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C₅×AGL(1,𝔽₅) #1041

order	100
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₄×AGL(1,𝔽₅) (order 80) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)³
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

C₅×AGL(1,𝔽₅)

Splittings

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

Subgroups

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

13 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×AGL(1,𝔽₅), C₂×C₅×AGL(1,𝔽₅), C₃×C₅×AGL(1,𝔽₅), …
maximal normal subgroup of	D₅×AGL(1,𝔽₅), C₂×C₅×AGL(1,𝔽₅), C₃×C₅×AGL(1,𝔽₅), …
maximal quotient of	C₅×#170, C₂×C₅×AGL(1,𝔽₅), C₃×C₅×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(100, 9);