Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = #72277883 (order 17496) · Inn(G) = H(3,𝔽3) · Out(G) = C2×D3×#272 |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C33 |
| direct product | C3×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C33 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3);(4,5,7)(6,8,11)(9,12,10);(5,8,9)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 51);