Finite Group Databasebeta

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#72323038

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

Aliases

No aliases for this group.

Subgroups

4822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=f,f^2=h,g^2=i,i^2=j,b^a=bhj,c^a=chj,c^b=chj,d^a=dg,d^b=dhj,e^c=ej,f^d=fj,g^a=gi,g^d=gij,i^a=ij,i^d=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

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