Finite Group DatabasebetaFinite Group Databasebeta| order | 60 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C4×D3 (order 48) · Inn(G) = D3 · Out(G) = C2×C4 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×#3811 |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic3 |
| unambiguous semidirect product | C3⋊(C4×C5) |
| derived, Sylow | C3 ⋊ C4×C5 |
| Hall | C3×C5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3 |
| commutator, hypocenter | C3 | C4×C5 |
| Frattini | C2 | C5×D3 |
| Fitting, socle, prime radical | C2×C3×C5 | C2 |
| cosocle | C2×C3 | C2×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C5 | |
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)(4,5,6,7)(8,9,10,11,12);(1,2,3)30 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic3, #1200, #1202, … |
| maximal normal subgroup of | D5×Dic3, #1200, #1202, … |
| maximal quotient of | C5×C3⋊C8, C2×C5×Dic3, C5×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(60, 1);