Finite Group Databasebeta

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#72265025

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

Aliases

No aliases for this group.

Subgroups

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=egi,d^2=h,e^2=i,i^2=j,b^a=bf,c^a=cd,c^b=ce,d^a=dh,d^b=dj,e^a=eh,e^b=ei,e^c=ej,f^c=fhj,g^c=gj,i^b=ij

Conjugacy classes & character table

94 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.7);;
gap> SetPower(coll, 3, F.5*F.7*F.9);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);