Groups of people with common political purposes are called

People usually belong to political parties. However, even within the parties themselves, there are smaller groups, called "factions". These "factions" are part of the larger group (parties) and they are committed to a common political purpose.

These people who are part of a political faction within a political party, band together to achieve a common goal or interest. They want to advance a certain agenda, such as certain policies that are important to the group within the political party.