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) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D5 |
| unambiguous semidirect product | C5⋊(C22×C3) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C3) |
| Fitting, socle | C2×C3×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C22×C3 |
| Hall | C3×C5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D5 |
| commutator, cosocle, hypocenter | C5 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2)(3,4,5)(7,8)(9,10);(1,2)(6,7,9,10,8)24 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic3, #1201, #1202, … |
| maximal normal subgroup of | D5×Dic3, #1201, #1202, … |
| maximal quotient of | C3×Dic10, C3×C4×D5, C3×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(60, 10);