Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #4208 (order 162) · Inn(G) = C32 · Out(G) = C3×D3 |
| holomorph | G ⋊ Aut(G) = C2×#72221828 |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C1 |
| direct product | C2×C27⋊C3 |
| unambiguous semidirect product | (C2×C27)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C32 |
| commutator | C3 | C2×C3×C9 |
| Frattini, cosocle | C9 | C2×C32 |
| socle | C2×C3 | C3×C9 |
| prime radical | C2×C32 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2)(3,4,7,5,9,15,11,18,23,6,10,14,12,19,22,20,26,28,13,8,16,21,17,24,27,25,29);(4,8,10)(7,14,16)(9,17,19)(15,22,24)(18,25,26)(23,28,29)66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 27);