Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C4×D3 (order 96) · Inn(G) = D3 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×Dic3 |
| unambiguous semidirect product | C3⋊(C3×C4×C5) |
| derived | C3 ⋊ C3×C4×C5 |
| Hall | C32×C5 ⋊ C4 |
| Sylow | C32 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | D3 |
| commutator, hypocenter | C3 | C3×C4×C5 |
| Frattini | C2 | C3×C5×D3 |
| Fitting, socle, prime radical | C2×C32×C5 | C2 |
| cosocle | C2×C3 | C2×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C3×C5 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(5,6)(7,10,9,8)(11,12,13,14,15);(4,5,6)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×Dic3, C3×#1200, C3×#1202, … |
| maximal normal subgroup of | C3×D5×Dic3, C3×#1200, C3×#1202, … |
| maximal quotient of | C3×C5×C3⋊C8, C2×C3×C5×Dic3, C3×C5×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 14);