Finite Group Databasebeta

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#72330723

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

Aliases

No aliases for this group.

Subgroups

At least 7713 conjugacy classes of subgroups; only the 1109 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=gj,c^2=h,d^2=f,e^2=fhij,b^a=bfg,c^a=cfh,c^b=cfgi,d^b=dfhj,d^c=dfh,e^a=efghi,e^b=efghi,e^c=efi,e^d=ehj

Conjugacy classes & character table

136 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, 1, F.6);;
gap> SetPower(coll, 2, F.7*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);