#72209891

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	#2032088 ⋊ D₃
derived	#15467⋊C₃ ⋊ C₂²

Subgroups

	subgroup	G/H
center	C₂	#72209661
commutator, cosocle, hypocenter	#15467⋊C₃	C₂²
Frattini	C₂⁴	C₂×#1020
Fitting	#2032088	D₃
socle	C₂³	C₂×#5934
hypercenter	C₂²	#11715753
LCS	G ▷ #15467⋊C₃
UCS	C₁ ◁ C₂ ◁ C₂²

4387 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 4387 conjugacy classes of subgroups — too many to draw. Only its 24 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:

2^2,3,2^8|b^2=k,f^2=hk,g^2=i,c^b=c^2,d^c=e,e^b=de,e^c=de,f^a=fhik,f^b=deg,f^c=gij,f^d=fh,f^e=fhi,g^a=gij,g^b=ef,g^c=fghijk,g^d=gj,g^e=gik,g^f=ghijk,h^b=j,h^c=ik,i^b=ik,i^c=j,j^b=h,j^c=hk

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209894
maximal quotient of	#72227917, #72228054, #72228096

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