Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of C3⋊((C3⋊D3)⋊C3) |
| automorphism | Aut(G) = #72222017 (order 5832) · Inn(G) = G · Out(G) = D32 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊((C3⋊D3)⋊C3) |
| Fitting, Sylow | C3×H(3,𝔽3) ⋊ C2 |
| derived | C33 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33 | C2×C3 |
| Frattini | C3 | C3×C3⋊D3 |
| Fitting | C3×H(3,𝔽3) | C2 |
| socle | C32 | C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3)(5,10,12,7,8,11)(6,9);(1,2,3);(4,5,7)(6,8,11)(9,12,10)Polycyclic presentation:
2,3^4|c^a=c^2,d^a=d^2,e^a=e^2,c^b=ce21 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 40);