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I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of order p and order p^2 are understood, as are some special dihedral groups. But, often, Krull-Schmidt does not hold making a full classification difficult. Does anyone know of any papers where explicit examples are calculated? I can only find MCR Butler's example of the Klein group. I'd be happy to know about some cyclic cases that are not of order p or p^2.

Thanks!

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    $\begingroup$ Try Curtis and Reiner's book, around 1981-1 $\endgroup$ – Geoff Robinson Jun 4 '14 at 14:56
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A couple of comments are combined here into a partial answer to the question:

As Geoff indicates, Methods of Representation Theory I by Curtis-Reiner has an extensive treatment of integral representations toward the end. In §34 there is a detailed discussion of examples, with the cautionary remark that a complete classification is essentially impossible for cyclic $p$-groups of higher order than $p^2$ due to infinite representation type.

Besides treating dihedral and metacyclic groups in some detail, they include at the end of §34 a careful summary of results (with references) about other groups studied in the literature. Keep in mind that this volume was published in 1981 and that Reiner specialized in integral representations but died around the time the second volume was published in 1987. I'm not sure what has been done in recent decades, but Curtis-Reiner cover thoroughly the methods in use at the time.

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