Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 629856 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C32×C9 |
| abelian | C3×C92 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C32 | C33 |
| socle, prime radical | C33 | C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5,6,7,8,9);(10,11,12,13,14,15,16,17,18);(19,20,21)243 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 31);