#72223477

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #20421

Subgroups

	subgroup	G/H
center	C₄	C₂×D₅⋊(C₂²⋊AGL(1,𝔽₅))
commutator	C₄×C₅²	C₂²×C₄²
Frattini	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×M₁₆	C₂×C₄
socle	C₂×C₅²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#11717471	C₂²
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×M₁₆	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#20421
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

3468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

196 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, 2, F.5);;
gap> SetPower(coll, 3, F.5*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);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 2, F.9^3);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);