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#72321944

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

Aliases

No aliases for this group.

Subgroups

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^11|a^2=f,c^2=fik,d^2=gik,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bhk,c^a=ce,c^b=ck,d^a=dgj,d^c=dgj,e^a=eg,e^b=ek,e^c=eg,f^c=fjk,f^d=fk,f^e=fk,g^a=gi,g^c=gi,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

152 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, 1, F.6);;
gap> SetPower(coll, 3, F.6*F.9*F.11);;
gap> SetPower(coll, 4, F.7*F.9*F.11);;
gap> SetPower(coll, 5, F.7*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10*F.11);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
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);