#72258937

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72258908 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂³×#560⋊C₃
commutator, cosocle	C₂×C₄²	C₂⁵×C₃
Frattini, socle	C₂³	C₂⁵×A₄
Fitting	#72258908	C₃
hypercenter	C₂×D₄	#560⋊C₃
hypocenter	C₄²	C₂×C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₄² ▷ C₄²
UCS	C₁ ◁ C₂ ◁ C₂×D₄

21968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^5,3,2^5|e^2=g,h^2=j,i^2=k,d^b=dg,e^b=eg,e^c=eg,f^e=fj,h^a=hj,h^b=hk,h^c=hj,h^f=ij,i^a=ik,i^b=ijk,i^c=ik,i^f=hi,j^f=k,k^f=jk

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72258954

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