# Symmetric designs and four dimensional projective special unitary groups

## TL;DR

This paper classifies symmetric designs with specific automorphism groups related to four-dimensional projective special unitary groups, identifying eight unique designs with particular parameters and automorphism groups.

## Contribution

It provides a complete classification of symmetric designs admitting flag-transitive, point-primitive automorphism groups with socle $PSU_{4}(q)$, including explicit parameter and group descriptions.

## Key findings

- Eight non-isomorphic designs found
- Designs have parameters with λ in {3,6,18}
- Automorphism groups are either $PSU_{4}(2)$ or $PSU_{4}(2):2$

## Abstract

In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is $PSU_{4}(q)$. We prove that there exist eight non-isomorphic such designs for which $\lambda\in\{3,6,18\}$ and $G$ is either $PSU_{4}(2)$, or $PSU_{4}(2):2$.