Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽5) (order 480) · Inn(G) = D15 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D15 |
| unambiguous semidirect product | (C2×C3)⋊(C3×D5) |
| unambiguous semidirect product | (C2×C5)⋊(C3×D3) |
| Fitting, socle | C2×C32×C5 ⋊ C2 |
| derived | C3×C5 ⋊ C22×C3 |
| Hall | C32×C5 ⋊ C22 |
| Sylow | C32 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D15 |
| commutator, cosocle, hypocenter | C3×C5 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(4,5)(6,7,8)(10,11)(12,13);(1,2)(3,4,5)(9,10,12,13,11)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#1200, C3×#1202, C3×#1203, … |
| maximal normal subgroup of | C3×#1200, C3×#1202, C3×#1203, … |
| maximal quotient of | C3×Dic30, C3×C4×D15, C3×D60, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 34);