#72268614

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72268609 ⋊ C₃
derived	#158817 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	#3549⋊C₃
commutator, cosocle, hypocenter	#158817	C₂×C₃
Frattini, socle	C₂⁵	C₂×C₂²⋊A₄
Fitting	#72268609	C₃
LCS	G ▷ #158817
UCS	C₁ ◁ C₂³

3305 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 3305 conjugacy classes of subgroups — too many to draw. Only its 106 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|c^2=g,d^2=g,e^2=hi,f^2=hj,c^a=chjk,c^b=d,d^a=dhik,d^b=cdg,d^c=dg,e^b=ef,e^c=egh,e^d=ehi,f^b=ehjk,f^c=fhj,f^d=fghk,f^e=fk,h^b=ik,i^b=j,j^b=hk

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268615
maximal normal subgroup of	#72268615
maximal quotient of	#72268645

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