Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = (#4178⋊C2)⋊C3 (order 972) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C3×D3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C32 |
No aliases for this group.
| 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 | |
16 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)(3,9,13)(4,8,14)(6,15,11)(7,18,12)(10,20,23)(16,26,27)(17,25,22)(19,24,21);(1,3,10,12,22,27,4,11,21)(2,6,16,7,9,19,8,17,20)(5,13,23,18,25,26,14,15,24)Polycyclic presentation:
3^4|b^3=d^2,b^a=bc,c^a=cd17 conjugacy classes — view the full character table.
| derived subgroup of | #4185, #66332, #4178⋊C2, … |
| Fitting subgroup of | #4178, #4179, #4185, … |
| maximal subgroup of | #4178, #4179, #4185, … |
| maximal normal subgroup of | #4178, #4179, #4185, … |
| maximal quotient of | C2×#732, #6856, #6859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 9);