Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D3×D15 |
| automorphism | Aut(G) = D32×AGL(1,𝔽5) (order 720) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D3×D15 |
| unambiguous semidirect product | (C3×D3)⋊D5 |
| unambiguous semidirect product | (C5×D3)⋊D3 |
| Fitting, socle, derived, Hall | C32×C5 ⋊ C22 |
| Sylow | C32 ⋊ C2×D5 |
| Sylow | C5 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C5 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2,3)(5,6)(8,9)(10,11);(2,3)(4,5,6)(7,8,10,11,9)27 conjugacy classes — view the full character table.
| maximal subgroup of | D32×D5, C2×D3×D15, D3×D45, … |
| maximal normal subgroup of | D32×D5, C2×D3×D15, C3×D3×D15, … |
| maximal quotient of | Dic3×D15, D3×Dic15, #67149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 29);