#72209727

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72209722 ⋊ C₃
derived	#31076 ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄	#8530⋊C₃
commutator, hypocenter	#31076	C₃×C₄
Frattini	C₂×C₄³	C₂×A₄
Fitting	#72209722	C₃
socle	C₂	#72209726
prime radical	#72209723	C₂
cosocle	#7622407	C₂×C₃
LCS	G ▷ #31076
UCS	C₁ ◁ C₄

589 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2^9|a^2=c,d^2=k,e^2=k,f^2=h,g^2=i,j^2=k,b^a=bik,d^a=df,d^b=e,d^c=dh,e^a=eg,e^b=dek,e^c=ei,e^d=ek,f^b=g,f^d=fh,f^e=fhjk,g^b=fghj,g^d=ghjk,g^e=gi,h^b=i,h^e=hk,i^b=hik,i^d=ik,j^b=hjk,j^d=jk

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209728
maximal quotient of	#72227442, #72227443, #72227995

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