Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #72277535 (order 8748) · Inn(G) = C32 · Out(G) = C3×#66432 |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C32 |
| direct product | C3×C27⋊C3 |
| unambiguous semidirect product | C27⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C32 |
| commutator | C3 | C32×C9 |
| Frattini, cosocle | C9 | C33 |
| socle | C32 | C3×C9 |
| prime radical | C33 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C9 ▷ C3 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3);(4,5,8,6,10,16,12,19,24,7,11,15,13,20,23,21,27,29,14,9,17,22,18,25,28,26,30);(5,9,11)(8,15,17)(10,18,20)(16,23,25)(19,26,27)(24,29,30)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 49);