# How many isomorphic 3SAT formulas?

For a 3SAT formula with $$n$$ variables and $$m$$ clauses, I am interested in counting the number of isomorphic formulas (isomorphic in the sense that they are logically equivalent and have the same number of variables and clauses). I assume that we can invert the sense of any variable (swap all $$x_i$$ with $$\lnot x_i$$) and permute any of the variables. The first should give $$2^n$$ and the second should give $$n!$$.

Is this all of the isomorphisms, or am I overlooking anything?

If this is correct, then I assume that the number of distinct 3SAT formulas with $$n$$ variables and “distinct clauses” is equal to $$\frac{2^m}{2^nn!}$$, where $$m$$ is the number of 3-clauses over $$n$$ variables in which all variables are distinct. I assume that $$m=2n(2n-2)(2n-4)$$. Does that sound right?