Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C3×#72224542 (order 8748) · Inn(G) = C32 · Out(G) = C3×#66432 |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C32 |
| direct product | C9×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C32 |
| commutator | C3 | C32×C9 |
| Frattini, socle, cosocle | C32 | C33 |
| prime radical | C33 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,4,3,5,7,6,8,9);(10,11,13,12,15,17,16,14,18);(11,14,15)(13,17,18)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 36);