This refers to the Gilded Age, first published in 1873 by Mark Twain.
The key issue dividing these three classes were mainly political.
The Mugwumps were members of the Republican party, who refused support to the nominee of 1884 and supported the Democratic candidate.
The Half-breed were rivals of the Stalwarts in the party control. They supported civil service reform and merit appointments to government posts while the Stalwarts favored the spoils system of political patronage. These were the result of a split in the Republican party.
