Finite Group Databasebeta

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#72265622

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

Aliases

No aliases for this group.

Subgroups

1576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=e,c^2=fij,d^2=hj,f^2=ij,i^2=j,b^a=bg,c^a=cd,c^b=cf,d^a=dh,e^c=ej,f^a=fhj,f^b=fi,g^c=ghj,i^b=ij

Conjugacy classes & character table

82 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.5);;
gap> SetPower(coll, 3, F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);