#72227334

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72222757 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#560⋊C₃
commutator, cosocle	C₂×#95	C₂⁴×C₃
Frattini, socle	C₂⁴	C₂⁴×A₄
Fitting	#72222757	C₃
hypercenter	C₂²⋊C₄	#560⋊C₃
hypocenter	#95	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×#95 ▷ #95
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

6082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^4,3,2^6|d^2=f,g^2=i,h^2=j,c^a=cf,c^b=ck,d^a=dk,d^b=dfk,d^c=dk,e^d=eij,g^a=gij,g^b=gik,g^e=gh,h^a=hik,h^b=hjk,h^e=gjk,h^g=hk,i^e=ijk,j^e=i

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227335

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