Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C22×C4×D32 (order 576) · Inn(G) = D32 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D3×Dic3 |
| unambiguous semidirect product | (C3×D3)⋊(C4×C5) |
| derived, Sylow | C32 ⋊ C2×C4×C5 |
| Hall | C32×C5 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D32 |
| commutator, hypocenter | C32 | C2×C4×C5 |
| Frattini | C2 | C5×D32 |
| Fitting, socle | C2×C32×C5 | C22 |
| prime radical | C2×C3×C5×D3 | C2 |
| cosocle | C2×C32 | C22×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C5 | |
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:
(1,2,3,4)(5,6,7)(9,10)(11,12,13,14,15);(6,7)(8,9,10)90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5×Dic3, D3×#1200, D3×#1202, … |
| maximal normal subgroup of | D3×D5×Dic3, D3×#1200, D3×#1202, … |
| maximal quotient of | C5×D3×C3⋊C8, C5×#3711, C5×Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 72);