#72220949

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#41 ⋊ C₂×C₄
Sylow	C₅² ⋊ C₂×#417

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅⋊(C₂²⋊AGL(1,𝔽₅))
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×#41	C₂×C₄
socle	C₂×C₅²	C₂²×C₂²⋊C₄
prime radical	D₅²×#41	C₂
cosocle	C₄×C₅⋊D₅	C₂⁴
hypercenter	#41	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₂×#417
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

1794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|b^2=e,d^2=ef,e^2=g,f^2=g,b^a=bg,c^b=cg,d^b=df,e^d=eg,f^d=fg,h^a=h^4,h^b=h^3,h^d=h^3,h^e=h^4,i^a=h^3i,i^b=i^3,i^d=i^3,i^e=i^4

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×D₅⋊AGL(1,𝔽₅), #72223005, #72223018
maximal normal subgroup of	AGL(1,ℤ/8)×D₅⋊AGL(1,𝔽₅), #72223005, #72223018

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