Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = #92285 (order 486) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C32 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C2×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C32 |
| prime radical | C2×C32 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32 ◁ G | |
26 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)(3,4,7,6,10,16,14,9,20)(5,11,15,13,8,17,24,19,27)(12,22,25,23,21,26,29,18,28);(3,5,12,14,24,29,6,13,23)(4,8,18,9,11,21,10,19,22)(7,15,25,20,27,28,16,17,26)34 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 31);