Our founding members chose "That Gay Group!" as our name for several reasons. When searching for ideas, one of their primary requirements was, as it still is for us, that the name be instantly identifiable with an LGBTQ organization. Their other criterion was that the name shouldn't sound too stuffy, since we are a social group. They settled on TGG because it met those criteria and, quite frankly, it still makes us laugh!
In this instance, we use the term "gay" as an umbrella word for the LGBTQ community and do not intend for it to signify a group which is exclusively cisgender, male, or homosexual.