Finite Group Databasebeta

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#72308703

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

Aliases

No aliases for this group.

Subgroups

5927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=gij,d^2=fg,e^2=h,i^2=j,c^a=ci,c^b=cj,d^b=df,d^c=dh,e^a=eghj,e^b=egh,e^d=egj,i^a=ij

Conjugacy classes & character table

118 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.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);