Finite Group DatabasebetaFinite Group Databasebeta| order | 6561 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C35 · Schur multiplier (H2) = C312 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C32×C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C35 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32×H(3,𝔽3) ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
36405 conjugacy classes of subgroups; only the 3519 normal subgroups are shown.
This group has 36405 conjugacy classes of subgroups — too many to draw. Only its 3519 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 — 5 generators on 81 points:
(1,2,6)(3,7,18)(4,8,19)(5,10,20)(9,28,17)(11,21,38)(12,22,39)(13,24,40)(14,23,41)(15,25,42)(16,26,43)(27,52,37)(29,44,58)(30,46,59)(31,47,60)(32,48,62)(33,49,61)(34,45,63)(35,50,64)(36,51,65)(53,66,73)(54,68,74)(55,70,75)(56,69,76)(57,67,77)(71,78,81)(72,79,80);(1,3,11)(2,7,21)(4,12,29)(5,14,30)(6,18,38)(8,22,44)(9,24,45)(10,23,46)(13,34,17)(15,31,53)(16,32,54)(19,39,58)(20,41,59)(25,47,66)(26,48,68)(27,49,67)(28,40,63)(33,57,37)(35,55,71)(36,56,72)(42,60,73)(43,62,74)(50,70,78)(51,69,79)(52,61,77)(64,75,81)(65,76,80);(1,4,15)(2,8,25)(3,12,31)(5,16,36)(6,19,42)(7,22,47)(9,27,50)(10,26,51)(11,29,53)(13,33,55)(14,32,56)(17,37,35)(18,39,60)(20,43,65)(21,44,66)(23,48,69)(24,49,70)(28,52,64)(30,54,72)(34,57,71)(38,58,73)(40,61,75)(41,62,76)(45,67,78)(46,68,79)(59,74,80)(63,77,81);(7,23,24)(18,40,41)(21,45,46)(22,48,49)(38,59,63)(39,61,62)(44,67,68)(47,69,70)(58,74,77)(60,75,76)(66,78,79)(73,80,81);(4,16,37)(8,26,27)(12,32,33)(15,35,36)(19,43,52)(22,48,49)(25,50,51)(29,54,57)(31,55,56)(39,62,61)(42,64,65)(44,68,67)(47,70,69)(53,71,72)(58,74,77)(60,75,76)(66,78,79)(73,81,80)Polycyclic presentation:
3^8|d^a=df,d^b=dg,e^c=eh^2,f^b=fh,g^a=gh465 conjugacy classes — view the full character table.
| maximal subgroup of | #72265118, #72265253 |
| maximal normal subgroup of | #72265118, #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, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);