Finite group such that $K_{-1} (\mathbb Z G)$ has non-trivial torsion

Question

According to Carters 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?

Answer 1

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, Commun. Algebra 41, No. 11, 4146-4160 (2013). ZBL1284.19004.

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 $s=1$.