Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #11757730 (order 972) · Inn(G) = C32 · Out(G) = C2×#272 |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C32 |
| direct product | C2×C32⋊C9 |
| unambiguous semidirect product | (C2×C32)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C32 |
| commutator | C3 | C2×C3×C9 |
| Frattini, cosocle | C32 | C2×C32 |
| prime radical | C2×C33 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(4,6,9)(7,11,10)(12,13,15,14,16,18,17,19,20);(3,4,7)(5,6,10)(8,9,11)66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 24);