Finite Group Databasebeta

← Group #13141659

Character table — Group #13141659

Order 1728.

C1C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C22C23C24C25C26C27C28C29C30C31C32C33C34C35C36C37C38C39C40C41C42C43C44C45C46C47C48C49C50C51C52C53C54C55C56C57C58C59C60C61C62C63C64C65C66C67C68C69C70C71C72C73C74C75C76C77C78C79C80C81C82C83C84C85C86C87C88C89C90C91C92C93C94C95C96C97C98C99C100C101C102C103C104C105C106C107C108C109C110
size12336669991233666999246612121218181824661212121818182466121212181818481212242424363636481212242424363636481212242424363636481212242424363636481212242424363636481212242424363636
order133399926626661818182662666181818266266618181826626661818182662666181818266412121236363641212412121236363641212412121236363641212412121236363641212412121236363641212
χ111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
χ21111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-1
χ311−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i32
χ411−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i32
χ511−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32
χ611−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32
χ72222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-1000
χ822−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i32000
χ922−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i32000
χ106-3000000006-3000000006-3000000006-3000000006-3000000006-3000000006-3000000006-3000000006-3000000006-3000000006-300000000
χ111111111111111111111111111111111111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
χ121111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-1-1-1-1-1-1-1-1111-1-1-1-1-1-1-1111-1-1-1-1-1-1-1111-1-1-1-1-1-1-1111
χ1311−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i32
χ1411−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32
χ1511−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i32
χ1611−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32
χ172222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-1000-2-2-2-2111000-2-2-2-2111000-2-2-2-2111000-2-2-2-2111000
χ1822−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i32000-2-21−i31+i31−1+i32−1−i32000-2-21−i31+i31−1+i32−1−i32000-2-21−i31+i31−1+i32−1−i32000-2-21−i31+i31−1+i32−1−i32000
χ1922−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i32000-2-21+i31−i31−1−i32−1+i32000-2-21+i31−i31−1−i32−1+i32000-2-21+i31−i31−1−i32−1+i32000-2-21+i31−i31−1−i32−1+i32000
χ206-3000000006-3000000006-3000000006-3000000006-3000000006-3000000006-300000000-6300000000-6300000000-6300000000-6300000000
χ2111111111111111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
χ221111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-1-1-1-1-1-1-1-1111-1-1-1-1-1-1-11111111111-1-1-11111111-1-1-1-1-1-1-1-1-1-1111-1-1-1-1-1-1-1111
χ2311−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i32
χ2411−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32
χ2511−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i32
χ2611−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32
χ272222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-1000-2-2-2-2111000-2-2-2-21110002222-1-1-10002222-1-1-1000-2-2-2-2111000-2-2-2-2111000
χ2822−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i32000-2-21−i31+i31−1+i32−1−i32000-2-21−i31+i31−1+i32−1−i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i32000-2-21−i31+i31−1+i32−1−i32000-2-21−i31+i31−1+i32−1−i32000
χ2922−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i32000-2-21+i31−i31−1−i32−1+i32000-2-21+i31−i31−1−i32−1+i3200022−1−i3−1+i3-11+i321−i3200022−1−i3−1+i3-11+i321−i32000-2-21+i31−i31−1−i32−1+i32000-2-21+i31−i31−1−i32−1+i32000
χ306-3000000006-3000000006-3000000006-3000000006-300000000-6300000000-63000000006-3000000006-300000000-6300000000-6300000000
χ3111111111111111111111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111111111111111
χ321111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-11111111-1-1-1-1-1-1-1-1-1-1111-1-1-1-1-1-1-1111-1-1-1-1-1-1-1111-1-1-1-1-1-1-11111111111-1-1-11111111-1-1-1
χ3311−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i32-1-11−i321+i32-11−i321+i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i321−1+i32−1−i32
χ3411−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i32-1-11−i321+i32-11−i321+i321−1+i32−1−i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i3211−1+i32−1−i321−1+i32−1−i32-11−i321+i32
χ3511−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32
χ3611−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32
χ372222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-10002222-1-1-1000-2-2-2-2111000-2-2-2-2111000-2-2-2-2111000-2-2-2-21110002222-1-1-10002222-1-1-1000
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χ521111111-1-1-11111111-1-1-11111111-1-1-1-1-1-1-1-1-1-1111-1-1-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1111−i−i−i−i−i−i−iiiiiiiiiii−i−i−i−i−i−i−i−i−i−iiiiiiiiiii−i−i−i
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χ7511−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i32-1-11+i321−i32-11+i321−i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i321−1−i32−1+i32−i−i−ζ11−ζ7−i−ζ11−ζ7−i−ζ11−ζ7iiζ11ζ7iζ11ζ7iζ11ζ7iiζ11ζ7iζ11ζ7iζ11ζ7−i−i−ζ11−ζ7−i−ζ11−ζ7−i−ζ11−ζ7
χ7611−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i32-1-11+i321−i32-11+i321−i321−1−i32−1+i3211−1−i32−1+i321−1−i32−1+i32-11+i321−i32−i−i−ζ11−ζ7−i−ζ11−ζ7iζ11ζ7iiζ11ζ7iζ11ζ7−i−ζ11−ζ7iiζ11ζ7iζ11ζ7−i−ζ11−ζ7−i−i−ζ11−ζ7−i−ζ11−ζ7iζ11ζ7
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χ7822−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i3200022−1+i3−1−i3-11−i321+i32000-2-21−i31+i31−1+i32−1−i32000-2-21−i31+i31−1+i32−1−i32000-2-21−i31+i31−1+i32−1−i3200022−1+i3−1−i3-11−i321+i32000−2i−2i−2ζ7−2ζ11iζ7ζ110002i2i2ζ72ζ11−i−ζ7−ζ110002i2i2ζ72ζ11−i−ζ7−ζ11000−2i−2i−2ζ7−2ζ11iζ7ζ11000
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χ806-3000000006-3000000006-300000000-6300000000-6300000000-63000000006-300000000−6i3i000000006i−3i000000006i−3i00000000−6i3i00000000
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χ9944−2−2i3−2+2i3-21+i31−i300044−2−2i3−2+2i3-21+i31−i3000-4-42+2i32−2i32−1−i3−1+i3000-4-42+2i32−2i32−1−i3−1+i300044−2−2i3−2+2i3-21+i31−i3000000000000000000000000000000000000000000000000000000000000000
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χ10344−2+2i3−2−2i34−2+2i3−2−2i34−2+2i3−2−2i3-4-42−2i32+2i3-42−2i32+2i3-42−2i32+2i3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
χ10444−2+2i3−2−2i34−2+2i3−2−2i3-42−2i32+2i3-4-42−2i32+2i3-42−2i32+2i34−2+2i3−2−2i3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
χ10544−2−2i3−2+2i34−2−2i3−2+2i34−2−2i3−2+2i3-4-42+2i32−2i3-42+2i32−2i3-42+2i32−2i3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
χ10644−2−2i3−2+2i34−2−2i3−2+2i3-42+2i32−2i3-4-42+2i32−2i3-42+2i32−2i34−2−2i3−2+2i3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
χ1078888-4-4-4000-8-8-8-8444000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
χ10888−4+4i3−4−4i3-42−2i32+2i3000-8-84−4i34+4i34−2+2i3−2−2i3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
χ10988−4−4i3−4+4i3-42+2i32−2i3000-8-84+4i34−4i34−2−2i3−2+2i3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
χ11024-1200000000-241200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Here ζ is a primitive root of unity of order 12 (ζ12 = 1); i = ζ3.