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D₄×AGL(1,𝔽₅) #4132

order	160
automorphism	Aut(G) = C₂×D₄×AGL(1,𝔽₅) (order 320) · Inn(G) = C₂²×AGL(1,𝔽₅) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

D₄×AGL(1,𝔽₅)

Splittings

Fitting	C₅×D₄ ⋊ C₄
Sylow	C₅ ⋊ C₄×D₄

Subgroups

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

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 3 generators on 9 points:

(2,3,4,5);(1,2,3,5,4)(7,9);(6,7)(8,9)

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₈×AGL(1,𝔽₅), #65731, SD₁₆×AGL(1,𝔽₅), …
maximal normal subgroup of	D₈×AGL(1,𝔽₅), #65731, SD₁₆×AGL(1,𝔽₅), …
maximal quotient of	#65692, #65693, #65694, …
automorphism group of	Dic₁₀, D₂₀, C₂×Dic₅, …
holomorph of	C₄×C₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 207);