Finite Group DatabasebetaFinite Group Databasebeta| order | 19683 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C37 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #72265462 |
| commutator, Frattini, cosocle | C33×H(3,𝔽3) | C33 |
| LCS | G ▷ C33×H(3,𝔽3) ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ C33×H(3,𝔽3) ◁ G | |
| LECS | G ▷ C33×H(3,𝔽3) ▷ C33 ▷ C3 ▷ C1 | |
7702 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 7702 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,8,21)(4,9,22)(5,11,23)(7,24,41)(10,37,19)(12,29,55)(13,30,20)(14,31,56)(15,32,57)(16,44,72)(17,34,58)(18,35,60)(25,54,38)(26,51,73)(27,62,47)(28,63,46)(33,53,42)(36,59,50)(39,69,80)(40,65,78)(43,70,79)(45,64,52)(48,71,76)(49,66,81)(61,74,68)(67,77,75);(1,3,12)(2,7,25)(4,13,38)(5,15,39)(6,20,51)(8,26,9)(10,28,64)(11,27,65)(14,43,19)(16,45,50)(17,41,73)(18,42,40)(21,54,58)(22,24,55)(23,53,61)(29,34,30)(31,67,36)(32,68,35)(33,69,66)(37,72,77)(44,70,71)(46,75,48)(47,74,49)(52,76,56)(57,78,81)(59,63,79)(60,62,80);(1,3,12)(2,8,29)(4,16,38)(5,15,39)(6,21,55)(7,26,66)(9,33,65)(10,31,70)(11,32,69)(13,40,18)(14,43,19)(17,47,73)(20,52,22)(23,57,80)(24,61,58)(25,36,30)(27,68,71)(28,67,34)(35,44,64)(37,56,79)(41,75,48)(42,45,50)(46,74,49)(51,76,63)(53,54,60)(59,72,78)(62,77,81)Polycyclic presentation:
3^9|b^a=bd,c^a=cf,c^b=ce,d^c=dg,e^a=eg^2h,e^b=eh^2i,e^c=ehi^2,e^d=ei^2,f^b=fhi^2,f^d=fi^2,g^a=gi,g^b=gi,g^c=gi,h^a=hi^2179 conjugacy classes — view the full character table.
| maximal subgroup of | UT(5,𝔽3) |
| maximal normal subgroup of | UT(5,𝔽3) |
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", "i");;
gap> coll := SingleCollector(F, [3, 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);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7^2*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8^2*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.9^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);