C₂²⋊D₅ #175

order	40
isoclinism	isoclinic class of Dic₁₀
automorphism	Aut(G) = C₂²×AGL(1,𝔽₅) (order 80) · Inn(G) = C₂×D₅ · Out(G) = C₄
holomorph	G ⋊ Aut(G) = #72213412
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

C₂²⋊D₅

Fitting	C₂²×C₅ ⋊ C₂
Sylow	C₅ ⋊ D₄

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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 8);