#72223398

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×H(5,𝔽₂) ⋊ C₂×C₄
Sylow	C₅² ⋊ #60174

Subgroups

	subgroup	G/H
center	C₂	C₂²×#11717375
commutator	C₄×C₅²	C₂⁴×C₄
Frattini	C₄	C₂³×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×H(5,𝔽₂)	C₂×C₄
socle	C₂×C₅²	C₂²×C₄×D₄
prime radical	#72221222	C₂
cosocle	C₄×C₅⋊D₅	C₂⁵
hypercenter	H(5,𝔽₂)	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#60174
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

7128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,5^2|d^2=f,g^2=h,b^a=bg,c^a=cgh,d^a=dh,e^a=eh,e^b=eh,g^a=gh,g^b=gh,g^c=gh,i^a=i^4,i^d=i^3,i^f=i^4,j^a=i^3j,j^d=j^3,j^f=j^4

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

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