Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of C9⋊AGL(1,𝔽5) |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,ℤ/9) (order 1080) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊AGL(1,𝔽5) |
| Fitting, derived, Hall | C5×C9 ⋊ C4 |
| Sylow | C9 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C5×C9 | C4 |
| Frattini | C3 | ΓL(1,𝔽16) |
| socle | C3×C5 | Dic3 |
| prime radical | C3×D5 | D3 |
| cosocle | C9×D5 | C2 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3,4,5)(7,9)(8,12)(10,11)(13,14);(1,2,5,3,4)(6,14,9,12,11,10,8,7,13)Polycyclic presentation:
2^2,3,5,3|a^2=b,c^3=e^2,c^a=c^2e,d^a=d^3,e^a=e^2,d^b=d^421 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 6);