Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = #13091641 (order 1458) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C33 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C9 | H(3,𝔽3) |
| commutator | C32 | C3×C9 |
| Frattini, cosocle | C3×C9 | C32 |
| socle | C3 | C32⋊C9 |
| prime radical | C33 | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,5,3,8,13,9,17,21,4,6,12,10,14,20,18,22,26,11,7,15,19,16,23,25,24,27);(2,6,7)(8,14,16)(17,22,24)Polycyclic presentation:
3^5|a^3=d,d^3=e,b^a=bc,c^a=ce51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 16);