Permutations acting on coordinates of codewords Let $\mathcal{C}$ be a binary code of length $n$.  The automorphism group of $\mathcal{C}$ is defined to be the set of permutations in $S_n$ that take $\mathcal{C}$ to itself.  The text by MacWilliams and Sloane (p. 229-230) elaborates further that if $c = (c_1,\ldots,c_n) \in \mathcal{C}$ and $\pi \in S_n$, then we can define $c\pi := (c_{\pi(1)}, \ldots, c_{\pi(n)})$.   As an example, they say if $\pi = (123), \rho=(14)$, then $(c_1,c_2,c_3,c_4) \pi = (c_2,c_3,c_1,c_4)$, and $(c_1,c_2,c_3,c_4) \pi \rho = (c_2,c_3,c_4,c_1)$. This latter equation is not clear to me.  For if we define $d = c \pi$, we get that $(c \pi) \rho = d \rho = d (14) = (d_4, d_2, d_3, d_1) = (c_4, c_3, c_1, c_2)$, which is not equal to their result. The action of $(1,4)$ is supposed to be to interchange the first and last components of a 4-tuple. And so the effect of $(123)$ followed by $(14)$ is not the same as the effect of $(123)(14)=(1234)$. Am I correct?
Perhaps we need to define the action of $S_n$ on $\{0,1\}^n$ by $c \pi = (c_{\pi^{-1}(1)}, \ldots, c_{\pi^{-1}(n)})$ to get an action, so that $(c \pi) \rho = c (\pi \rho)$?
 A: They are defining an action of $S_n$ from the right on the set of binary $n$-tuples. The key condition is that the equality
$$
(c\cdot\alpha)\cdot\beta=c\cdot(\alpha\beta)
$$
should hold for all $c$ and all $\alpha,\beta\in S_n$.
Your mistake is in calculating the product
$$
(123)(14)=(1423)
$$
that still should be read from right to left, because the group operation is composition of functions and this is how we define composition. In other words, when you calculate the product $(123)(14)$ of permutations, you first apply $\beta=(14)$ to the numbers $1,2,3,4$ and then apply $\alpha=(123)$. But when you apply these
to binary $n$-tuples (instead of their indices) you must reverse the order.
But...
You are correct in that with their definition
$$
(c_1,c_2,c_3,c_4)(123)(14)=(c_2,c_3,c_1,c_4)(14)=(c_4,c_3,c_1,c_2).
$$
Do notice that this equals $(c_1,c_2,c_3,c_4)(1423)$ as it should.
The alternative definition in your last paragraph would give a left action of $S_n$, when the above condition is replaced with
$$
(\alpha\beta)\cdot c=\alpha\cdot(\beta\cdot c).
$$
