Finite Group DatabasebetaFinite Group Databasebeta| order | 171 |
| isoclinism | isoclinic class of C19⋊C3 |
| automorphism | Aut(G) = D3×AGL(1,𝔽19) (order 2052) · Inn(G) = C19⋊C3 · Out(G) = C2×C3×D3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
| direct product | C3×C19⋊C3 |
| unambiguous semidirect product | C19⋊C32 |
| unambiguous semidirect product | (C3×C19)⋊C3 |
| Fitting, socle | C3×C19 ⋊ C3 |
| derived, Sylow | C19 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C19⋊C3 |
| commutator, cosocle, hypocenter | C19 | C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C19 | C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C3 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 57 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,3,5)(4,7,8)(6,10,9)(11,15,13)(12,16,14)(17,19,18);(1,2,4,8,13,18,15,3,6,11,16,5,9,14,7,12,17,19,10)(20,21,22)27 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D19⋊C3, D3×C19⋊C3, C3⋊(D19⋊C3), … |
| maximal normal subgroup of | C3×D19⋊C3, D3×C19⋊C3, C3⋊(D19⋊C3), … |
| maximal quotient of | C2×C3×C19⋊C3, C9×C19⋊C3, #10587022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(171, 4);