Finite Group DatabasebetaFinite Group Databasebeta| order | 60 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C22×AGL(1,𝔽5) (order 80) · Inn(G) = D5 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = D3×#11717375 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic5 |
| derived, Sylow | C5 ⋊ C3×C4 |
| Hall | C3×C5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D5 |
| commutator, hypocenter | C5 | C3×C4 |
| Frattini | C2 | C3×D5 |
| Fitting, socle, prime radical | C2×C3×C5 | C2 |
| cosocle | C2×C5 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,3)(4,5)(6,9,8,7)(10,11,12);(1,2,4,5,3)24 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#170, C3⋊#170, D3×Dic5, … |
| maximal normal subgroup of | C3×#170, C3⋊#170, D3×Dic5, … |
| maximal quotient of | C3×#168, C2×C3×Dic5, C9×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(60, 2);