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I present the complete calculation here:

In cgs units, Planck's constant, the speed of light and the Newtonian gravitational constant are, respectively:

H = 6.6260755E-27 g.cm^2/sec = M.L^2/T

C = 2.99792458E10 cm/sec = L/T

G = 6.6873E-8 cm^3/g.sec^2=L^3/M.T^2

Solving these equations or the Planck units of mass, length and time, repectively, we get:

M = SQRT(H.C/G) = 5.45020877E-5 g

L = SQRT(H.G/C^3) = 4.055503739E-30 cm

T = SQRT(H.C/C^5) = 1.352770435E-40 sec

Now, the charge on a electron is:

Q = 1.60217733E-20 SQRT(g.cm/mu) = 4.803206799E-10 SQRT(epsilon.g.cm^3)/sec,

where mu and epsilon are magnetic permeability and electrical permittivity, repectively. Multiplying these two values, we get:

Q^2 = 7.695589045E-30 (g.cm^2/sec).SQRT(epsilon/mu)

for the cgs value of the square of the charge on an electron.

Now, (g.cm^2/sec) is the dimension of Planck's constant and SQRT(epsilon/mu) is the dimension of conductivity or the reciprocal of resistance. So, the dimension of the square of charge is the same as the dimension of Planck's constant divided by resistance:

Q^2 = H/R

However

alpha.H/2.Pi = 7.695589115E-30

if

1/alpha = 137.0359895.

So, if the fine structure constant is dimensionless, and

Q^2 = 7.695589E-30 = alpha.M.L^2/2.Pi.T.R,

then R must equal one in Planck units. It makes more sense to me to say that alpha has the dimension of electrical conductivity.