Flowers look like Gymnocalycium, the pups remind me to this genus as well, but not the mother-plant (though there are "chinlike" structures). Apparently the two pictures show different plants (10 respectively 13 ribs)! What about the dimensions?
Perhaps it's a "Gymnopsis", a so called "Piltdown-Cactus"?!
Hi midhungirish
To be serious again (and helpful), I'd like to mention the washboard-like profile of the ribs which remind me to Gymnocalycium mihanovichii or friedrichii. This feature, however is only weekly marked at your plants. Perhaps a hybrid?