Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×C4×D3 (order 144) · Inn(G) = C1 · Out(G) = C2×C3×C4×D3 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×S4×AGL(1,ℤ/9) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C22×C5×C9 |
| multiplicative | (ℤ/209)ˣ, (ℤ/297)ˣ, (ℤ/418)ˣ, (ℤ/594)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
30 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)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18);(3,4)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 12);