I don't get group texts with cellular data off
A one-to-one text message uses SMS. A group message uses MMS and this requires data. You would likely have the same issue if someone texts you a picture directly as this also uses MMS.

Check with your service provider though, as some include picture messaging (i.e. MMS) for "free" and don't count it as data. In which case you could leave iMessage enabled for data. YMMV.