Finite Group Databasebeta

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#72224865

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

Aliases

No aliases for this group.

Subgroups

3131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,8,12,9,13,19,28,10,14,20,27,21,29,33,26)(7,11,15,22,17,23,30,35,16,24,18,25,31,36,32,34);(1,2,3,4)(5,7)(6,11,14,24)(8,18)(9,17)(10,16)(12,25,27,22)(13,23,29,36)(15,20)(19,32)(21,31)(26,35,28,34)(30,33);(2,4)(6,12,14,27)(8,19)(9,21)(11,22,24,25)(13,26,29,28)(15,30)(17,31)(18,32)(20,33)(23,34,36,35);(6,13,14,29)(8,20)(11,23,24,36)(12,26,27,28)(15,18)(19,33)(22,34,25,35)(30,32)

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=f,d^2=f,g^2=i,i^2=j,b^a=bh,c^a=cgij,c^b=ce,d^a=dij,e^a=ej,f^a=fj,g^b=gj,g^c=gi,g^d=gj,h^a=hj,h^b=hj,i^c=ij

Conjugacy classes & character table

106 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.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);