Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4181 |
| automorphism | Aut(G) = #72210336 (order 2916) · Inn(G) = C3⋊D9 · Out(G) = #272 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | C32⋊C9 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊D9 |
| commutator, Fitting, cosocle, hypocenter | C32⋊C9 | C2 |
| Frattini, socle | C32 | C3⋊D3 |
| LCS | G ▷ C32⋊C9 | |
| UCS | C1 ◁ C3 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,8)(4,12)(5,7)(6,13)(9,16)(10,14)(11,17)(15,18);(3,10,11)(6,15,16)(8,17,14)(9,18,13);(1,11,13,5,10,9,4,3,18)(2,15,8,12,16,14,7,6,17)Polycyclic presentation:
2,3^4|b^3=d^2,b^a=b^2d,c^a=c^2,d^a=d^2,c^b=ce21 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C9)⋊C2, C2×#4181, #92160, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C9)⋊C2, C2×#4181, #92160, … |
| maximal quotient of | #66328, C2×#4181, #92197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 17);