#72267657

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₄²

solvable

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72267648 ⋊ C₃
derived	#17309 ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	#3549⋊C₃
commutator, hypocenter	#17309	C₃×C₄
Frattini, socle	C₂⁵	C₂×C₂²⋊A₄
Fitting	#72267648	C₃
prime radical	C₂×#11709340	C₂
cosocle	C₂×#17309	C₂×C₃
LCS	G ▷ #17309
UCS	C₁ ◁ C₂³

2988 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 2988 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2,3,2^9|a^2=c,d^2=h,e^2=hij,f^2=k,g^2=k,b^a=bij,d^a=di,d^b=ei,e^a=ej,e^b=dehi,e^d=ehi,f^a=fi,f^b=fg,f^d=fijk,f^e=fj,g^a=gij,g^b=f,g^d=gi,g^e=gijk,g^f=gk,h^b=hij,i^b=j,j^b=ij

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267660, #72267671
maximal normal subgroup of	#72267671

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