Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #72222023 (order 23328) · Inn(G) = C32 · Out(G) = D3×AGL(2,𝔽3) |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C34 |
| direct product | C3×H(3,𝔽3) |
| unambiguous semidirect product | C32⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32 |
| commutator, Frattini, cosocle | C3 | C33 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
56 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)(6,9,11)(8,12,10);(5,8,9)(7,11,12)33 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 12);