Finite Group Databasebeta

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#72231889

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

Aliases

No aliases for this group.

Subgroups

787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bj,c^b=ci,d^a=dh,e^d=ej,f^a=fj

Conjugacy classes & character table

220 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, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);