#72242255

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#7333 ⋊ C₄
Sylow	C₅ ⋊ #72223644

Subgroups

	subgroup	G/H
center	C₂	#72242073
commutator	C₅×C₁₆	C₂×C₄×C₈
Frattini	#346	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#7333	C₄
socle	C₂×C₅	#129165
prime radical	C₂²×D₅	AGL(1,ℤ/16)
cosocle	D₅×#346	C₂³
hypercenter	#7333	AGL(1,𝔽₅)
hypocenter	C₅	#72223644
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1313 ◁ #7333

614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^10,5|a^2=f,b^2=d,c^2=de,d^2=g,f^2=h,h^2=i,i^2=j,b^a=bfhij,c^a=chij,d^a=dij,e^a=ej,f^b=fh,f^c=fi,f^d=fj,g^a=gj,h^b=hi,h^c=hj,i^b=ij,k^b=k^3,k^d=k^4,k^e=k^4

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×AGL(1,ℤ/32)
maximal normal subgroup of	AGL(1,𝔽₅)×AGL(1,ℤ/32)

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