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I have a finite subgroup, which I determined to be finite using the is_finite() function. The cardinality() function helped me establish how many matrices are present in this subgroup. Now, I am curious to know if there is a way to identify this specific finite subgroup in SageMath, for example, is it actually the XX* subgroup? Alternatively, to which group does this subgroup belong? Perhaps there exists a database where we can conveniently check the nature of this subgroup.

I am also wondering if we can directly check if the 'MatrixGroup(new_matrices_)' group is actually an XX* group using a function like if MatrixGroup(new_matrices) == XX. Additionally, I have a related question: How can I locate a specific subgroup in SageMath? For example, I have several subgroups defined in this paper: A Monograph on the Classification of the Discrete Subgroups of SU (4), and if I want to access, let's say the XX subgroup, how can I do it in SageMath? Sorry for the many little questions. I thought that they are all related, so instead of asking a new question for each little question, I combined them into one question and asked them all together.

Here is the minimal code for my matrix group:

addition = matrix([[1,0,0,0],[0,1,0,0],[1,0,0,1],[1,0,1,0]]) 
new_matrices_ = []
gens_ = MatrixGroup(SL(2,5)).gens() #the subgroup I am interested
for m in gens_:
    tensor_mat_= m.matrix().tensor_product(identity_matrix(2))
    new_matrices_.append(tensor_mat_)
 new_matrices_.append(addition)
MatrixGroup(new_matrices_)
MatrixGroup(new_matrices_).is_finite()
MatrixGroup(new_matrices_).cardinality()

I have updated the question

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