Archimylacris was a Carboniferous genus of insect. Even though it looked like one, it was not a cockroach, but an ancestor. The first true cockroaches appeared in the Jurassic period, and the first modern cockroach appeared in the Cretaceous period. But the paleozoic "cockroaches" are not cockroaches but Blattopterans, which are ancestors of cockroaches, termites, and mantises.
Archimylacris was very large, reaching up to 9 cm in length and 4 cm in width. It looked like a modern cockroach, except it was much bigger. It had slight differences, such as sticky pads on its legs called euplantulae and a long ovipositor. (Only the female had the ovipositor). The euplantulae may have helped it crawl on smooth surfaces, such as leaves, or possibly help it walk upside down like some modern insects do. This may have allowed it to escape from forest floor predators such as amphibians and early reptiles and allow it to climb tall plants.
Archimylacris lived in the swamp forests of Europe and North America about 300 million years ago. It probably scurried along the forest floor and acted like a modern cockroach. The video below shows an animated CT scan of one of the fossils.