According to Carters [Lower K-theory of finite groups](https://www.researchgate.net/publication/233130753_Lower_K-theory_of_finite_groups) for a finite group $G$ we have
$$ K\_{-1} (\mathbb Z G) = \mathbb Z^r \oplus \mathbb Z\_2^s $$
where $s$ is the sum over all irreducible representations over $\mathbb Q$ which have even Schur index, but odd Schur index over $\mathbb Q\_p$ for every finite prime which divides the order of $G$.


The integer $s$ seems to be $0$ for a pretty wide class of groups and its hard for me to even find a single concrete example of a finite group with non-trivial $s$. Does anyone know of a result in that direction?



Finite group such that $K_{-1} (\mathbb Z G)$ has non-trivial torsion
Many results in this direction can be found in the paper


B. A. Magurn: [Negative (K)-theory of generalized quaternion groups and binary polyhedral groups](http://dx.doi.org/10.1080/00927872.2012.692005), *Commun. Algebra* **41**, No. 11, 4146-4160 (2013). [ZBL1284.19004](https://zbmath.org/?q=an:1284.19004).


In particular, you can have a look at



> 
> **Corollary (p.4155)** Given the dicyclic group $\mathsf{Q}\_n$, the group $K\_{-1}(\mathbb{Z} \mathsf{Q}\_n)$ is torsion free if and only if $n$ is a power
> of a prime in $3 + 4 \mathbb{Z}$ or $n=2$.
> 
> 
> 


All the other occurrences for $n$ provide an infinite series of examples with $s\neq 0$. The paper also contains explicit calculations for $s$.



> 
> **Theorem 5 (p.4155)** The group $K\_{-1}$ for binary polyhedral groups is as follows: $$K\_{-1}(\mathbb{Z} \tilde{\mathsf{A}}\_4)= \mathbb{Z},
>  \quad K\_{-1}(\mathbb{Z} \tilde{\mathsf{S}}\_4)= \mathbb{Z}\oplus
>  \mathbb{Z}\_2, \quad K\_{-1}(\mathbb{Z} \tilde{\mathsf{A}}\_5)=
>  \mathbb{Z}^2\oplus \mathbb{Z}\_2$$
> 
> 
> 


This yields two further examples with $s=1$.


