#72309372

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×#386
commutator	C₂³×C₄	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁶
prime radical	C₂⁴×#447	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

At least 33438 conjugacy classes of subgroups; only the 12371 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|a^2=jl,c^2=jl,d^2=j,f^2=g,g^2=j,i^2=l,b^a=bi,c^b=cl,d^a=dl,f^a=fh,f^c=fjl,f^d=fj,f^e=fjl,g^a=gl,h^b=hk,h^f=hl,i^a=il,i^b=il,i^f=ik

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

Relationships

automorphism group of

#12165, #12167

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