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#72313211

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

Aliases

No aliases for this group.

Subgroups

3586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,5,7)(3,8,4,6);(3,4)(6,8)(10,14,16,21)(11,17)(15,20,23,22)(18,24);(1,3,5,4)(2,6,7,8)(10,15,16,23)(11,18)(12,19)(14,20,21,22)(17,24);(1,4,5,3)(2,6,7,8);(9,10,11,15,12,14,18,22,13,16,17,23,19,21,24,20)

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fg,d^2=g,e^2=h,h^2=i,i^2=j,c^a=cg,c^b=cg,d^b=dg,e^b=ei,e^c=eh,f^e=fj,h^b=hj,h^c=hi,i^c=ij

Conjugacy classes & character table

112 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.7);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);