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 | #14548209 |
| 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 | |
2510 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 2510 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6)(3,12,38)(4,9,22)(5,11,24)(7,25,18)(8,29,15)(10,37,21)(13,31,57)(14,34,70)(16,33,23)(17,47,56)(19,35,54)(20,36,55)(26,60,49)(27,59,51)(28,62,52)(30,64,50)(32,53,43)(39,58,45)(40,63,42)(41,67,78)(44,69,77)(46,68,72)(48,71,76)(61,79,74)(65,80,75)(66,81,73);(2,7,9)(6,22,25)(8,30,34)(10,28,35)(11,27,36)(12,39,47)(14,43,17)(23,56,58)(24,55,59)(26,32,33)(29,53,60)(31,66,71)(37,54,62)(38,70,64)(40,61,67)(41,44,48)(45,49,50)(57,78,79)(63,77,80)(65,69,68)(72,76,81)(73,75,74);(1,3,13)(2,8,31)(4,17,48)(5,16,42)(6,23,57)(7,26,61)(9,34,71)(10,33,68)(11,12,40)(14,44,19)(15,46,21)(18,50,74)(20,43,41)(22,53,76)(24,29,63)(25,58,79)(27,30,65)(28,39,66)(32,69,35)(36,47,67)(37,38,72)(45,75,51)(49,73,52)(54,56,77)(55,70,78)(59,60,80)(62,64,81)Polycyclic presentation:
3^8|b^a=bd,c^a=cf^2,c^b=ce,d^b=df,d^c=dh,e^b=eg,e^d=eh,f^c=fh,g^a=gh129 conjugacy classes — view the full character table.
| maximal subgroup of | #72265253 |
| maximal normal subgroup of | #72265253 |
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.6^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);