D₅⋊AGL(1,𝔽₅) #6055

order	200
isoclinism	isoclinic class of D₅⋊AGL(1,𝔽₅)
automorphism	Aut(G) = AGL(1,𝔽₅)²⋊C₂ (order 800) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₅⋊AGL(1,𝔽₅)
unambiguous semidirect product	#1043⋊C₂

C₅² ⋊ C₂×C₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²	C₂×C₄
prime radical	D₅²	C₂
cosocle	C₅⋊D₅	C₂²
LCS	G ▷ C₅²
UCS	C₁

38 conjugacy classes of subgroups.

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

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

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

Polycyclic presentation:

2^3,5^2|a^2=c,d^a=d^3,e^a=e^3,d^b=d^4,d^c=d^4,e^c=e^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 42);