Finite Group Databasebeta

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#72303100

• p-group
• nilpotent
• supersolvable
• solvable
• monolithic
• monomial

Aliases

No aliases for this group.

Subgroups

4585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13,16,40)(4,18)(5,10)(6,26,28,11)(7,12)(8,17)(9,21)(14,41,43,39)(15,34,46,64)(19,35)(20,33,53,49)(22,54)(23,32)(24,36,27,38)(25,37)(29,57)(30,56)(31,47)(42,63,45,60)(44,48)(50,58)(51,62,52,61)(55,59);(1,3)(2,8)(4,14)(5,22,25,56)(6,17,28,47)(7,16)(9,29)(10,34,37,64)(11,13,26,40)(12,31)(15,27)(18,48)(19,50,23,55)(20,44,53,57)(21,43)(24,46)(30,38)(32,60,35,63)(33,39,49,41)(36,54)(42,52)(45,51)(58,62)(59,61);(1,4,7,21)(2,9,12,18)(3,14,16,43)(5,23,25,19)(6,20,28,53)(8,29,31,48)(10,35,37,32)(11,33,26,49)(13,39,40,41)(15,45,46,42)(17,44,47,57)(22,55,56,50)(24,52,27,51)(30,59,54,58)(34,63,64,60)(36,62,38,61);(1,5)(2,10)(3,15)(4,19)(6,27)(7,25)(8,30)(9,32)(11,38)(12,37)(13,34)(14,42)(16,46)(17,22)(18,35)(20,52)(21,23)(24,28)(26,36)(29,58)(31,54)(33,62)(39,60)(40,64)(41,63)(43,45)(44,50)(47,56)(48,59)(49,61)(51,53)(55,57)

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=j,f^2=j,b^a=bf,c^a=cj,d^a=deij,d^b=dfgj,d^c=dj,e^b=eh,f^a=fhj,f^b=fi,f^d=fi,f^e=fj,g^a=gij,g^d=gi,g^f=gj,h^a=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

70 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);