C₅²×D₄ #6051

order	200
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = D₄×GL(2,𝔽₅) (order 3840) · Inn(G) = C₂² · Out(G) = C₂×GL(2,𝔽₅)
homology	Abelianization (H₁) = C₂²×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₅²×D₄
unambiguous semidirect product	C₄⋊(C₂×C₅²)
unambiguous semidirect product	C₂²⋊(C₂×C₅²)

	subgroup	G/H
center, socle	C₂×C₅²	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₅²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ G

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 38);