Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊D3)) |
| automorphism | Aut(G) order 1965150720 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C36 |
| unambiguous semidirect product | C3⋊(C3⋊(C3⋊D3)) |
| Fitting, socle, derived, Sylow | C34 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C34 | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 5 generators on 12 points:
(2,3)(5,6)(8,9)(11,12);(10,11,12);(7,8,9);(4,5,6);(1,2,3)Polycyclic presentation:
2,3^4|b^a=b^2,c^a=c^2,d^a=d^2,e^a=e^242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 54);