#72243234

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#8534 ⋊ C₄
Sylow	C₅ ⋊ #72242885

Subgroups

	subgroup	G/H
center	C₂²	#13028920
commutator	C₂⁵×C₅	C₂³×C₄
Frattini	C₂⁵	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#8534	C₄
socle	C₂²×C₅	#14693
prime radical	D₅×#3484	C₂²
cosocle	C₂⁵×D₅	C₂⁴
hypercenter	#8534	AGL(1,𝔽₅)
hypocenter	C₅	#72242885
LCS	G ▷ C₂⁵×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #8534

16422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^10,5|b^2=g,c^2=i,d^2=e,b^a=bh,c^b=cf,d^a=dh,d^b=dg,d^c=dfi,f^a=fj,f^b=fi,g^c=gi,h^c=hj,k^d=k^3,k^e=k^4

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#347042
maximal normal subgroup of	AGL(1,𝔽₅)×#347042

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