Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #72212528 (order 2916) · Inn(G) = C32 · Out(G) = #66432 |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C32 |
| direct product | C2×C3×3−1+2 |
| unambiguous semidirect product | (C2×C9)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C32 |
| commutator, Frattini, cosocle | C3 | C2×C33 |
| prime radical | C2×C33 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,4,5);(6,7,9,8,11,13,12,10,14);(7,10,11)(9,13,14)66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 49);