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#72322063

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

Aliases

No aliases for this group.

Subgroups

1395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 4 generators on 128 points:

(2,10)(3,18)(4,9)(5,27)(7,30)(8,36)(11,42)(13,51)(14,55)(15,54)(16,50)(19,43)(20,69)(21,72)(22,74)(23,64)(24,75)(25,80)(26,34)(28,35)(29,78)(31,82)(32,89)(33,87)(37,77)(38,65)(39,66)(40,53)(41,60)(44,101)(45,103)(46,100)(47,104)(48,109)(49,106)(52,93)(56,117)(57,59)(58,119)(61,107)(62,97)(63,98)(67,115)(68,114)(70,125)(71,124)(73,126)(76,95)(79,128)(81,86)(83,84)(85,91)(88,127)(90,94)(92,123)(96,99)(102,118)(105,112)(108,116)(110,122)(111,113)(120,121);(1,2,6,10)(3,19)(4,12,9,17)(5,14,31,13)(7,15,33,40)(8,16,34,41)(11,23)(18,43)(20,67)(21,68)(22,63)(24,48,29,47)(25,49,37,61)(26,50,36,60)(27,51,82,55)(28,52)(30,53,87,54)(32,58,88,56)(35,93)(38,96)(39,46)(42,64)(44,71)(45,73)(57,85)(59,91)(62,92)(65,99)(66,100)(69,115)(70,121)(72,114)(74,98)(75,104,78,109)(76,105)(77,106,80,107)(79,111,83,108)(81,122)(84,113,128,116)(86,110)(89,117,127,119)(90,118)(94,102)(95,112)(97,123)(101,124)(103,126)(120,125);(1,3)(2,11)(4,22)(5,28)(6,23)(7,35)(8,38)(9,39)(10,19)(12,46)(13,52)(14,57)(15,59)(16,62)(17,63)(18,27)(20,30)(21,36)(24,76)(25,81)(26,73)(29,86)(31,85)(32,90)(33,91)(34,92)(37,95)(40,93)(41,96)(42,51)(43,55)(44,53)(45,60)(47,105)(48,110)(49,112)(50,68)(54,67)(56,118)(58,120)(61,122)(64,82)(65,89)(66,78)(69,83)(70,77)(71,87)(72,84)(74,75)(79,124)(80,94)(88,125)(97,117)(98,109)(99,119)(100,104)(101,108)(102,106)(103,113)(107,121)(111,115)(114,116)(123,127)(126,128);(1,4,6,9)(2,12,10,17)(5,29,31,24)(7,25,33,37)(8,26,34,36)(13,48,14,47)(15,49,40,61)(16,50,41,60)(18,64)(27,75,82,78)(28,85)(30,77,87,80)(32,83,88,79)(42,43)(51,104,55,109)(52,57)(53,106,54,107)(56,111,58,108)(65,123)(66,74)(69,124)(72,126)(76,86)(84,89,128,127)(90,125)(97,99)(98,100)(101,115)(103,114)(105,110)(113,117,116,119)(118,120)

Polycyclic presentation:

2^11|b^2=g,d^2=g,e^2=hi,f^2=hi,h^2=ij,i^2=jk,j^2=k,b^a=bg,c^a=ce,c^b=cefh,d^a=dk,d^c=dj,e^a=eh,e^b=ek,e^c=eh,e^d=ek,f^a=fh,f^c=fh,f^d=fk,g^c=gk,h^a=hi,h^c=hi,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

140 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);