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C₂×C₅×AGL(1,𝔽₅) #6058

order	200
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×C₄×AGL(1,𝔽₅) (order 160) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×AGL(1,𝔽₅)
unambiguous semidirect product	D₅⋊(C₄×C₅)

Splittings

Fitting, socle	C₂×C₅² ⋊ C₄
derived	C₅ ⋊ C₂×C₄×C₅
Sylow	C₅² ⋊ C₂×C₄

Subgroups

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

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₅×AGL(1,𝔽₅), (C₂×AGL(1,𝔽₅))⋊D₅, #87825, …
maximal normal subgroup of	Dic₅×AGL(1,𝔽₅), (C₂×AGL(1,𝔽₅))⋊D₅, #87825, …
maximal quotient of	C₅×D₅⋊C₈, C₅×#700, C₄×C₅×AGL(1,𝔽₅), …
automorphism group of	D₅×C₁₁, C₁₁×AGL(1,𝔽₅)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 45);