Finite Group DatabasebetaFinite Group Databasebeta| order | 6561 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #14547534 |
| commutator, Frattini, cosocle | C32×H(3,𝔽3) | C33 |
| LCS | G ▷ C32×H(3,𝔽3) ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ C32×H(3,𝔽3) ◁ G | |
| LECS | G ▷ C32×H(3,𝔽3) ▷ C33 ▷ C3 ▷ C1 | |
2646 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 2646 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,7)(3,12,34)(4,10,22)(5,21,51)(6,11,24)(8,25,18)(9,29,16)(13,30,54)(14,41,63)(15,32,23)(17,31,52)(19,33,53)(20,50,73)(26,59,43)(27,57,48)(28,58,49)(35,56,46)(36,62,40)(37,55,44)(38,69,74)(39,65,66)(42,64,77)(45,68,76)(47,61,75)(60,79,72)(67,78,70)(71,80,81);(2,8,10)(7,22,25)(9,26,31)(11,28,33)(12,35,37)(21,27,50)(23,52,56)(24,53,58)(29,55,75)(30,60,64)(32,61,41)(34,63,59)(36,67,68)(51,73,57)(54,74,78)(62,77,81)(65,80,69)(66,76,79);(1,3,13)(2,9,30)(4,17,45)(5,15,40)(6,16,39)(7,23,54)(8,26,60)(10,31,64)(11,32,65)(12,36,21)(14,42,19)(18,47,71)(20,44,38)(22,52,74)(24,34,66)(25,56,78)(27,35,67)(28,61,80)(29,62,51)(33,41,69)(37,68,50)(43,72,48)(46,70,49)(53,63,76)(55,77,73)(57,75,81)(58,59,79)Polycyclic presentation:
3^8|b^a=bd,c^a=ce,d^a=df,d^c=dg,e^b=egh,e^c=eh,e^d=eh^2,f^c=fh,g^a=gh^2129 conjugacy classes — view the full character table.
| maximal subgroup of | #72265328 |
| maximal normal subgroup of | #72265328 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);