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#72321955

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

Aliases

No aliases for this group.

Subgroups

707 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^11|a^2=f,b^2=fjk,d^2=gik,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bhjk,c^a=ce,c^b=ci,d^a=dgj,d^b=djk,d^c=dg,e^a=eg,e^b=ej,e^c=egjk,f^c=fjk,f^d=fk,f^e=fk,g^a=gi,g^b=gk,g^c=gik,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

128 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, 2, F.6*F.10*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.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10*F.11);;
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, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.11);;
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);