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C₂×AGL(1,𝔽₅)×AGL(1,ℤ/8) #13044735

order	1280
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product	C₂×AGL(1,𝔽₅)×AGL(1,ℤ/8)
unambiguous semidirect product	D₅⋊(C₄×AGL(1,ℤ/8))

Splittings

Fitting	C₂×C₅×AGL(1,ℤ/8) ⋊ C₄
Sylow	C₅ ⋊ C₂×C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂⁴×C₄
Frattini	C₄	C₂⁴×AGL(1,𝔽₅)
Fitting	C₂×C₅×AGL(1,ℤ/8)	C₄
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×D₅×AGL(1,ℤ/8)	C₂
cosocle	C₄×D₅	C₂⁵
hypercenter	C₂×AGL(1,ℤ/8)	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

3242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×AGL(1,𝔽₅)×AGL(1,ℤ/8), C₂×C₃×AGL(1,𝔽₅)×AGL(1,ℤ/8), C₂×AGL(1,ℤ/8)×#1043, …
maximal normal subgroup of	C₂×C₃×AGL(1,𝔽₅)×AGL(1,ℤ/8), AGL(1,𝔽₅)×#3585, AGL(1,𝔽₅)×#3003, …
maximal quotient of	D₇×AGL(1,𝔽₅)×AGL(1,ℤ/8), C₂×C₃×AGL(1,𝔽₅)×AGL(1,ℤ/8), C₂×AGL(1,ℤ/8)×#1043, …
automorphism group of	#3951, #4050, D₅×D₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1115003);