Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #11757730 (order 972) · Inn(G) = C32 · Out(G) = C2×#272 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C32⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C32 | C32 |
| commutator | C3 | C3×C9 |
| prime radical | C33 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
23 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,13,12,14,16,15,17,18);(2,5,6)(4,8,9)Polycyclic presentation:
3^4|a^3=d,b^a=bc33 conjugacy classes — view the full character table.
| derived subgroup of | #4181, #66328, ((C3⋊D3)⋊C9)⋊C2, … |
| Fitting subgroup of | (C3⋊D3)⋊C9, #4169, #4181, … |
| maximal subgroup of | (C3⋊D3)⋊C9, #4169, #4181, … |
| maximal normal subgroup of | (C3⋊D3)⋊C9, #4169, #4181, … |
| maximal quotient of | C2×C32⋊C9, #6855, C32⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 3);