#72215896

order	9216
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	#62817 ⋊ D₃²
derived	#72215828 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#72215828	C₂²
Frattini, socle	C₂²	#72215889
Fitting	#62817	D₃²
LCS	G ▷ #72215828
UCS	C₁

2962 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 2962 conjugacy classes of subgroups — too many to draw. Only its 16 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:

2^2,3^2,2^8|c^b=c^2,d^a=d^2,e^a=f,e^d=f,f^a=e,f^d=ef,g^c=gij,g^d=h,g^e=gl,g^f=gl,h^a=gh,h^c=hi,h^d=gh,h^f=hl,i^a=j,i^b=hi,i^c=h,i^d=j,i^f=ik,j^a=i,j^b=ghj,j^c=gh,j^d=ij,j^e=jk,k^b=kl,k^c=l,l^c=kl

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72215896, C₃×#72215896
maximal normal subgroup of	C₂×#72215896, C₃×#72215896
maximal quotient of	C₂×#72215896, #72248221, #72253623, …
automorphism group of	C₂²×Q₈

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.5);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 10, 3, F.7*F.8);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.12);;
gap> SetConjugate(coll, 12, 3, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);