[THOUGHT] We need to understand the problem. We have a ternary linear code C of dimension 4 (since generator matrix G is 4×7 over F_3). The group of monomial automorphisms of C has order 24. We are asked: how many generator matrices produce codes that are monomially equivalent to C?

Interpretation: Monomial automorphisms of a code C are permutations of coordinates combined with nonzero scalar multiplications on each coordinate (scalars in F_3^* = {1,2}). So the monomial automorphism group M(C) is a subgroup of the monomial group M = (F_3^*)^7 ⋊ S_7 (size 2^7 * 7! = 2^7 * 5040 = 128 * 5040 = 645120). The given order of M(C) is 24.

Now, "generator matrices produce codes that are monomially equivalent to C" means: two generator matrices G and G' produce codes C and C' that are monomially equivalent (i.e., there exists a monomial transformation mapping C to C'). We want to count the number of generator matrices (over F_3) that produce a code monominally equivalent to C.

But careful: There are many generator matrices for the same code (different bases). The question likely asks: How many generator matrices produce codes that are monominally equivalent to C? That is, consider all generator matrices G' (4×7 over F_3) such that the code C' = span(rows of G') is monominally equivalent to C. But also G' must be a generator matrix (full rank 4). Two generator matrices produce the same code if they are related by row operations (GL(4,F_3)). So the number of generator matrices for a given code C is |GL(4,F_3)| = (3^4 - 1)(3^4 - 3)(3^4 - 3^2)(3^4 - 3^3) = (81-1)(81-3)(81-9)(81-27) = 80787254.

But we need generator matrices for codes monominally equivalent to C. That is, the set of codes in the monomial equivalence class of C. For each such code C', the number of generator matrices for C' is also |GL(4,F_3)|. So total number of generator matrices producing codes in the equivalence class = (number of codes in equivalence class) * |GL(4,F_3)|.

But maybe the intended interpretation: "How many generator matrices produce codes that are monominally equivalent to C?" could mean: count all generator matrices G' (4×7 over F_3) such that C' is monominally equivalent to C. That includes all bases for each code in the equivalence class.

Alternatively, maybe they mean: consider the set of all generator matrices for codes monominally equivalent to C, up to something? Possibly they want the number of distinct generator matrices (as matrices) that yield such codes. But then we must consider that two different matrices could yield the same code, which is fine because we count matrices, not codes.

Given the problem, likely they want: total number of generator matrices (4×7 full rank) for codes monominally equivalent to C = (size of monomial equivalence class of C) * |GL(4,F_3)|.

But we need to compute size of monomial equivalence class of C. The monomial group M acts on the set of all 4-dimensional codes in F_3^7. The stabilizer of C in M is M(C), the monomial automorphism group of C, of order 24. The orbit of C under M has size |M| / |M(C)| = (2^7 * 7!) / 24.

Compute: 2^7 = 128, 7! = 5040, product = 128*5040 = 645120. Divide by 24: 645120 / 24 = 26880.

So there are 26880 codes monominally equivalent to C.

Thus total number of generator matrices for these codes = 26880 * |GL(4,F_3)|.

Compute |GL(4,F_3)|: as above: (3^4 - 1)(3^4 - 3)(3^4 - 3^2)(3^4 - 3^3) = (81-1)=80, (81-3)=78, (81-9)=72, (81-27)=54. Multiply: 8078 = 6240, 624072 = 449280, 449280*54 = 24261120. Wait, compute carefully:

8078 = 6240. 624072 = 449280. 44928054 = 44928050 + 449280*4 = 22464000 + 1797120 = 24261120.

Yes, |GL(4,F_3)| = 24261120.

Now total = 26880 * 24261120.

Compute product: 26880 * 24261120.

Let's compute with Python to verify.

But wait: Is that what they ask? Possibly they want the number of generator matrices up to monomial equivalence? Or maybe they want the number of generator matrices that produce codes monominally equivalent to C, but considering that two generator matrices that are related by a monomial transformation (on coordinates and scaling) produce the same code? Actually, if we apply a monomial transformation to the generator matrix (permute columns and multiply each column by nonzero scalar), we get a generator matrix for a code monominally equivalent to the original. So the set of all generator matrices for codes in the equivalence class is exactly the orbit of G under monomial transformations (but careful: monomial transformations act on columns of G, not rows). However, applying a monomial transformation to a generator matrix yields a generator matrix for a code monominally equivalent to the original. But also, different monomial transformations might yield the same generator matrix up to row operations? Actually, if we apply a monomial transformation M to G (as in permute columns and scale), we get a matrix G' = G * M (since M acts on columns). Then G' is a generator matrix for C' = M(C). But also, any generator matrix for C' is of the form G' * A for some A in GL(4,F_3). So the set of all generator matrices for codes in the equivalence class is the orbit of G under the group (M ⋊ GL(4,F_3))? Wait: monomial transformations act on columns, but row operations are separate. So the group acting on generator matrices by monomial equivalence (i.e., producing monominally equivalent codes) is: first, apply a monomial transformation to columns (which gives a new code monominally equivalent), then apply row operations (which preserve the code). So the total number of distinct generator matrices (as matrices) that produce codes in the equivalence class is the size of the orbit of G under the group H = (monomial group) × GL(4,F_3) acting by: (M, A) sends G to A * G * M? But careful: M acts on columns (right multiplication by permutation and scaling matrix). So the orbit size is |M| * |GL(4,F_3)| / |Stab(G)|, where Stab(G) is the set of (M, A) such that A * G * M = G. But M(C) is the stabilizer of C under monomial transformations, i.e., monomials M such that M(C) = C. That is exactly the stabilizer of C in M. However, for the stabilizer of G under (M, A), we need A * G * M = G. If M fixes C (i.e., M(C) = C), then there exists A in GL(4,F_3) such that A * G = G * M^{-1}? Actually, if M(C) = C, then M maps C to itself, so there exists a linear automorphism A of C (i.e., A in GL(4,F_3)) such that M(g) = A(g) for all g in C. But that means M acts as A on columns of G? More precisely, if we have generator matrix G, then applying M to columns gives GM. For GM to equal G up to row operations, we need there exists A in GL(4,F_3) such that GM = AG. So Stab(G) = {(M, A) ∈ M(C) × GL(4,F_3) | GM = AG}. But note that A is determined by M? Not necessarily unique. However, for a given M in M(C), there may be multiple A satisfying that equation? Possibly only one if G is in systematic form? But we can compute size of stabilizer of G under the group action.

Alternatively, maybe the intended answer is simpler: The number of generator matrices producing codes monominally equivalent to C = (size of monomial equivalence class) * |GL(4,F_3)|. That seems plausible.

But let's verify with given data: order of monomial automorphism group of C is 24. That is the stabilizer of C in M. So orbit size = |M|/24 = 128*5040/24 = 645120/24 = 26880. Multiply by |GL(4,F_3)| = 24261120 gives huge number: 26880 * 24261120 = 26880 * 24261120.

Compute with Python to be sure.

We'll compute product.

Also, we should double-check if there is any nuance: The generator matrix G given is in standard form? It is 4×7, but looks like first 4 columns are identity, so systematic form. So G = [I | A] where A is 4×3 matrix. But that doesn't affect counts.

Let's compute product.

But maybe the problem expects answer in terms of product, but they want integer digits. So we compute product.

We'll compute with Python.

But before finalizing, let's consider if there is any overcount: Two different generator matrices could produce the same code, but we count matrices, so that's fine. However, are we counting each generator matrix for each code in the equivalence class? Yes.

But is it possible that some generator matrix for a code in the equivalence class could also be obtained as a generator matrix for another code in the equivalence class? That would require the code to be the same, so they are the same matrix only if same code and same basis. But we count all distinct matrices (different rows or different columns? Actually generator matrices are 4×7, so each matrix is a list of rows. Two matrices that produce the same code are different if there is a row operation between them. So we count each basis separately.

Thus total = orbit size * |GL(4,F_3)|.

But wait: The orbit of G under monomial transformations (acting on columns) has size |M| / |M(C)| = 26880. However, that orbit consists of matrices GM for M in M. But GM is a generator matrix for C' = M(C). However, for each such code C', there are |GL(4,F_3)| generator matrices. But note: G*M is just one generator matrix for C'. So the total number of generator matrices for all codes in the orbit is orbit size * |GL(4,F_3)|? But careful: The orbit of G under monomial transformations (only column operations) gives one generator matrix per code. But we also need all bases for each code. So indeed total = orbit size * |GL(4,F_3)|.

But is there overlap? Could GM1 and GM2 be row-equivalent? That would mean they are generator matrices for the same code. But if M1 and M2 give different codes, then GM1 and GM2 generate different codes, so they cannot be row-equivalent unless M1 and M2 are in the same left coset? Actually, if GM1 = AGM2 for some A in GL(4,F_3), then M1 = AGM2G^{-1}? But G is not square. However, since G is full rank, there exists a linear transformation on rows. But it's possible that GM1 and GM2 are row-equivalent even if M1 ≠ M2. That would imply that M1(M2)^{-1} preserves C up to row operations. But if M1(M2)^{-1} is in M(C), then M1 and M2 are in the same coset of M(C). So indeed, GM1 and GM2 are row-equivalent iff M1^{-1}M2 ∈ M(C) * (maybe something). Actually, let's think: GM1 = AGM2 => G(M1 M2^{-1}) = AG. So M1 M2^{-1} is a monomial transformation such that there exists A with AG = G*(M1 M2^{-1}). That is exactly the condition for M1 M2^{-1} to be in the stabilizer of G under monomial transformations (i.e., there exists A such that AG = G(M1 M2^{-1})). That stabilizer is the set of monomials M such that GM is row-equivalent to G. But that's not necessarily M(C). However, if M ∈ M(C), then there exists A such that AG = GM. So M(C) is contained in the stabilizer of G. But there could be more: monomials M such that GM is row-equivalent to G but M(C) ≠ C? Possibly, but if GM is row-equivalent to G, then the code generated by GM is the same as code generated by G, so M maps C to C, so M ∈ M(C). So indeed, stabilizer of G under monomial transformations (allowing row operations) is exactly M(C). So GM1 and GM2 are row-equivalent iff M1^{-1}M2 ∈ M(C). Thus the orbit of G under the group M (acting on columns) has size |M|/|M(C)| = 26880, and each element corresponds to a distinct code (since if GM1 and GM2 are row-equivalent, they generate same code). So indeed, the orbit under monomial transformations gives distinct codes.

Now, for each such code C', there are |GL(4,F_3)| generator matrices. But note: Among those |GL(4,F_3)| generator matrices for C', one is G*M (for a specific M). The others are different bases. So total generator matrices for codes in the equivalence class = orbit size * |GL(4,F_3)|.

Thus answer = 26880 * 24261120.

Compute product.

Let's compute with Python.

We'll also verify that the order 24 is indeed the stabilizer size. But we trust given.

Thus final answer is that product.

But maybe the problem expects the answer as integer digits.

We'll compute.

Also, we should check if there is any other factor: The generator matrix G is given, but we are counting generator matrices for codes monominally equivalent to C. That includes G itself? Yes.

Thus we compute.

Let's do Python.

We'll compute product and print.

But also, we need to ensure that the product fits within Python's integer range (it will). Let's compute. </think>

[THOUGHT] I'll compute the product: orbit size = |M|/|M(C)| = (2^7 * 7!) / 24, and multiply by |GL(4,F_3)|. Let's verify with Python.

First compute |M| = 2^7 * 7! = 128 * 5040 = 645120. Divide by 24: 645120 / 24 = 26880. |GL(4,F_3)| = (3^4 - 1)(3^4 - 3)(3^4 - 3^2)(3^4 - 3^3) = 80787254. Compute product and multiply.

Let's compute with Python. [/PYTHON] [PYTHON] import math

Compute orbit size

M_size = 2**7 * math.factorial(7) # 128 * 5040 orbit_size = M_size // 24 # integer division

Compute |GL(4, F_3)|

Order of GL(n, q) = (q^n - 1)(q^n - q)...(q^n - q^{n-1})

q = 3 n = 4 gl_size = 1 for k in range(n): gl_size *= (qn - qk)

Total number of generator matrices

total = orbit_size * gl_size

print("|M| =", M_size) print("Orbit size = |M|/|M(C)| =", orbit_size) print("|GL(4, F_3)| =", gl_size) print("Total = orbit_size * |GL(4, F_3)| =", total) [/PYTHON]

[THOUGHT] The total number of generator matrices is 652138905600.

\boxed{652138905600}