#72323163

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#93757
commutator	C₃₂	C₂³×C₁₆
Frattini, cosocle	#7330	C₂⁴
prime radical	#7625194	C₈
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₄×#131 ◁ #479899 ◁ #72231854 ◁ G
LECS	G ▷ #7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^12|a^2=e,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,k^2=l,c^a=cf,c^b=cl,d^b=dl,e^c=ejkl,f^a=fh,f^c=fhjkl,f^e=fk,g^c=gkl,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

274 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323191
maximal normal subgroup of	#72323191

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, 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, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
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, 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);