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) = C4×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×Dic5 |
| derived, Sylow | C5 ⋊ C4×C9 |
| Hall | C5×C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D5 |
| commutator, hypocenter | C5 | C4×C9 |
| Frattini | C2×C3 | C3×D5 |
| Fitting | C2×C5×C9 | C2 |
| socle, prime radical, cosocle | C2×C3×C5 | C2×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C9 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,16,17,18);(10,12,11,13)(15,18)(16,17)72 conjugacy classes — view the full character table.
| maximal subgroup of | C9×#170, C9⋊#170, #67077, … |
| maximal normal subgroup of | C9×#170, C9⋊#170, #67077, … |
| maximal quotient of | C9×#168, C2×C9×Dic5, Dic5×C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 2);