A group of matrices satisfying a particular constraint(definition) of the group

Suppose that one wants to have a group of matrices that satisfy some constraints. (As for a similar example, Pauli matrices satisfy some particular constraints.)

The constraint goes like following: (A matrix in the group is denoted $A_{ij}$; $ij$ is not referring to entries; it is used to label each matrix.

1) Matrices in the group commute.

2) For any matrix multiplication $B = A_{ij} \, A_{kl} \, A_{mn} \, ...$ if some particular number is used more than twice in $i,j,k,l,m,n,....$, the eigenvalue of $B$ becomes zero, and zero is the sole eigenvalue.

Can anyone provide me some hints?

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A set $A_{ji}$ with $B = A_{ij} \, A_{kl} \, A_{mn} \, ...$ where the eigenvalue of B becomes zero, and zero is the sole eigenvalue is not a group, since the zero matrix is not invertible.