#72220910

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×SD₁₆ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	D₄×D₅⋊AGL(1,𝔽₅)
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×SD₁₆	C₂×C₄
socle	C₂×C₅²	C₂×C₄×D₄
prime radical	#13105831	C₂
cosocle	C₄×C₅⋊D₅	C₂⁴
hypercenter	SD₁₆	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=f,b^2=e,d^2=e,f^2=g,c^a=cfg,d^a=dg,f^c=fg,h^b=h^2,h^d=h^3,h^e=h^4,i^b=h^3i^3,i^d=i^3,i^e=i^4

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×D₅⋊AGL(1,𝔽₅), #72222999, #72223006
maximal normal subgroup of	AGL(1,ℤ/8)×D₅⋊AGL(1,𝔽₅), #72222999, #72223006

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, 1, F.6);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^4);;
gap> SetConjugate(coll, 9, 2, F.8^3*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);