#72218147

order	8192
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#177751
commutator	C₂×#1744	C₂³×C₄
Frattini, cosocle	C₂²×#1744	C₂⁴
socle	C₂³	#72217679
prime radical	#72213854	C₂
LCS	G ▷ C₂×#1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ C₂×#353159 ◁ G
LECS	G ▷ C₂²×#1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁

At least 14050 conjugacy classes of subgroups; only the 962 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^13|a^2=f,b^2=jm,c^2=e,d^2=hjkm,g^2=il,h^2=k,i^2=l,j^2=m,b^a=bfj,c^a=cim,c^b=ck,d^a=dg,d^b=dh,d^c=dk,e^a=el,g^a=gi,g^b=gijm,g^c=gl,g^d=gj,h^a=hij,h^b=hk,h^g=hm,i^a=il,i^b=ilm,i^d=im,j^a=jm,k^a=klm

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

automorphism group of

#7052

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