This is book 3 of the "Skylark" series.

This note contains spoilers for the book.

Book opens showing how DuQuesne's immolation in the previous book was a trick. Then things get complicated.

The party gets rotated into the 4th dimension, even -- to escape those pure intellectuals from book 1. They survive until they snap back into our space, and find they're so far from home they can't find the galaxy, let alone the sun. So they look for a planet, and run into a human - chloran war, help the humans win that, build a spaceship with sixth order capability and a brain, and map the First Universe. And head home.

This one seems very much to have been written as a series ender. And indeed it was more than 30 years before the next book turned up.