Finite Group Databasebeta

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#72280994

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

Aliases

No aliases for this group.

Subgroups

At least 9663 conjugacy classes of subgroups; only the 3765 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=hi,c^2=i,d^2=h,e^2=gi,f^2=g,b^a=bh,c^b=ch,d^a=dhi,d^b=di,d^c=dhij,e^a=eg,e^b=egj,e^c=eg,e^d=eg,f^b=fi,f^e=fg

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