Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | Aut(G) = C23×D3×AGL(1,𝔽5) (order 960) · Inn(G) = D3×D5 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1200 |
| derived | C3×C5 ⋊ C2×C3×C4 |
| Hall | C32×C5 ⋊ C2×C4 |
| Sylow | C32 ⋊ C4×D5 |
| Sylow | C5 ⋊ C3×C4×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×D5 |
| commutator, hypocenter | C3×C5 | C2×C3×C4 |
| Frattini | C2 | C3×D3×D5 |
| Fitting, socle | C2×C32×C5 | C22 |
| prime radical | C2×C3×D15 | C2 |
| cosocle | C2×C3×C5 | C22×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3 | |
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)(10,11)(12,13,15,14);(2,3)(4,5)(9,10,11)(12,14,15,13)72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#6738, C3×#6740, #10623974, … |
| maximal normal subgroup of | C3×#6738, C3×#6740, #10623974, … |
| maximal quotient of | C3×#6648, C3×#6651, C3×Dic3×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 60);