Finite Group DatabasebetaFinite Group Databasebeta| order | 6561 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C35 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #14548193 |
| 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 | |
21005 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 21005 conjugacy classes of subgroups — too many to draw. Only its 99 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)(3,8,22)(4,9,21)(5,11,23)(7,24,17)(10,36,20)(12,29,52)(13,30,51)(14,31,53)(15,32,54)(16,33,48)(18,34,49)(19,35,50)(25,55,45)(26,56,46)(27,59,47)(28,60,41)(37,63,76)(38,64,77)(39,67,74)(40,66,75)(42,65,78)(43,68,72)(44,58,73)(57,79,70)(61,80,71)(62,81,69);(2,7,9)(6,21,24)(8,28,33)(10,27,34)(11,26,35)(13,39,16)(22,51,55)(23,50,56)(25,30,31)(29,62,68)(32,58,67)(36,49,59)(37,40,43)(41,44,45)(48,60,54)(52,76,79)(53,74,73)(57,63,64)(61,66,65)(69,71,70)(72,81,78)(75,80,77);(1,3,12)(2,8,29)(4,16,43)(5,15,38)(6,22,52)(7,25,57)(9,33,68)(10,31,65)(11,32,64)(13,40,18)(14,42,20)(17,45,70)(19,39,37)(21,48,72)(23,54,77)(24,55,79)(26,28,61)(27,58,62)(30,66,34)(35,67,63)(36,53,78)(41,71,46)(44,69,47)(49,51,75)(50,74,76)(56,60,80)(59,73,81)Polycyclic presentation:
3^8|b^a=bd,c^b=ce,d^b=df,d^c=dh,e^b=eg,e^d=eh,f^c=fh,g^a=gh273 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, 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);