Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×Dic15 |
| unambiguous semidirect product | (C2×C3)⋊(C3×Dic5) |
| derived | C3×C5 ⋊ C2×C3×C4 |
| Hall | C32×C5 ⋊ C2×C4 |
| Sylow | C32 ⋊ C2×Dic5 |
| Sylow | C5 ⋊ C2×C3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D15 |
| commutator, hypocenter | C3×C5 | C2×C3×C4 |
| Frattini | C2 | C2×C3×D15 |
| Fitting, socle, prime radical | C22×C32×C5 | C2 |
| cosocle | C2×C3×C5 | C22×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22×C3 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2)(4,5)(6,7,8)(9,10,11,12)(14,15)(16,17);(1,2)(3,4,5)(13,14,16,17,15)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic3×Dic5, C3×#6665, C3×(C2×D3)⋊Dic5, … |
| maximal normal subgroup of | C3×Dic3×Dic5, C3×#6665, C3×(C2×D3)⋊Dic5, … |
| maximal quotient of | C2×C3×C3⋊#168, C3×#6710, C3×C4×Dic15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 103);