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#72234252

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

Aliases

No aliases for this group.

Subgroups

487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 136 points:

(9,10,13,19,14,20,33,49,15,21,34,50,36,52,73,95,16,22,35,51,37,53,74,96,38,54,75,97,76,98,113,128)(11,17,24,40,26,42,61,83,27,43,62,84,65,87,108,123,25,41,60,82,63,85,106,121,64,86,107,122,109,124,132,136)(12,18,29,45,31,47,69,91,32,48,70,92,72,94,112,127,30,46,68,90,71,93,111,126,55,77,99,114,100,115,129,133)(23,39,56,78,58,80,102,117,59,81,103,118,105,120,131,135,57,79,101,116,104,119,130,134,28,44,66,88,67,89,110,125);(1,2,3,5,4,6,7,8)(9,11)(10,17)(12,28,30,59)(13,24)(14,26)(15,27)(16,25)(18,44,46,81)(19,40)(20,42)(21,43)(22,41)(23,55,57,32)(29,66,68,103)(31,67,71,105)(33,61)(34,62)(35,60)(36,65)(37,63)(38,64)(39,77,79,48)(45,88,90,118)(47,89,93,120)(49,83)(50,84)(51,82)(52,87)(53,85)(54,86)(56,99,101,70)(58,100,104,72)(69,110,111,131)(73,108)(74,106)(75,107)(76,109)(78,114,116,92)(80,115,119,94)(91,125,126,135)(95,123)(96,121)(97,122)(98,124)(102,129,130,112)(113,132)(117,133,134,127)(128,136);(9,12)(10,18)(11,23,25,57)(13,29)(14,31)(15,32)(16,30)(17,39,41,79)(19,45)(20,47)(21,48)(22,46)(24,56,60,101)(26,58,63,104)(27,59,64,28)(33,69)(34,70)(35,68)(36,72)(37,71)(38,55)(40,78,82,116)(42,80,85,119)(43,81,86,44)(49,91)(50,92)(51,90)(52,94)(53,93)(54,77)(61,102,106,130)(62,103,107,66)(65,105,109,67)(73,112)(74,111)(75,99)(76,100)(83,117,121,134)(84,118,122,88)(87,120,124,89)(95,127)(96,126)(97,114)(98,115)(108,131,132,110)(113,129)(123,135,136,125)(128,133)

Polycyclic presentation:

2^10|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,i^2=j,c^b=cij,e^c=ej,f^b=fj

Conjugacy classes & character table

352 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);