#72268722

order	6144
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³	#11711328
commutator	C₄×#95⋊C₃	C₄²
Frattini	C₂⁵×C₄	C₂×S₄
Fitting	C₂×#153709	D₃
socle	C₂⁵	(C₂×S₄)⋊C₄
prime radical	C₂³×#95⋊C₃	D₄
cosocle	C₂²×C₄×#95⋊C₃	C₂²
hypercenter	C₂×#95	#857
hypocenter	#95⋊C₃	#339
LCS	G ▷ C₄×#95⋊C₃ ▷ C₂×#95⋊C₃ ▷ #95⋊C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#95

9372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^6,3,2^5|a^2=c,b^2=d,d^2=fl,e^2=fl,h^2=j,i^2=k,b^a=be,c^b=cf,d^a=dl,e^a=el,e^b=ef,g^a=gkl,g^b=g^2,g^e=gjl,h^b=hikl,h^g=ijl,i^b=ik,i^g=hil,i^h=il,j^b=jkl,j^g=k,k^g=jkl

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72268759

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