#72238851

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	#2679
commutator	C₂²×C₄²	C₂²×C₄²
Frattini, cosocle	C₂⁴×C₄²	C₂⁴
socle	C₂³	#153953
prime radical	C₂⁶	H(3,𝔽₄)
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄² ▷ C₂² ▷ C₁

53387 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^12|a^2=e,b^2=f,c^2=ghkl,d^2=ghk,e^2=k,f^2=kl,g^2=kl,h^2=kl,i^2=l,j^2=l,c^a=ci,c^b=cg,d^a=dhijl,d^b=dhikl,d^c=dghl,g^a=gl,g^b=gkl,g^c=gkl,g^d=gk,h^a=hl,h^b=hkl,h^c=hkl,h^d=hk,i^a=il,i^b=il,i^c=il,j^a=jkl,j^b=jk,j^c=jk,j^d=jl

Conjugacy classes & character table

424 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72238855
Fitting subgroup of	#72238855
maximal subgroup of	#72238855
maximal normal subgroup of	#72238855

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