Finite Group Databasebeta

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#72321870

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

Aliases

No aliases for this group.

Subgroups

4907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,7)(5,9)(8,12)(10,13)(11,14)(15,16)(19,29)(20,30)(21,22)(25,38)(26,39)(27,28)(31,44)(32,43)(33,45)(34,35)(36,46)(37,42)(40,48)(41,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,15)(9,13)(12,16)(17,18)(19,25)(20,26)(21,27)(22,28)(23,24)(29,38)(30,39)(31,40)(32,41)(33,36)(34,42)(35,37)(43,47)(44,48)(45,46);(3,10)(6,13)(11,16)(14,15)(18,24)(25,36)(26,37)(27,28)(38,46)(39,42)(40,41)(47,48);(18,24)(25,36)(26,37)(27,28)(38,46)(39,42)(40,41)(47,48);(2,7,9,8)(6,14,13,15)(17,19,30,44,22,32,35,45,23,33,34,43,21,31,20,29)(18,25,39,48,28,41,37,46,24,36,42,47,27,40,26,38)

Polycyclic presentation:

2^10|e^2=fgi,f^2=hj,g^2=i,h^2=j,c^b=ci,d^b=dj,e^a=ef,f^a=fh,g^a=gj,h^a=hj

Conjugacy classes & character table

172 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, 5, F.6*F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);