#72239009

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72238970 ⋊ C₃
derived	#8093167 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72209985
commutator, cosocle, hypocenter	#8093167	C₂×C₃
Frattini, socle	C₂³	C₂×C₂²⋊(C₂²⋊A₄)
Fitting	#72238970	C₃
LCS	G ▷ #8093167
UCS	C₁ ◁ C₂

17619 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 17619 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 24

Polycyclic presentation:

2,3,2^9|c^2=j,d^2=jk,e^2=jk,f^2=k,g^2=jk,h^2=k,b^a=bj,c^a=cj,c^b=dj,d^a=djk,d^b=cd,e^b=fjk,e^c=ei,e^d=eijk,f^b=ef,f^c=fk,f^d=fik,g^b=hjk,g^c=gk,g^d=gjk,h^b=gh,h^c=hjk,h^d=hj,i^b=ik,j^b=jk,k^b=j

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72239011

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