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#72323172

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

Aliases

No aliases for this group.

Subgroups

2312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^12|a^2=e,d^2=h,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,k^2=l,b^a=bi,c^a=cf,c^b=cl,d^a=dh,d^c=dhjkl,e^c=ejkl,e^d=ekl,f^a=fh,f^b=fl,f^c=fhjkl,f^e=fk,g^c=gkl,g^d=gl,g^f=gl,h^a=hj,h^c=hjkl,h^e=hl,i^c=il,j^a=jk,j^c=jkl,k^a=kl,k^c=kl

Conjugacy classes & character table

202 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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);