#72209784

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72209644
commutator, cosocle, hypocenter	C₂²⋊(#95⋊C₃)	C₂³
Frattini, socle	C₂⁵	C₂²×S₄
Fitting	#156699	D₃
prime radical	#72209777	C₂
LCS	G ▷ C₂²⋊(#95⋊C₃)
UCS	C₁ ◁ C₂

5131 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 5131 conjugacy classes of subgroups — too many to draw. Only its 54 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:

2^3,3,2^7|c^2=ijk,e^2=i,f^2=j,d^a=djk,d^c=d^2jk,e^a=eik,e^b=eg,e^c=fi,e^d=fik,f^a=fjk,f^b=fh,f^c=ej,f^d=efk,f^e=fk,g^c=h,g^d=h,h^c=g,h^d=gh,i^c=j,i^d=j,j^c=i,j^d=ijk

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209785
maximal quotient of	#72227741

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