Smith Normal Form of a Cayley Graph of the Symmetric Group Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of this Cayley graph is a conjugacy class, it is not too difficult to use the representation theory of $S_n$ to elegantly count the number of nonzero eigenvalues of $A$:
$$ \text{rank}(A_n) = \binom{n-1}{0}^2 + \binom{n-1}{1}^2 + \cdots + \binom{n-1}{n-1}^2 = \binom{2(n-1)}{n-1}.$$
I am interested in the rank of $A_n$ modulo $p$ where $p$ is an odd prime for all $n$. One way to determine this would be to compute the Smith Normal Form of $A_n$ (over $\mathbb{Z}$). Let $D_n = \text{diag}(s_1,s_2,\cdots,s_r,0,\cdots,0)$ such that $s_i | s_{i+1}$ for all $1 \leq i < r := \text{rank}(A)$ be the Smith Normal Form of $A_n$. Computations for small $n$ show that the nonzero $s_i$'s are all powers of 2, which might suggest that $\text{rank}_p(A_n) = \text{rank}(A_n)$ for all $n$ and odd primes $p$. 
It seems unlikely that one can divine unimodular matrices $U_n,V_n$ such that $D_n = U_nA_nV_n$ for all $n$, so I would like to think of $A_n$ as an endomorphism of the group algebra $\mathbb{F}_p[S_n]$ and perhaps use $p$-modular representation theory of $S_n$ to say something about the image of $A_n$. (Here, we are assuming $p$ is small, i.e., $p \mid n!$, so $\mathbb{F}_p[S_n]$ is not semisimple.)
Generally speaking, working with modular representations of $S_n$ is also difficult; however, the image of $A_n$ (in the characteristic 0 case) is the direct sum of the hook-shaped Specht modules, which are pretty well-understood, even in the modular case. In particular, Peel (1971) showed for odd primes $p$ that the hook-shaped Specht modules $S^{(n-k,1^k)}_{\mathbb{F}_p}$ are simple when $p \not \mid n$ and determined their composition series when $p \mid n$. 
Experimentally, if one picks $b \in S^{(n-k,1^k)}_{\mathbb{F}_p}$ to be a $(n-k,1^k)$-standard polytabloid (which is a $\{0,\pm1 \}$-valued vector well-defined for any Specht module over any field), then $A_nx = b$ indeed has a solution over $\mathbb{F}_p$ for small $k$, odd primes $p$, and $n$. In the case that $p \not \mid n$, because $A_nx = b$ has a solution, it follows that $A_nx = b'$ for any $b' \in S^{(n-k,1^k)}_{\mathbb{F}_p}$, as $S^{(n-k,1^k)}_{\mathbb{F}_p}$ is irreducible by Peel's result. Here, we are "using the modular representation theory of $S_n$", but the problem is that showing a solution $x$ for $A_nx = b$ exists over $\mathbb{F}_p$ for all $0 \leq k < n$ and odd primes $p$ seems to involve similar row-operation-type calculations as putting $A_n$ into Smith Normal Form.
My (open-ended) question is whether there is a more clever way to leverage such information about the modular representation theory of $S_n$ that would circumvent row and column operations to say something about the Smith Normal Form of $A_n$ or $\text{rank}_p(A)$ for odd primes $p$.
EDIT Here's the SNF for small $n$ (thanks Dima for verifying these):
$n = 2$ the SNF is $1^2$
$n = 3$ the SNF is $1^42^2$
$n = 4$ the SNF is $1^{8}2^{12}$
$n = 5$ the SNF is $1^{16}2^{52}8^2$
$n = 6$ the SNF is $1^{32}2^{200}8^{20}$
$n = 7$ the SNF is $1^{64}2^{728}4^{2}8^{128}16^2$
(I have not gone beyond $n=7$, as this would take some time.)
 A: I would think of $A_n$ as an element of the commutative algebra spanned by the  conjugacy class sums of $S_n$. It is known that the multiplication coefficients of such an algebra are determined by the character table of the underlying group, see e.g. the documentation on the GAP system function ClassMultiplicationCoefficient. 
An advantage of working in this algebra is that its dimension equals the number of conjugacy classes rather than the order of the group (i.e. the multiplicies of eigenvalues of $A_n$ will be taken care of once you pass to the image of $A_n$ in this algebra via the natural algebra isomorphism).
You'd need to keep track of the eigenvalue multiplicites, but this is all well-understood, e.g. in terms of the underlying association scheme (an object in algberaic combinatorics that generalises this sort of setting).

This does not seem to give one a direct (or any?) way to compute the SNF of $A_n$, but as you are interested in the $p$-rank of $A_n$ rather than its SNF, this looks reasonable. 
