#72323120

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₆ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#12067
commutator	C₁₆	C₂⁴×C₁₆
Frattini, cosocle	#1311	C₂⁵
prime radical	C₂×#33982	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #72231875 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 5 generators on 80 points:

(1,2,4,8,5,9,11,15,6,10,14,3,7,12,16,13)(17,18,21,27,22,29,34,30,23,31,44,46,45,32,43,60)(19,25,36,28,37,24,33,48,39,49,38,59,68,50,62,52)(20,26,41,54,42,56,63,57,40,53,69,72,70,58,71,78)(35,51,65,55,66,47,61,74,64,75,67,77,80,73,79,76);(18,24)(19,33)(21,43)(22,45)(25,29)(26,47)(27,59)(28,46)(30,48)(31,50)(32,49)(34,44)(35,61)(36,37)(38,68)(39,62)(41,71)(42,70)(51,56)(52,60)(53,73)(54,77)(55,72)(57,74)(58,75)(63,69)(64,79)(65,66)(67,80)(76,78);(1,3,11,2,7,15,4,12,6,8,16,10,5,13,14,9)(17,19,34,33,45,68,21,36,23,39,43,62,22,37,44,38)(18,25,30,48,32,50,27,28,31,49,60,52,29,24,46,59)(20,35,63,61,70,80,41,65,40,64,71,79,42,66,69,67)(26,51,57,74,58,73,54,55,53,75,78,76,56,47,72,77);(17,20)(18,26)(19,35)(21,41)(22,42)(23,40)(24,47)(25,51)(27,54)(28,55)(29,56)(30,57)(31,53)(32,58)(33,61)(34,63)(36,65)(37,66)(38,67)(39,64)(43,71)(44,69)(45,70)(46,72)(48,74)(49,75)(50,73)(52,76)(59,77)(60,78)(62,79)(68,80);(20,40)(26,53)(35,64)(41,69)(42,70)(47,73)(51,75)(54,72)(55,77)(56,58)(57,78)(61,79)(63,71)(65,67)(66,80)(74,76)

Polycyclic presentation:

2^12|a^2=f,c^2=fh,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bg,c^b=cikl,e^d=el,f^b=fkl,g^a=gi,g^b=gikl,g^c=gk,g^f=gl,h^b=hl,i^a=ik,i^b=ikl,i^c=il,k^a=kl,k^b=kl

Conjugacy classes & character table

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