#72215868

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	#62817 ⋊ C₃⋊D₃
derived	#72215828 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#72215828	C₂
Frattini, socle	C₂²	#11923851
Fitting	#62817	C₃⋊D₃
LCS	G ▷ #72215828
UCS	C₁

1230 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 18 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:

2,3^2,2^8|b^a=b^2,c^a=c^2,d^a=e,d^c=e,e^a=d,e^c=de,f^a=g,f^b=fi,f^c=fg,f^d=fk,g^a=f,g^b=ghi,g^c=f,g^e=gk,h^a=fgh,h^b=fg,h^c=hi,h^d=hj,h^e=hj,i^a=ghi,i^b=f,i^c=h,i^d=ij,j^a=jk,j^b=k,k^b=jk

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215896, C₂×#72215868, #72240863, …
maximal normal subgroup of	#72215896, C₂×#72215868, C₃×#72215868
maximal quotient of	C₂×#72215868, #72247788, C₃×#72215868

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