Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C3×#4183 (order 486) · Inn(G) = C32 · Out(G) = #272 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C32 | C32 |
| commutator | C3 | C3×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
17 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,4,7)(5,9,8)(10,11,13,12,14,16,15,17,18);(1,2,5,3,4,8,6,7,9)Polycyclic presentation:
3^4|a^3=d,b^3=c,b^a=bc33 conjugacy classes — view the full character table.
| Fitting subgroup of | D9⋊C9 |
| maximal subgroup of | D9⋊C9, C2×C9⋊C9, #6875, … |
| maximal normal subgroup of | D9⋊C9, C2×C9⋊C9, #6875, … |
| maximal quotient of | C2×C9⋊C9, #6855, C9⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 4);