Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | Aut(G) = C3×#72224543 (order 8748) · Inn(G) = C33 · Out(G) = C3×D3×C3⋊D3 |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C9⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C32 | C33 |
| prime radical | C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,4,7)(5,9,8)(10,11,13)(12,15,17)(14,18,16);(1,2,5,3,4,8,6,7,9);(11,14,15)(13,17,18)Polycyclic presentation:
3^5|b^3=d,b^a=bd,c^a=ce51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 39);