D₅²⋊C₂ #6056

order	200
isoclinism	isoclinic class of D₅²⋊C₂
automorphism	Aut(G) = AGL(1,𝔽₅)²⋊C₂ (order 800) · Inn(G) = G · Out(G) = C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₅²⋊C₂

C₅² ⋊ D₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₅⋊D₅	C₂²
Fitting, socle, hypocenter	C₅²	D₄
LCS	G ▷ C₅⋊D₅ ▷ C₅²
UCS	C₁

29 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 43);