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#72321969

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

Aliases

No aliases for this group.

Subgroups

657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^11|d^2=f,e^2=gi,f^2=h,g^2=ij,h^2=jk,i^2=jk,j^2=k,b^a=bhijk,c^a=ce,c^b=ck,d^a=dgj,d^c=dgij,e^a=eg,e^b=ek,e^c=eg,e^d=ejk,f^a=fij,f^c=fi,f^e=fk,g^a=gi,g^c=gi,g^d=gk,h^a=hj,h^c=hjk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

104 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, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, 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.9*F.10*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.9*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, 5, 4, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
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, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*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);