Finite Group Databasebeta

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#72297883

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

Aliases

No aliases for this group.

Subgroups

30767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 6 generators on 28 points:

(8,12)(10,11)(13,14)(15,16)(17,18)(19,20)(22,25)(26,28);(1,2,4,3)(5,6,9,7)(8,10,12,11)(14,18)(16,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,9)(8,10)(11,12)(23,27)(26,28);(1,2,4,3)(6,7)(10,11)(13,15)(14,16)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,19)(18,20)(21,23)(22,26)(24,27)(25,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,18)(17,20)

Polycyclic presentation:

2^10|b^2=gj,d^2=ghj,e^2=gj,b^a=bg,c^b=ch,d^a=dhi,d^b=dh,d^c=dh,e^c=ei,e^d=eh,f^a=fgi,f^b=fghi,f^d=fghi,f^e=fghi

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