Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×#4208 (order 324) · Inn(G) = C1 · Out(G) = C2×#4208 |
| holomorph | G ⋊ Aut(G) = C2×#72221873 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C3 |
| abelian | C2×C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C9 | C2×C32 |
| socle, prime radical | C2×C32 | C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29);(30,31,32)162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 26);