all in the family
The permutation group of 3 by 3 permutation matrices are M(123), M(132), M(213), M(231), M(312), and M(321). For brevity, these six matrices can be denoted respectively as M₁, M₂, M₃, M₄, M₅, and M₆. Although they all satisfy the three familial group properties of associativity, identity, and inversibility, in general they do not satisfy the intrinsically familial property of commutativity under the operation of matrix multiplication. This non-commutativity can be shown by matrix product of any two permutation matrices giving a set of 30 possible products.

Six pairs of the 30 possible products do commute. These are: (1) M₁M₂=M₂M₁=M₂, (2) M₁M₃=M₃M₁=M₃, (3) M₁M₄=M₄M₁=M₄, (4) M₁M₅=M₅M₁=M₅, (5) M₁M₆=M₆M₁=M₆, and (6) M₄M₅=M₅M₄=M₁. The nine pairs that do not commute are: (1) M₂M₃=M₅ while M₃M₂=M₄, (2) M₂M₄=M₆ while M₄M₂=M₃, (3) M₂M₅=M₃ while M₅M₂=M₆, (4) M₂M₆=M₄ while M₆M₂=M₅, (5) M₃M₄=M₂ while M₄M₃=M₆, (6) M₃M₅=M₆ while M₅M₃=M₂, (7) M₃M₆=M₅ while M₆M₃=M₄, ( 8 ) M₄M₆=M₂ while M₆M₄=M₃, and (9) M₅M₆=M₃ while M₆M₅=M₂. clearly these 30 matrix products are all in the family of these six permutation matrices. As products, M₁ occurs twice, M₂ occurs six times, M₃ occurs six times, M₄ occurs five times, M₅ occurs five times, and M₆occurs six times. However, if self-products are included then there are full totalof 36 products and as products each permutation matrix occurs exactly six times. The self-products are: M₁M₁=M₁, M₂M₂=M₁, M₃M₃=M₁, M₄M₄=M₅, M₅M₅=M₄, and M₆M₆=M₁. These form concisely a table of a permutation algebra under the operation of matrix multiplication. Hypothetically, for any positive integer N greater than 3 there are N! of N by N permutation matrices and these also form a permutation group.