C₅×D₄ #177

order	40
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₄×D₄ (order 32) · Inn(G) = C₂² · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×#453
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₄
unambiguous semidirect product	C₄⋊(C₂×C₅)
unambiguous semidirect product	C₂²⋊(C₂×C₅)

	subgroup	G/H
center, socle	C₂×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ G

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 10);