#72267202

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72267107 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₄	C₂²⋊#5464
commutator, cosocle, hypocenter	#8093119	C₂×C₃
Frattini, socle	C₂³	C₂×C₂²⋊(C₂²⋊A₄)
Fitting	#72267107	C₃
LCS	G ▷ #8093119
UCS	C₁ ◁ C₄

19235 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 19235 conjugacy classes of subgroups — too many to draw. Only its 103 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:

2,3,2^9|a^2=k,c^2=k,d^2=k,e^2=ijk,f^2=jk,b^a=bi,c^b=d,d^b=cdk,d^c=dk,e^b=fij,e^c=ei,e^d=ej,f^b=efk,f^c=fj,f^d=fij,f^e=fk,g^b=gh,g^c=gk,g^d=gk,g^e=gk,h^b=g,h^c=hk,h^f=hk,i^b=ij,j^b=i

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72267211

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