Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #4208 (order 162) · Inn(G) = C32 · Out(G) = C3×D3 |
| holomorph | G ⋊ Aut(G) = #72221828 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C9 | C32 |
| commutator, socle | C3 | C3×C9 |
| prime radical | C32 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ G | |
| LECS | G ▷ C9 ▷ C3 ▷ C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,5,3,7,13,9,16,21,4,8,12,10,17,20,18,24,26,11,6,14,19,15,22,25,23,27);(2,6,8)(5,12,14)(7,15,17)(13,20,22)(16,23,24)(21,26,27)Polycyclic presentation:
3^4|a^3=c,c^3=d,b^a=bd33 conjugacy classes — view the full character table.
| Fitting subgroup of | D27⋊C3 |
| maximal subgroup of | D27⋊C3, C2×C27⋊C3, #6869, … |
| maximal normal subgroup of | D27⋊C3, C2×C27⋊C3, #6869, … |
| maximal quotient of | C2×C27⋊C3, #6864, C32⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 6);