#72341123

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#63011 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72218857
commutator, cosocle	C₂×#5464	C₂³
Frattini, socle	C₂³	C₂²×#1020
Fitting	C₂×#63011	D₃
hypocenter	#5464	C₂×D₄
LCS	G ▷ C₂×#5464 ▷ #5464
UCS	C₁ ◁ C₂

11134 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 11134 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 48

Polycyclic presentation:

2^4,3,2^6|c^2=d,h^2=k,i^2=jk,c^a=cd,e^c=e^2,f^b=fj,f^c=fgjk,f^e=g,g^b=gk,g^c=gk,g^e=fg,h^a=hjk,h^b=hk,h^c=fi,h^e=ik,h^f=hk,h^g=hj,i^a=ij,i^b=ijk,i^c=fghjk,i^e=hi,i^f=ij,i^g=ijk,j^c=jk,j^e=k,k^e=jk

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341172, C₂×#72341123
maximal normal subgroup of	#72341172, C₂×#72341123
maximal quotient of	C₂×#72341123

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