Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4169 |
| automorphism | Aut(G) = (((C3⋊D3)⋊C9)⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | C32⋊C9 ⋊ C2 |
| derived | C3×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C9 | C2×C3 |
| Frattini, socle | C32 | C3×D3 |
| Fitting | C32⋊C9 | C2 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,7,9,4,5,8)(3,6)(11,13)(12,16)(14,15)(17,18);(1,2,4)(3,5,8)(6,9,7)(10,18,13,16,15,14,12,11,17)Polycyclic presentation:
2,3^4|c^3=e^2,c^a=c^2e,d^a=d^2,e^a=e^2,c^b=cd21 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C9)⋊C2, C2×#4169, #92198, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C9)⋊C2, C2×#4169, C3×#4169, … |
| maximal quotient of | #66316, C2×#4169, #92158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 5);