Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D45 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×AGL(1,ℤ/9) (order 2160) · Inn(G) = D45 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D90 |
| direct product | C2×D45 |
| unambiguous semidirect product | (C2×C5)⋊D9 |
| unambiguous semidirect product | (C2×C9)⋊D5 |
| Fitting | C2×C5×C9 ⋊ C2 |
| derived, Hall | C5×C9 ⋊ C22 |
| Sylow | C9 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D45 |
| commutator, cosocle, hypocenter | C5×C9 | C22 |
| Frattini | C3 | C2×D15 |
| Fitting | C2×C5×C9 | C2 |
| socle | C2×C3×C5 | D3 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C2 | |
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)(3,4,5,6,7,8,9,10,11)(12,13,14,15,16);(4,11)(5,10)(6,9)(7,8)(13,16)(14,15)48 conjugacy classes — view the full character table.
| maximal subgroup of | #67079, #67080, #67083, … |
| maximal normal subgroup of | #67079, #67080, #67083, … |
| maximal quotient of | Dic90, C4×D45, D180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 11);