Finite Group DatabasebetaFinite Group Databasebeta| order | 19683 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C37 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #72265324 |
| commutator, Frattini, cosocle | C32×H(3,𝔽3) | C34 |
| LCS | G ▷ C32×H(3,𝔽3) ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ C33×H(3,𝔽3) ◁ G | |
| LECS | G ▷ C32×H(3,𝔽3) ▷ C33 ▷ C3 ▷ C1 | |
38746 conjugacy classes of subgroups; only the 394 normal subgroups are shown.
This group has 38746 conjugacy classes of subgroups — too many to draw. Only its 394 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 — 4 generators on 81 points:
(1,2,7)(3,9,22)(4,10,21)(5,20,47)(6,11,23)(8,24,17)(12,28,50)(13,30,53)(14,36,48)(15,31,51)(16,40,52)(18,32,49)(19,46,70)(25,57,42)(26,55,44)(27,56,45)(29,63,39)(33,62,74)(34,65,71)(35,64,72)(37,61,75)(38,60,54)(41,67,73)(43,59,76)(58,79,69)(66,80,77)(68,81,78);(2,8,10)(7,21,24)(9,25,29)(11,27,32)(20,26,46)(22,48,54)(23,49,56)(28,58,61)(30,59,36)(31,60,40)(47,70,55)(50,71,77)(51,63,76)(52,57,53)(62,81,65)(64,80,67)(72,75,78)(73,79,74);(1,3,12)(2,9,28)(4,16,41)(5,13,35)(6,15,33)(7,22,50)(8,25,58)(10,29,61)(11,31,62)(14,37,18)(17,43,68)(19,39,34)(20,30,64)(21,48,71)(23,51,74)(24,54,77)(26,59,80)(27,60,81)(32,40,65)(36,67,46)(38,69,44)(42,66,45)(47,53,72)(49,63,73)(52,75,70)(55,57,78)(56,76,79);(3,13,15)(12,33,35)(14,16,39)(22,51,53)(34,41,37)(38,42,43)(48,63,52)(50,72,74)(54,76,57)(66,69,68)(71,75,73)(77,78,79)Polycyclic presentation:
3^9|b^a=be,c^a=cf,d^a=dg,d^c=dh^2,e^a=eg,e^c=eh,f^b=fhi,f^e=fi^2,g^c=gi,h^a=hi^2355 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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9^2);;
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);