Representation dimension of isoclinic groups with isomorphic centers

Determine whether isoclinic groups of order p^6 with isomorphic centers necessarily have equal representation dimensions, that is, whether δ(G)=δ(H) whenever G and H are isoclinic groups of order p^6 and Z(G)≅Z(H).

Background

The paper computes the representation dimension δ(G) for groups of order p6, with p≥5, by organizing the groups according to their isoclinism families. The resulting table shows that, in the computed cases, isoclinic groups having isomorphic centers also have the same representation dimension.

The authors explicitly state that they cannot prove whether this observed equality holds in general, and the Problems section reformulates the observation as a question for isoclinic groups of equal order. Establishing the claim would identify an invariance property of representation dimension under isoclinism subject to preservation of the center's isomorphism type.

References

We conclude by the observation that in Table \ref{t:delta G p6}, if $G$ and $H$ are isoclinic groups of order $p6$ such that $\mathcal{Z}(G) \cong \mathcal{Z}(H)$, then $\delta(G) = \delta(H)$. However, we are unable to prove it at this moment.

— On the representation dimension of finite $p$-groups  (2608.18814 - Kaur et al., 19 Aug 2026) in Section 3, immediately following Table 1 (Representation dimension of groups of order p^6); revisited in Section 4, Problems, item 2