Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) order 26244 |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C33 |
| direct product | C3×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C33 |
| prime radical | C33 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
66 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);(4,5,8,7,11,17,15,10,21)(6,12,16,14,9,18,25,20,28)(13,23,26,24,22,27,30,19,29);(4,6,13,15,25,30,7,14,24)(5,9,19,10,12,22,11,20,23)(8,16,26,21,28,29,17,18,27)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 54);