semidirect product | C4⋊(C3×D15) |
| Fitting | C32×C4×C5 ⋊ C2 |
| Hall | C32×C5 ⋊ D4 |
| Sylow | C32 ⋊ D20 |
| Sylow | C5 ⋊ C3×D12 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D15 |
| commutator, cosocle | C2×C3×C5 | C22×C3 |
| Frattini | C2 | C2×C3×D15 |
| Fitting | C32×C4×C5 | C2 |
| socle | C2×C32×C5 | C22 |
| hypercenter | C3×C4 | D15 |
| hypocenter | C3×C5 | C3×D4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,4,5,3)(6,7,8)(9,10,11)(12,15,14,13);(2,3)(4,5)(10,11)(13,15)99 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#6653, C3×#6654, C3×#6658, … |
| maximal normal subgroup of | C3×#6653, C3×#6654, C3×#6658, … |
| maximal quotient of | C3×#6706, C3×D120, C3×Dic60, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 102);