Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽5) (order 240) · Inn(G) = D5 · Out(G) = C23×C3 |
| holomorph | G ⋊ Aut(G) = AGL(1,ℤ/9)×#11717375 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C9×D5 |
| unambiguous semidirect product | C5⋊(C22×C9) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C9) |
| Fitting | C2×C5×C9 ⋊ C2 |
| derived, Sylow | C5 ⋊ C22×C9 |
| Hall | C5×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D5 |
| commutator, hypocenter | C5 | C22×C9 |
| Frattini | C3 | C2×C3×D5 |
| Fitting | C2×C5×C9 | C2 |
| socle | C2×C3×C5 | C2×C3 |
| prime radical | C2×C3×D5 | C3 |
| cosocle | C3×C5 | C22×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C9 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(4,5)(6,7)(8,12,16,10,15,11,13,9,14);(1,2)(3,4,6,7,5)72 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic9, #67082, #67083, … |
| maximal normal subgroup of | D5×Dic9, #67082, #67083, … |
| maximal quotient of | C9×Dic10, C4×C9×D5, C9×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 9);