Finite Group Databasebeta

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#72280928

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

Aliases

No aliases for this group.

Subgroups

At least 8981 conjugacy classes of subgroups; only the 3749 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=h,c^2=i,d^2=hi,e^2=g,f^2=gi,b^a=bh,c^b=chi,d^a=dh,d^b=di,d^c=dhj,e^b=ej,e^c=egi,e^d=egi,f^b=fi,f^e=fgi

Conjugacy classes & character table

166 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.7);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);