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AGL(1,𝔽₅)×C₂₅ #92529

order	500
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₄×C₅×AGL(1,𝔽₅) (order 400) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₄×C₅
homology	Abelianization (H₁) = C₄×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

AGL(1,𝔽₅)×C₂₅

Splittings

Fitting, Sylow	C₅×C₂₅ ⋊ C₄
derived	C₅ ⋊ C₄×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₂₅
Frattini	C₅	C₅×AGL(1,𝔽₅)
Fitting	C₅×C₂₅	C₄
socle	C₅²	C₄×C₅
prime radical, cosocle	C₅×D₅	C₂×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₂₅

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 30 points:

(2,3,4,5)(6,7,9,12,16,8,10,13,17,21,11,14,18,22,25,15,19,23,26,28,20,24,27,29,30);(1,2,3,5,4)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×D₂₅, C₂×AGL(1,𝔽₅)×C₂₅, C₃×AGL(1,𝔽₅)×C₂₅, …
maximal normal subgroup of	AGL(1,𝔽₅)×D₂₅, C₂×AGL(1,𝔽₅)×C₂₅, C₃×AGL(1,𝔽₅)×C₂₅, …
maximal quotient of	C₂₅×#170, C₂×AGL(1,𝔽₅)×C₂₅, C₃×AGL(1,𝔽₅)×C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 15);