Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C3×((C3⋊D3)⋊C3)⋊C2 (order 324) · Inn(G) = H(3,𝔽3) · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = #72222000 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
| affine special semilinear | AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2,4)(3,5,8)(6,9,7);(2,5,6)Polycyclic presentation:
3^4|b^a=bc,c^a=cd17 conjugacy classes — view the full character table.
| derived subgroup of | #4183, #66330, #4174⋊C2, … |
| Fitting subgroup of | #4174, #4175, #4183, … |
| maximal subgroup of | #4174, #4175, #4183, … |
| maximal normal subgroup of | #4174, #4175, #4183, … |
| maximal quotient of | C2×AΣL(1,𝔽27), #6856, #6857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 7);