Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C33×#272 (order 1458) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C33 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | 3−1+2⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | H(3,𝔽3) |
| commutator | C32 | C3×C9 |
| Frattini, prime radical, cosocle | C33 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4,3,5,7,6,8,9)(10,11,14)(12,16,13)(15,18,17);(10,12,16,13,15,18,17,11,14)Polycyclic presentation:
3^5|a^3=d,b^3=e,b^a=bc,c^a=ce,c^b=ce51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 18);