#72338996

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72338790
commutator, Frattini, cosocle	#6628901	C₂⁴
prime radical	#72338906	C₂
LCS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628901 ◁ G
LECS	G ▷ #6628901 ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

At least 9120 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^13|a^2=j,d^2=ej,e^2=m,f^2=j,g^2=l,i^2=m,b^a=bj,c^a=cg,c^b=ch,d^a=dgjl,d^b=df,d^c=de,e^a=ei,e^b=eh,e^c=em,e^d=el,f^b=fj,f^c=fh,f^d=fhjl,f^e=fklm,g^b=gj,g^c=gl,g^d=gim,g^e=gklm,g^f=gjl,h^a=hj,h^d=hk,h^f=hl,h^g=hl,i^b=ij,i^c=iklm,i^d=ikl,i^e=im,i^h=im,j^c=jl,j^d=jl,j^e=jm,k^b=klm,k^f=km,l^d=lm

Conjugacy classes & character table

89 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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 4, F.5*F.10);;
gap> SetPower(coll, 5, F.13);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.12);;
gap> SetPower(coll, 9, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.13);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.11*F.12*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.13);;
gap> SetConjugate(coll, 7, 5, F.7*F.11*F.12*F.13);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.11*F.12*F.13);;
gap> SetConjugate(coll, 9, 4, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.13);;
gap> SetConjugate(coll, 9, 8, F.9*F.13);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 10, 4, F.10*F.12);;
gap> SetConjugate(coll, 10, 5, F.10*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 6, F.11*F.13);;
gap> SetConjugate(coll, 12, 4, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);