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AGL(1,𝔽₅)×#123 #10617303

order	640
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

AGL(1,𝔽₅)×#123

Splittings

Fitting	C₅×#123 ⋊ C₄
Sylow	C₅ ⋊ C₄×#123

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#123	C₄
prime radical	C₂×D₄×D₅	C₂²
cosocle	C₂²×D₅	C₂⁴
hypercenter	#123	AGL(1,𝔽₅)
hypocenter	C₅	C₄×#123
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #123

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#352, AGL(1,𝔽₅)×#458, C₂×AGL(1,𝔽₅)×#123, …
maximal normal subgroup of	AGL(1,𝔽₅)×#352, AGL(1,𝔽₅)×#458, C₂×AGL(1,𝔽₅)×#123, …
maximal quotient of	AGL(1,𝔽₅)×#386, AGL(1,𝔽₅)×#388, AGL(1,𝔽₅)×#385, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 19385);