[THOUGHT] To find the number of generator matrices that produce codes monomially equivalent to $C$, we need to understand the concept of monomial equivalence and how it relates to the automorphism group of the code.

Monomial equivalence between two codes $C$ and $C'$ means that there exists a monomial matrix $M$ such that $C' = C \cdot M$. A monomial matrix is a product of a permutation matrix and a diagonal matrix with non-zero entries. In the case of a ternary code, these diagonal entries can be from $\mathbb{F}_3^*$, the non-zero elements of $\mathbb{F}_3$, which are $\{1, 2\}$.

Given that the order of the group of monomial automorphisms of $C$ is 24, this means there are 24 different monomial matrices that, when multiplied by the generator matrix $G$, produce another generator matrix for the same code $C$.

However, we want to count the number of different generator matrices that produce codes monomially equivalent to $C$, and not just the automorphisms of $C$. The total number of possible monomial matrices (permutation matrices times diagonal matrices with entries from $\mathbb{F}_3^*$) in $\mathbb{F}_3^7$ is calculated as follows:

1. The number of permutation matrices of size $7 \times 7$ is $7!$.
2. For each permutation, there are $2^7$ diagonal matrices with entries from $\{1, 2\}$, because each of the 7 diagonal entries can independently be either 1 or 2.

So, the total number of monomial matrices is $7! \times 2^7$.

Now, each of the 24 monomial automorphisms of $C$ corresponds to a generator matrix of $C$ itself, so the number of unique generator matrices for codes monomially equivalent to $C$ is given by dividing the total number of monomial matrices by the order of the automorphism group:

Let's calculate this expression. [/THOUGHT] [PYTHON] import math

Calculate 7!

factorial_7 = math.factorial(7)

Calculate 2^7

power_2_7 = 2 ** 7

Calculate the total number of monomial matrices

total_monomial_matrices = factorial_7 * power_2_7

The order of the automorphism group is 24

automorphism_group_order = 24

Calculate the number of unique generator matrices

unique_generator_matrices = total_monomial_matrices // automorphism_group_order print(unique_generator_matrices)

[THOUGHT] The calculation shows that the number of unique generator matrices that produce codes monomially equivalent to $\mathcal{C}$ is 26880. Therefore, the final answer is: $\boxed{26880}$ [/THOUGHT]