#72258954

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72258915 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂³×#560⋊C₃
commutator, cosocle	C₂²×C₄²	C₂⁵×C₃
Frattini, socle	C₂⁴	C₂⁵×A₄
Fitting	#72258915	C₃
hypercenter	#127	#560⋊C₃
hypocenter	C₄²	C₂×C₃×#560
LCS	G ▷ C₂²×C₄² ▷ C₄²
UCS	C₁ ◁ C₂² ◁ #127

37222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^5,3,2^6|d^2=g,e^2=h,i^2=k,j^2=l,d^b=dh,d^c=dg,e^b=egh,e^c=eh,f^e=fkl,i^a=ik,i^b=ikl,i^c=ik,i^f=ij,j^a=jl,j^b=jk,j^c=jl,j^f=il,k^f=kl,l^f=k

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258960
maximal normal subgroup of	#72258960

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