Finite Group Databasebeta

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#72323263

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

Aliases

No aliases for this group.

Subgroups

At least 7098 conjugacy classes of subgroups; only the 3861 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|c^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=cgh,c^b=cgj,d^a=di,d^c=di,e^a=egi,e^b=eg,e^c=egi,e^d=ei,f^a=fgh,f^b=fg,f^c=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

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