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H(3,𝔽₁₆) #72228239

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial
• stem
• rational
• ambivalent

Aliases

Subgroups

545252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 8 generators on 128 points:

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

Polycyclic presentation:

2^12|e^a=ei,e^b=eik,e^c=eij,e^d=el,f^a=fik,f^b=fij,f^c=fl,f^d=fk,g^a=gij,g^b=gl,g^c=gk,g^d=gjk,h^a=hl,h^b=hk,h^c=hjk,h^d=hijl

Conjugacy classes & character table

271 conjugacy classes — view the full character table.

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