Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C4×D3×D4 (order 384) · Inn(G) = D3 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C5×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C3×C4×C5) |
| derived | C3 ⋊ C2×C3×C4×C5 |
| Hall | C32×C5 ⋊ C2×C4 |
| Sylow | C32 ⋊ C2×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C5 | D3 |
| commutator, hypocenter | C3 | C2×C3×C4×C5 |
| Frattini | C2 | C2×C3×C5×D3 |
| Fitting, socle, prime radical | C22×C32×C5 | C2 |
| cosocle | C2×C3 | C22×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C3×C5 | |
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)(13,14,15,16,17);(1,2)(3,4,5)180 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic3×Dic5, C3×#6665, C3×#6667, … |
| maximal normal subgroup of | C3×Dic3×Dic5, C3×#6665, C3×#6667, … |
| maximal quotient of | C2×C3×C5×C3⋊C8, C3×C5×#208, C3×C4×C5×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 98);