#72262239

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

#10586797 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	#85647	C₂³
Frattini	C₂³	C₂²×#1020
Fitting	#10586797	D₃
socle	C₂²	C₂×#87524
hypocenter	C₂²⋊(C₄²⋊C₃)	C₂×D₄
LCS	G ▷ #85647 ▷ C₂²⋊(C₄²⋊C₃)
UCS	C₁

8638 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 8638 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 32

Polycyclic presentation:

2^4,3,2^6|h^2=j,i^2=k,c^a=cd,c^b=cj,e^b=ek,e^c=e^2,f^b=fjk,f^c=fg,f^e=g,g^b=gj,g^e=fg,h^a=hk,h^b=hk,h^c=ghj,h^d=hj,h^e=ij,i^a=ijk,i^b=ijk,i^c=fhi,i^d=ik,i^e=hi,j^e=k,k^c=jk,k^e=jk

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

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