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C₂×AGL(1,𝔽₅)×H(5,𝔽₂) #13046016

order	1280
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product	C₂×AGL(1,𝔽₅)×H(5,𝔽₂)
unambiguous semidirect product	D₅⋊(C₄×H(5,𝔽₂))

Splittings

Fitting	C₂×C₅×H(5,𝔽₂) ⋊ C₄
Sylow	C₅ ⋊ C₂×C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²	C₂⁴×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂⁵×C₄
Frattini	C₂	C₂⁵×AGL(1,𝔽₅)
Fitting	C₂×C₅×H(5,𝔽₂)	C₄
socle	C₂²×C₅	C₂⁴×C₄
prime radical	C₂×D₅×H(5,𝔽₂)	C₂
cosocle	C₂×D₅	C₂⁶
hypercenter	C₂×H(5,𝔽₂)	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

12378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213412, C₂×H(5,𝔽₂)×#1043, AGL(1,𝔽₅)×#3484, …
maximal normal subgroup of	AGL(1,𝔽₅)×#3484, AGL(1,𝔽₅)×#3498, AGL(1,𝔽₅)×#3585, …
maximal quotient of	C₂×#72213412, C₂×H(5,𝔽₂)×#1043, AGL(1,𝔽₅)×#3484, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1116284);