Finite Group DatabasebetaFinite Group Databasebeta| order | 60 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽5) (order 240) · Inn(G) = D15 · Out(G) = C2×C4 |
| holomorph | G ⋊ Aut(G) = #72224650 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D30 |
| direct product | C2×D15 |
| unambiguous semidirect product | (C2×C3)⋊D5 |
| unambiguous semidirect product | (C2×C5)⋊D3 |
| Fitting, socle | C2×C3×C5 ⋊ C2 |
| derived, Hall | C3×C5 ⋊ C22 |
| Sylow | C5 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D15 |
| commutator, cosocle, hypocenter | C3×C5 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2 | |
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)(6,7,8,9,10);(4,5)(7,10)(8,9)18 conjugacy classes — view the full character table.
| maximal subgroup of | #1200, #1202, #1203, … |
| maximal normal subgroup of | #1200, #1202, #1203, … |
| maximal quotient of | Dic30, C4×D15, D60, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(60, 12);