#72210727

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72210670
commutator	C₂×#1312	C₂×C₈
Frattini, cosocle	#72210241	C₂²
prime radical	C₂³	#92837
LCS	G ▷ C₂×#1312 ▷ C₂×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72210241 ◁ G
LECS	G ▷ #72210241 ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

617 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|a^2=hij,b^2=d,c^2=ehl,d^2=g,e^2=hk,f^2=ik,g^2=jl,h^2=k,i^2=l,b^a=bc,c^a=ce,c^b=cf,d^a=defhk,d^c=dijkl,e^a=eh,e^b=ei,e^d=el,f^a=fikl,f^b=fj,f^c=fkl,g^a=ghikl,g^c=gl,h^a=hk,h^b=hl,i^a=il,j^a=jkl

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210811
maximal quotient of	#72210738

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