Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C3×AGL(2,𝔽3) (order 1296) · Inn(G) = C32 · Out(G) = C3×GL(2,𝔽3) |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C9 | C32 |
| commutator, Frattini, socle, cosocle | C3 | C33 |
| prime radical | H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,5)(3,7,13)(4,8,12)(6,14,11)(9,16,21)(10,17,20)(15,22,19)(18,24,26)(23,27,25);(2,6,8)(5,12,14)(7,15,17)(13,20,22)(16,23,24)(21,26,27);(1,3,9,4,10,18,11,19,25)(2,7,16,8,17,24,6,15,23)(5,13,21,12,20,26,14,22,27)Polycyclic presentation:
3^4|c^3=d,b^a=bd33 conjugacy classes — view the full character table.
| derived subgroup of | #4207⋊C2, #4207⋊C4, #10619908, … |
| Fitting subgroup of | #4207, #4208, #66419, … |
| maximal subgroup of | #4207, #4208, C2×#737, … |
| maximal normal subgroup of | #4207, #4208, C2×#737, … |
| maximal quotient of | C2×#737, #6887, C9×H(3,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 14);