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 | #14547277 |
| 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 | |
1050 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 57 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)(3,12,37)(4,9,21)(5,11,23)(7,24,18)(8,28,16)(10,36,20)(13,30,56)(14,42,63)(15,32,22)(17,33,53)(19,34,54)(25,60,48)(26,58,50)(27,59,51)(29,57,44)(31,55,45)(35,70,52)(38,72,49)(39,62,46)(40,69,78)(41,67,71)(43,68,76)(47,66,77)(61,80,73)(64,79,75)(65,81,74);(2,7,9)(6,21,24)(8,29,33)(10,26,35)(11,27,34)(12,38,42)(14,17,45)(22,55,57)(23,54,59)(25,31,32)(28,53,72)(30,65,69)(36,70,58)(37,63,60)(39,61,66)(40,47,43)(44,49,48)(56,78,79)(62,77,81)(64,68,67)(71,76,80)(73,74,75);(1,3,13)(2,8,30)(4,17,47)(5,16,41)(6,22,56)(7,25,61)(9,33,69)(10,12,39)(11,32,67)(14,43,52)(15,46,20)(18,49,74)(19,45,40)(21,53,76)(23,37,71)(24,57,79)(26,29,64)(27,38,65)(28,62,36)(31,68,34)(35,42,66)(44,75,51)(48,73,50)(54,55,77)(58,72,81)(59,60,80)(63,78,70)Polycyclic presentation:
3^8|b^a=bd,c^a=ce,c^b=cef^2,d^a=df,d^b=df^2,d^c=dg,e^b=egh^2,e^c=eh,e^d=eh^2,f^c=fh,g^a=gh^297 conjugacy classes — view the full character table.
| maximal subgroup of | #72265376 |
| maximal normal subgroup of | #72265376 |
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, 3, 2, F.3*F.5*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8^2);;
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);