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#72323103

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

No aliases for this group.

Subgroups

2006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,10,8,9,12,17,16)(7,11,15,14,13,18,20,19)(22,27,28,43)(23,31)(24,36)(25,38)(29,46,45,52)(30,47,40,44)(32,42,50,37)(33,39)(34,51)(35,49)(41,48);(5,7)(6,11)(8,14)(9,13)(10,15)(12,18)(16,19)(17,20);(1,2)(3,4)(7,13)(11,18)(14,19)(15,20);(2,4)(5,8,10,12,9,16,17,6)(7,14,15,18,13,19,20,11)(21,22,25,40,26,28,38,30)(23,32,34,52,35,50,48,46)(24,37,36,29,39,42,33,45)(27,41,47,49,43,51,44,31)

Polycyclic presentation:

2^11|a^2=g,d^2=gj,f^2=hjk,g^2=i,h^2=jk,j^2=k,c^b=ci,d^a=df,d^c=de,f^a=fh,f^d=fh,g^d=gk,h^a=hj,h^d=hj,j^a=jk,j^d=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

GAP

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);