An Enumeration of the Supercharacter Theories of $C_p \times C_2 \times C_2$ for Prime $p$

TL;DR

This paper provides a detailed enumeration and classification of supercharacter theories for the group $C_p imes C_2 imes C_2$, including counts, construction methods, and an alternative proof approach.

Contribution

It offers a precise count of supercharacter theories for $C_p imes C_2 imes C_2$ and clarifies the methods of their construction, extending prior classifications.

Findings

Every nontrivial supercharacter theory can be constructed via wedge product, direct product, or automorphisms.

The paper counts the total number of supercharacter theories for the group.

It identifies when multiple construction methods yield the same supercharacter theory.

Abstract

The supercharacter theories of $C_p \times C_2 \times C_2$ were classified in the language of Schur rings by Evdokimov, Kov\'acs, and Ponomarenko in [EKP16]. It was shown that every nontrivial supercharacter theory of $C_p \times C_2 \times C_2$ can be constructed as a wedge product, a direct product, or is generated by automorphisms. We use this classification to give a precise count of the distinct supercharacter theories of $C_p\times C_2\times C_2$ and describe when a supercharacter theory can be constructed by more than one method. We also present an alternative proof of the classification using the language of supercharacter theories.