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The family -- a mother and father, two daughters aged 9 and 12, and two sons aged 16 and 18 -- were linked to local extremist network Jamaah Ansharut Daulah, which supports IS, Karnavian said.
Indonesian president Joko Widodo slammed the attacks, telling reporters: "We must unite against terrorism."
"The state will not tolerate this act of cowardice," he added.