The vSphere Client (C#) is only available for Windows systems while running vCenter Server 6.0 or earlier and it is not available for Linux systems.
For vCenter Server 6.0, the vSphere Web Client (Flex/Flash) was introduced providing full administrative abilities through a web based user interface. Additionally, in vCenter Server 6.5 and higher the vSphere Client (HTML5) was introduced allowing the administration of vCenter Server without the requirement of Flash.
For more specific details on the clients available, see below:
Note: With the release of vSphere 6.5 the vSphere Client (C#) has been deprecated.