#72323095

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72323093
commutator	C₂²×C₃₂	C₂³×C₄
Frattini, cosocle	C₂×#1397	C₂⁴
prime radical	#72320927	C₂
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #60204 ◁ #7622657 ◁ #72231859 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

3075 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:

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

Polycyclic presentation:

2^12|a^2=g,f^2=hjkl,g^2=i,h^2=jkl,j^2=kl,k^2=l,c^b=cil,d^a=df,d^b=dl,d^c=de,e^b=el,f^a=fh,f^d=fhkl,g^d=gkl,g^f=gl,h^a=hj,h^d=hjl,i^d=il,j^a=jk,j^d=jk,k^a=kl,k^d=kl

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323149
maximal normal subgroup of	#72323149

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