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#72285128

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

Aliases

No aliases for this group.

Subgroups

2574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=fhi,d^2=hij,c^a=cfh,c^b=cg,d^a=dhij,d^b=dghj,d^c=dfgi,e^a=eij,e^b=ei,e^d=ej,f^c=fi,f^d=fij,g^a=gj,g^b=gi,g^c=gj,g^d=gj,h^b=hj,h^c=hi

Conjugacy classes & character table

67 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.6);;
gap> SetPower(coll, 3, F.6*F.8*F.9);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);