#72222932

order	20736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ #492
Sylow	C₃⁴ ⋊ #25774

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#72213315
commutator, cosocle	C₂×#10620198	C₂⁴
Fitting	C₂²×C₃⁴	#492
socle	C₂×C₃⁴	C₂×#492
prime radical	#72220744	C₂
hypercenter	C₂²	#72213315
hypocenter	C₃⁴	#25774
LCS	G ▷ C₂×#10620198 ▷ C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

6140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^4|a^2=g,d^2=e,e^2=h,b^a=bf,c^a=cg,d^c=de,e^c=eh,i^a=j^2,i^c=i^2,i^d=ij,i^e=j^2,i^g=i^2,i^h=i^2,j^a=i,j^d=i^2j,j^e=i,j^g=j^2,j^h=j^2,k^a=i^2j^2k,k^c=k^2,k^d=l,k^e=kl^2,k^g=ik,k^h=k^2,l^a=jl,l^c=kl,l^d=kl^2,l^e=k^2l^2,l^g=ijl,l^h=l^2

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

holomorph of

C₂×#157

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