Lewaa al-Thawra surfaced in 2016 to claim several attacks in Cairo, including the assassination of an army brigadier general outside his suburban home in October.
Police have accused the blacklisted Muslim Brotherhood movement of ousted president Mohamed Morsi of directing the group.
The interior ministry said in a statement police carried out raids against members of the group in the provinces of Cairo, Giza, Beheira, and Kafr el-Sheikh.
In one of the raids in Beheira on a hideout used to make bombs the suspects opened fire as police approached, triggering a shootout in which three "Brotherhood terrorist elements" were killed, the ministry said.
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Lewaa al-Thawra, and another group named Hasam, appeared after the military ousted in July 2013 Morsi in July 2013, unleashing a deadly crackdown on his supporters.
The crackdown crushed the Brotherhood, once the country's best organised political movement, with its leaders now mostly jailed or in exile.
Jihadists have killed hundreds of soldiers and policemen since then, mostly in the Sinai Peninsula where an affiliate of the Islamic State group is based.
In its statement, the interior ministry said it had confirmed "links between elements who were arrested with some of the leaders of the terrorist organisation (the Muslim Brotherhood) who have fled abroad."