S3's elements

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The results of

sage: G = SymmetricGroup(3)
sage: G.list()
[(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)]

are given in cycle notation. So the book's equivalent of (1,3,2) is one where 1 becomes 3, 3 becomes 2, and 2 becomes 1, which is P6, if I'm reading correctly.

If you want to match the bottom three elements of your matrix, you can simply convert each to a list, or maybe a dict would make the mapping more explicit:

sage: for g in G:
....:     print g, g.list(), g.dict()
....:     
() [1, 2, 3] {1: 1, 2: 2, 3: 3}
(2,3) [1, 3, 2] {1: 1, 2: 3, 3: 2}
(1,2) [2, 1, 3] {1: 2, 2: 1, 3: 3}
(1,2,3) [2, 3, 1] {1: 2, 2: 3, 3: 1}
(1,3,2) [3, 1, 2] {1: 3, 2: 1, 3: 2}
(1,3) [3, 2, 1] {1: 3, 2: 2, 3: 1}