H(3,ℤ/16) #72210724

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₁₆² · Schur multiplier (H₂) = C₈×C₁₆

Aliases

unitriangular	UT(3,ℤ/16)
Heisenberg	H(3,ℤ/16)

Subgroups

	subgroup	G/H
center, commutator	C₁₆	C₁₆²
Frattini, cosocle	#72210602	C₂²
socle	C₂	#72210719
prime radical	C₂³	H(3,ℤ/8)
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ #72210602 ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|a^2=c,b^2=d,c^2=f,d^2=g,e^2=hkl,f^2=i,g^2=j,h^2=k,k^2=l,b^a=be,c^b=ch,d^a=dhkl,d^c=dkl,f^b=fk,f^d=fl,g^a=gkl,g^c=gl,i^b=il,j^a=jl

Conjugacy classes & character table

376 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234359
maximal normal subgroup of	#72234359
maximal quotient of	#72234314, #72234319

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