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#72249759

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

Aliases

No aliases for this group.

Subgroups

4820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=h,d^2=hj,e^2=g,f^2=j,g^2=i,i^2=j,c^a=ch,d^a=dh,d^c=dhj,e^a=ej,e^b=eg,e^c=ef,e^d=efij,f^a=fj,f^b=fj,f^d=fj,g^b=gi,g^c=gj,h^e=hj,i^b=ij

Conjugacy classes & character table

88 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, 2, F.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);