Finite Group Databasebeta

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#72322729

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

Aliases

No aliases for this group.

Subgroups

1358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^11|a^2=f,d^2=g,f^2=h,h^2=i,i^2=j,j^2=k,b^a=bk,c^a=ce,c^b=cg,d^a=djk,e^b=ek,f^d=fk,g^a=gk

Conjugacy classes & character table

320 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, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);