Finite Group Databasebeta

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#72227201

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

Aliases

No aliases for this group.

Subgroups

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=d,c^2=fj,d^2=g,e^2=hj,f^2=j,g^2=j,h^2=i,b^a=be,c^a=cj,c^b=cf,d^b=dij,e^a=eh,e^b=ehij,f^b=fj,h^a=hi,h^b=hi

Conjugacy classes & character table

124 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.4);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);