Finite Group Databasebeta

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#72334262

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

Aliases

No aliases for this group.

Subgroups

At least 8904 conjugacy classes of subgroups; only the 3045 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=g,e^2=hij,b^a=bi,c^b=cgh,d^b=dghij,e^a=eij,e^b=eghij,e^c=eij,e^d=eghij,f^b=fghi,f^c=fij,f^d=fhi,f^e=fg

Conjugacy classes & character table

124 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, 2, F.7);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);