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C₄×C₅×AGL(1,𝔽₅) #87843

order	400
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₄×D₄×AGL(1,𝔽₅) (order 640) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₄×D₄
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₄

Aliases

C₄×C₅×AGL(1,𝔽₅)

Splittings

Fitting	C₄×C₅² ⋊ C₄
derived	C₅ ⋊ C₄²×C₅
Sylow	C₅² ⋊ C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄²×C₅
Frattini	C₂	C₂×C₅×AGL(1,𝔽₅)
Fitting	C₄×C₅²	C₄
socle	C₂×C₅²	C₂×C₄
prime radical	C₂×C₅×D₅	C₂²
cosocle	C₂×D₅	C₂²×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₄×C₅

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 14 points:

(1,2,3,4,5)(7,8,9,10);(6,7,8,10,9)(11,12,13,14)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#168, #11716732, #11716739, …
maximal normal subgroup of	AGL(1,𝔽₅)×#168, #11716732, #11716739, …
maximal quotient of	C₅×C₈×AGL(1,𝔽₅), C₅×#3992, C₄×C₅×#170, …
automorphism group of	D₅×C₂₅, AGL(1,𝔽₅)×C₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 137);