Finite Group Databasebeta

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#72260397

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

Aliases

No aliases for this group.

Subgroups

4869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=hi,e^2=f,f^2=h,g^2=i,h^2=j,i^2=j,c^a=chi,c^b=chi,d^b=dj,e^a=eg,f^a=fi,g^a=gij,h^a=hj,i^a=ij

Conjugacy classes & character table

172 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.8*F.9);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);