Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #72221863 (order 2916) · Inn(G) = C32 · Out(G) = C2×#4208 |
| homology | Abelianization (H1) = C3×C27 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C32⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C3×C9 | C32 |
| commutator | C3 | C3×C27 |
| socle | C32 | C3×C9 |
| prime radical | C33 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36);(2,5,6)(4,8,9)Polycyclic presentation:
3^5|a^3=d,d^3=e,b^a=bc99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 12);