Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) order 78732 |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C33 |
| direct product | C3×C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C33 | C32 |
| commutator | C3 | C32×C9 |
| Frattini, cosocle | C32 | C33 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
90 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);(5,7,10)(8,12,11)(13,14,16,15,17,19,18,20,21);(4,5,8,6,7,11,9,10,12)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 33);