[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:

[ \frac{7! \times 2^7}{24} ]

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]