#72246846

order	5200
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅²×C₁₃ ⋊ C₄²
Sylow	C₅² ⋊ C₄×#261
Sylow	C₁₃ ⋊ #87830

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₃	C₄²
prime radical, cosocle	D₁₃×C₅⋊D₅	C₂²
LCS	G ▷ C₅²×C₁₃
UCS	C₁

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 130

Polycyclic presentation:

2^4,5^2,13|a^2=c,b^2=d,e^a=e^4,e^b=e^3,e^d=e^4,f^a=e^2f,f^b=f^3,f^d=f^4,g^a=g^8,g^c=g^12

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#261×D₅⋊AGL(1,𝔽₅), #72246910
maximal normal subgroup of	#261×D₅⋊AGL(1,𝔽₅), #72246910

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 5, 13]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 5, 2, F.5^3);;
gap> SetConjugate(coll, 5, 4, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);