Finite Fields -- Field Extensions

Please see the attached PDF file. I would like someone other than OTA #101767 to provide a solution. Thank you :)

Groups of Order 56

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Quotient Ring

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Irreducible Polynomial

Irreducible Polynomial

Groups of Order 56

Groups of Order 56. The problem is out of Dummit and Foote in the section on Semidirect Products, which says that there are 13 isomorphism types for such a group.