Finite Group DatabasebetaFinite Group Databasebeta| order | 171 |
| isoclinism | isoclinic class of C19⋊C3 |
| automorphism | Aut(G) = C3×AGL(1,𝔽19) (order 1026) · Inn(G) = C19⋊C3 · Out(G) = C2×C32 |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C19 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C19⋊C3 |
| commutator, hypocenter | C19 | C9 |
| Fitting, socle, prime radical, cosocle | C3×C19 | C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C3 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 171 |
A minimal faithful permutation representation — 2 generators on 28 points:
(2,3,5)(4,7,8)(6,10,9)(11,15,13)(12,16,14)(17,19,18)(20,21,23,22,24,26,25,27,28);(1,2,4,8,13,18,15,3,6,11,16,5,9,14,7,12,17,19,10)Polycyclic presentation:
3^2,19|a^3=b,c^a=c^727 conjugacy classes — view the full character table.
| maximal subgroup of | #4294⋊C2, C2×#4294, #10587020, … |
| maximal normal subgroup of | #4294⋊C2, C2×#4294, #10587020, … |
| maximal quotient of | C2×#4294, #10587018, C3×#4294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(171, 1);