Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) order 1889568 |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C35 |
| direct product | C32×3−1+2 |
| unambiguous semidirect product | C9⋊C33 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | C32 |
| commutator, Frattini, cosocle | C3 | C34 |
| prime radical | C34 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3);(4,5,6);(7,8,10,9,12,14,13,11,15);(8,11,12)(10,14,15)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 63);