Finite Group Databasebeta

← Prev (#92554) Next (#92556) →

C₅²×AGL(1,𝔽₅) #92555

order	500
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅² · Schur multiplier (H₂) = C₅

Aliases

C₅²×AGL(1,𝔽₅)

Splittings

Fitting, socle, Sylow	C₅³ ⋊ C₄
derived	C₅ ⋊ C₄×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₅²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₅²
Frattini	C₁	G
Fitting, socle	C₅³	C₄
prime radical	C₅²×D₅	C₂
cosocle	D₅	C₂×C₅²
LCS	G ▷ C₅
UCS	C₁ ◁ C₅²

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 15 points:

(6,7,8,9,10)(12,13,14,15);(1,2,3,4,5)(11,12,13,15,14)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₅×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₅⋊D₅, C₂×C₅²×AGL(1,𝔽₅), …
maximal normal subgroup of	C₅×D₅×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₅⋊D₅, C₂×C₅²×AGL(1,𝔽₅), …
maximal quotient of	C₅²×#170, C₂×C₅²×AGL(1,𝔽₅), C₃×C₅²×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 41);