Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C4×C5×C3⋊D3 |
| Fitting | C32×C4×C5 ⋊ C2 |
| derived, Sylow | C32 ⋊ C2×C4×C5 |
| Hall | C32×C5 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | C3⋊D3 |
| commutator, hypocenter | C32 | C2×C4×C5 |
| Frattini | C2 | C2×C5×C3⋊D3 |
| Fitting | C32×C4×C5 | C2 |
| socle | C2×C32×C5 | C22 |
| prime radical | C2×C5×C3⋊D3 | C2 |
| cosocle | C2×C32 | C22×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ 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:
(1,2,3,4,5)(11,12)(14,15);(6,7,8,9)(13,14,15);(10,11,12)120 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#168, #10623755, C5×#3710, … |
| maximal normal subgroup of | C3⋊D3×#168, #10623755, C5×#3710, … |
| maximal quotient of | C5×C8×C3⋊D3, C5×#3743, C4×C5×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 106);