Finite Group Databasebeta

← Group #13168423

Character table — Group #13168423

Order 1728.

C1C2C3C4C5C6C7C8C9C10C11C12C13C14C15C16C17C18C19C20C21C22C23C24C25C26C27C28C29C30C31C32C33C34C35C36C37C38C39C40C41C42C43C44C45C46C47C48C49C50C51C52C53C54C55C56C57C58C59C60C61C62C63C64C65C66C67C68C69C70C71C72C73C74C75C76C77C78C79C80C81C82C83C84C85C86C87C88C89C90C91C92C93C94C95C96C97C98C99C100C101C102C103C104C105C106C107C108C109C110C111C112C113C114C115C116C117C118C119C120C121C122C123C124C125C126C127C128C129C130C131C132C133C134C135C136C137C138C139C140C141C142C143C144
size111122222222222222222233334444444444444444444444446666666666666666666666666688888888881212121212121212121212121212121212121212121212121212121218181818181818181818181824242424242436363636363636367272108108
order12223334466666666612121212444433366666666612121212121212121212121244444412121212121212121212121212121212121212123666121212121212226666121212121212121212121212121212121212121212122244666612121212666666444412121212121244
χ1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
χ211111111111111111111111111111111111111111111111111111111111111111111111111111111111111-1-1-1-1-1-11111111111111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
χ311111111111111111111111111111111111111111111111111-1-111-1-111111111-1-1-1-1-1-1-1-11111111111111111111111-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111111111-1-11111-1-1-1-1
χ411111111111111111111111111111111111111111111111111-1-111-1-111111111-1-1-1-1-1-1-1-111111111111111-1-1-1-1-1-111-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111-1-1-1-11111
χ51111111111111111111111-1-1-1-111111111111111111111111111-1-111-1-1-1-1-1-1-1-111111111-1-1-1-11111111111111111-1-11111-1-1-1-1-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1-1-1111111-1-111-1-1-1-111-1-1
χ61111111111111111111111-1-1-1-111111111111111111111111111-1-111-1-1-1-1-1-1-1-111111111-1-1-1-11111111111-1-1-1-1-1-1-1-11111-1-1-1-1-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-111-1-11111-1-111
χ71111111111111111111111-1-1-1-1111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111111111111-1-1-1-1-1-1-1-1-1-111
χ81111111111111111111111-1-1-1-1111111111111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-11111111111-1-1
χ91111111-1-1111111111-1-1-1-11111111111111111-1-1-1-1-1-1-1-1-1-1-1-111-1-1-1-11111-1-1-1-11111-1-1-1-111111111-1-1-1-1-1-11-111-1-1-1-111-1-11111-1-1-1-11111-1-1-1-111-1-11111-1-1-1-11-111-1-11-11-111-1-11-11-1
χ101111111-1-1111111111-1-1-1-11111111111111111-1-1-1-1-1-1-1-1-1-1-1-111-1-1-1-11111-1-1-1-11111-1-1-1-111111111-1-1-1-1-1-1-11-1-111-1-111-1-11111-1-1-1-11111-1-1-1-111-1-11111-1-1-1-1-11-1-111-11-11-1-111-11-11
χ111111111-1-1111111111-1-1-1-11111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-11-111-1-1-1-1-1-1111111-1-1-1-1-1-1-1-11111-1-111-1-1-1-111111-111-1-11-1-1111-1-1-11-11
χ121111111-1-1111111111-1-1-1-11111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111111-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-1-11-1-111-1-1-1-1111111-1-1-1-1-1-1-1-11111-1-111-1-1-1-11111-11-1-111-111-1-1-1111-11-1
χ131111111-1-1111111111-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-1-1-111111111-1-1-1-1-1-1-1-11111-1-1-1-1-1-11-111-1-11111-1-1-1-1-1-111111111-1-1-1-1-1-111-1-1-1-111111-111-1-1-111-1-1-1111-1-11
χ141111111-1-1111111111-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-1-1-111111111-1-1-1-1-1-1-1-11111-1-1-1-1-1-1-11-1-1111111-1-1-1-1-1-111111111-1-1-1-1-1-111-1-1-1-11111-11-1-1111-1-1111-1-1-111-1
χ151111111-1-1111111111-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-11111-1-1-1-11111-1-1-1-11111-1-1-1-1-1-11-111-1-111-1-111-1-1-1-11111-1-1-1-1111111-1-11111-1-1-1-11-111-1-1-11-11-1-111-111-1
χ161111111-1-1111111111-1-1-1-1-1-1-1-1111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1-11111-1-1-1-11111-1-1-1-11111-1-1-1-11111-1-1-1-1-1-1-11-1-11111-1-111-1-1-1-11111-1-1-1-1111111-1-11111-1-1-1-1-11-1-1111-11-111-1-11-1-11
χ17222222-1222222-1-1-122-1-1-1-12222-1-1222-1-12-1-1-1-12222-1-1-1-1-1-1-1-1222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-12222-1-1-1-122-1-1-1-1000000222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-12222-1-1-1-1-1-1-1-1000000000000000000
χ18222222-1222222-1-1-122-1-1-1-12222-1-1222-1-12-1-1-1-12222-1-1-1-1-1-1-1-1-2-222-2-2-1-1-1-1-1-1-1-1111111112222-1-1-1-122-1-1-1-100000022-2-2-2-2-1-1-1-1-1-1-1-111111111-2-2-2-211111111000000000000000000
χ19222222-1222222-1-1-122-1-1-1-1-2-2-2-2-1-1222-1-12-1-1-1-12222-1-1-1-1-1-1-1-122-2-22211111111-1-1-1-1-1-1-1-1-2-2-2-2-1-1-1-122-1-1-1-1000000-2-2222211111111-1-1-1-1-1-1-1-1-2-2-2-211111111000000000000000000
χ20222222-1222222-1-1-122-1-1-1-1-2-2-2-2-1-1222-1-12-1-1-1-12222-1-1-1-1-1-1-1-1-2-2-2-2-2-21111111111111111-2-2-2-2-1-1-1-122-1-1-1-1000000-2-2-2-2-2-211111111111111112222-1-1-1-1-1-1-1-1000000000000000000
χ21222222-1-2-22222-1-1-12211112222-1-1222-1-12-1-1-1-1-2-2-2-21111111122-2-2-2-2-1-1-1-11111-1-1-1-111112222-1-1-1-1-2-21111000000-2-222-2-2-1-1-1-11111-1-1-1-1111122-2-2-1-1-1-11111000000000000000000
χ22222222-1-2-22222-1-1-12211112222-1-1222-1-12-1-1-1-1-2-2-2-211111111-2-2-2-222-1-1-1-111111111-1-1-1-12222-1-1-1-1-2-21111000000-2-2-2-222-1-1-1-111111111-1-1-1-1-2-2221111-1-1-1-1000000000000000000
χ23222222-1-2-22222-1-1-1221111-2-2-2-2-1-1222-1-12-1-1-1-1-2-2-2-2111111112222-2-21111-1-1-1-1-1-1-1-11111-2-2-2-2-1-1-1-1-2-211110000002222-2-21111-1-1-1-1-1-1-1-11111-2-2221111-1-1-1-1000000000000000000
χ24222222-1-2-22222-1-1-1221111-2-2-2-2-1-1222-1-12-1-1-1-1-2-2-2-211111111-2-222221111-1-1-1-11111-1-1-1-1-2-2-2-2-1-1-1-1-2-2111100000022-2-2221111-1-1-1-11111-1-1-1-122-2-2-1-1-1-11111000000000000000000
χ2522222-122222-1-1222-12222200002-1-1-1-12-1-122-1-122-1-12222-1-1-1-122002200000000222222220000-1-1-1-1-1-1-1-1-1-122222200-1-1-1-100000000-1-1-1-1-1-1-1-1000000000000-1-1-1-1-1-100220000-1-100
χ2622222-122222-1-1222-12222200002-1-1-1-12-1-122-1-122-1-12222-1-1-1-122002200000000222222220000-1-1-1-1-1-1-1-1-1-1-2-2-2-2-2-200-1-1-1-100000000-1-1-1-1-1-1-1-100000000000011111100-2-200001100
χ2722222-122222-1-1222-12222200002-1-1-1-12-1-122-1-122-1-12222-1-1-1-1-2-200-2-200000000-2-2-2-2-2-2-2-20000-1-1-1-1-1-1-1-1-1-12222220011110000000011111111000000000000-1-1-1-1-1-100-2-200001100
χ2822222-122222-1-1222-12222200002-1-1-1-12-1-122-1-122-1-12222-1-1-1-1-2-200-2-200000000-2-2-2-2-2-2-2-20000-1-1-1-1-1-1-1-1-1-1-2-2-2-2-2-2001111000000001111111100000000000011111100220000-1-100
χ2922222-12-2-222-1-1222-12-2-2-2-200002-1-1-1-12-1-122-1-1-2-211-2-2-2-211112200-2-2000000002222-2-2-2-20000-1-1-1-11111112-222-2-200-1-11100000000-1-1-1-11111000000000000-11-1-111002-20000-1100
χ3022222-12-2-222-1-1222-12-2-2-2-200002-1-1-1-12-1-122-1-1-2-211-2-2-2-211112200-2-2000000002222-2-2-2-20000-1-1-1-1111111-22-2-22200-1-11100000000-1-1-1-111110000000000001-111-1-100-2200001-100
χ3122222-12-2-222-1-1222-12-2-2-2-200002-1-1-1-12-1-122-1-1-2-211-2-2-2-21111-2-2002200000000-2-2-2-222220000-1-1-1-11111112-222-2-20011-1-1000000001111-1-1-1-1000000000000-11-1-11100-2200001-100
χ3222222-12-2-222-1-1222-12-2-2-2-200002-1-1-1-12-1-122-1-1-2-211-2-2-2-21111-2-2002200000000-2-2-2-222220000-1-1-1-1111111-22-2-2220011-1-1000000001111-1-1-1-10000000000001-111-1-1002-20000-1100
χ332222-12222-1-1222222-122222222-12-1-1-1-12-1-1-122-1-122-1-1-1-122220022002222222200000000-1-1-1-1-1-1-1-1-1-1-1-1-1-122-1-1-1-1-1-10000-1-1-1-1-1-1-1-10000000000000000000022-1-1-1-12200-1-1-1-10000
χ342222-12222-1-1222222-122222222-12-1-1-1-12-1-1-122-1-122-1-1-1-122220022002222222200000000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-2-21111-1-10000-1-1-1-1-1-1-1-100000000000000000000-2-21111-2-20011110000
χ352222-12222-1-1222222-12222-2-2-2-2-12-1-1-1-12-1-1-122-1-122-1-1-1-1222200-2-200-2-2-2-2-2-2-2-2000000001111-1-1-1-1-1-1-1-1-1-122-1-1-1-1110000111111110000000000000000000022-1-1-1-1-2-20011110000
χ362222-12222-1-1222222-12222-2-2-2-2-12-1-1-1-12-1-1-122-1-122-1-1-1-1222200-2-200-2-2-2-2-2-2-2-2000000001111-1-1-1-1-1-1-1-1-1-1-2-211111100001111111100000000000000000000-2-211112200-1-1-1-10000
χ372222-122-2-2-1-1222222-1-2-2-2-22222-12-1-1-1-12-1-1-12211-2-21111-2-2-2-200-2-2002222-2-2-2-200000000-1-1-1-1-1-1-1-11111112-2-1-111110000-1-1-1-11111000000000000000000002-2-1-1112-200-1-1110000
χ382222-122-2-2-1-1222222-1-2-2-2-22222-12-1-1-1-12-1-1-12211-2-21111-2-2-2-200-2-2002222-2-2-2-200000000-1-1-1-1-1-1-1-1111111-2211-1-1110000-1-1-1-1111100000000000000000000-2211-1-1-220011-1-10000
χ392222-122-2-2-1-1222222-1-2-2-2-2-2-2-2-2-12-1-1-1-12-1-1-12211-2-21111-2-2-2-2002200-2-2-2-22222000000001111-1-1-1-11111112-2-1-111-1-100001111-1-1-1-1000000000000000000002-2-1-111-220011-1-10000
χ402222-122-2-2-1-1222222-1-2-2-2-2-2-2-2-2-12-1-1-1-12-1-1-12211-2-21111-2-2-2-2002200-2-2-2-22222000000001111-1-1-1-1111111-2211-1-1-1-100001111-1-1-1-100000000000000000000-2211-1-12-200-1-1110000
χ4122-2-222200-2-22-22-2-2-22000022-2-22222-2-2-2-22-22-200000000000000002i−2i22-2-2000000002i2i−2i−2i22-2-222-2-200000000000000002i−2i22-2-2000000002i2i−2i−2i002i−2i00002i2i−2i−2i000000000000000000
χ4222-2-222200-2-22-22-2-2-22000022-2-22222-2-2-2-22-22-20000000000000000−2i2i22-2-200000000−2i−2i2i2i22-2-222-2-20000000000000000−2i2i22-2-200000000−2i−2i2i2i00−2i2i0000−2i−2i2i2i000000000000000000
χ4322-2-222200-2-22-22-2-2-220000-2-2222222-2-2-2-22-22-200000000000000002i−2i-2-222000000002i2i−2i−2i-2-22222-2-200000000000000002i−2i-2-222000000002i2i−2i−2i00−2i2i0000−2i−2i2i2i000000000000000000
χ4422-2-222200-2-22-22-2-2-220000-2-2222222-2-2-2-22-22-20000000000000000−2i2i-2-22200000000−2i−2i2i2i-2-22222-2-20000000000000000−2i2i-2-22200000000−2i−2i2i2i002i−2i00002i2i−2i−2i000000000000000000
χ4522-2-222-100-2-22-2-111-2233−3−322-2-2-1-122-211-2-11-11000033−3−333−3−300002i−2i-1-11133−3−3−i3−i3i3i3−i−iii22-2-2-1-1110033−3−300000000002i−2i-1-11133−3−3−i3−i3i3i3−i−iii002i−2i−i3−i3i3i3−i−iii000000000000000000
χ4622-2-222-100-2-22-2-111-2233−3−322-2-2-1-122-211-2-11-11000033−3−333−3−30000−2i2i-1-11133−3−3i3i3−i3−i3ii−i−i22-2-2-1-1110033−3−30000000000−2i2i-1-11133−3−3i3i3−i3−i3ii−i−i00−2i2ii3i3−i3−i3ii−i−i000000000000000000
χ4722-2-222-100-2-22-2-111-2233−3−3-2-222-1-122-211-2-11-11000033−3−333−3−300002i−2i11-1-1−3−333−i3−i3i3i3−i−iii-2-222-1-1110033−3−300000000002i−2i11-1-1−3−333−i3−i3i3i3−i−iii00−2i2ii3i3−i3−i3ii−i−i000000000000000000
χ4822-2-222-100-2-22-2-111-2233−3−3-2-222-1-122-211-2-11-11000033−3−333−3−30000−2i2i11-1-1−3−333i3i3−i3−i3ii−i−i-2-222-1-1110033−3−30000000000−2i2i11-1-1−3−333i3i3−i3−i3ii−i−i002i−2i−i3−i3i3i3−i−iii000000000000000000
χ4922-2-222-100-2-22-2-111-22−3−33322-2-2-1-122-211-2-11-110000−3−333−3−33300002i−2i-1-111−3−333i3i3−i3−i3−i−iii22-2-2-1-11100−3−33300000000002i−2i-1-111−3−333i3i3−i3−i3−i−iii002i−2ii3i3−i3−i3−i−iii000000000000000000
χ5022-2-222-100-2-22-2-111-22−3−33322-2-2-1-122-211-2-11-110000−3−333−3−3330000−2i2i-1-111−3−333−i3−i3i3i3ii−i−i22-2-2-1-11100−3−3330000000000−2i2i-1-111−3−333−i3−i3i3i3ii−i−i00−2i2i−i3−i3i3i3ii−i−i000000000000000000
χ5122-2-222-100-2-22-2-111-22−3−333-2-222-1-122-211-2-11-110000−3−333−3−33300002i−2i11-1-133−3−3i3i3−i3−i3−i−iii-2-222-1-11100−3−33300000000002i−2i11-1-133−3−3i3i3−i3−i3−i−iii00−2i2i−i3−i3i3i3ii−i−i000000000000000000
χ5222-2-222-100-2-22-2-111-22−3−333-2-222-1-122-211-2-11-110000−3−333−3−3330000−2i2i11-1-133−3−3−i3−i3i3i3ii−i−i-2-222-1-11100−3−3330000000000−2i2i11-1-133−3−3−i3−i3i3i3ii−i−i002i−2ii3i3−i3−i3−i−iii000000000000000000
χ532-22-22222-22-2-2-2-2-222-22-22-22i−2i2i−2i222-2-2222-2-2-2-22-22-22-22-22-22-2002i−2i002i−2i2i−2i2i−2i2i−2i000000002i−2i2i−2i2-2-222-22-22-20000002i−2i00002i−2i2i−2i2i−2i2i−2i00000000000000000000000000000000000000
χ542-22-22222-22-2-2-2-2-222-22-22-2−2i2i−2i2i222-2-2222-2-2-2-22-22-22-22-22-22-200−2i2i00−2i2i−2i2i−2i2i−2i2i00000000−2i2i−2i2i2-2-222-22-22-2000000−2i2i0000−2i2i−2i2i−2i2i−2i2i00000000000000000000000000000000000000
χ552-22-2222-222-2-2-2-2-222-2-22-222i−2i2i−2i222-2-2222-2-2-2-2-22-22-22-22-22-2200−2i2i002i−2i2i−2i−2i2i−2i2i000000002i−2i2i−2i2-2-22-22-22-22000000−2i2i00002i−2i2i−2i−2i2i−2i2i00000000000000000000000000000000000000
χ562-22-2222-222-2-2-2-2-222-2-22-22−2i2i−2i2i222-2-2222-2-2-2-2-22-22-22-22-22-22002i−2i00−2i2i−2i2i2i−2i2i−2i00000000−2i2i−2i2i2-2-22-22-22-220000002i−2i0000−2i2i−2i2i2i−2i2i−2i00000000000000000000000000000000000000
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χ1274-4-444-2-200-442-22-222-4−232323−230000-21-22-22-122-2-110000−232323−233−3−334i−4i000000000000−2i2i2i−2i−2i32i32i3−2i300001-11-1003−3−3300000000−2i2i0000000000i−i−iii3−i3−i3i3000000000000000000000000000000
χ1284-4-444-2-200-442-22-222-4−232323−230000-21-22-22-122-2-110000−232323−233−3−33−4i4i0000000000002i−2i−2i2i2i3−2i3−2i32i300001-11-1003−3−33000000002i−2i0000000000−iii−i−i3i3i3−i3000000000000000000000000000000
χ1294-4-44-244002-2-44-44-4-4200004i−4i−4i4i-24-22-22-422-2-440000000000000000004i−4i−4i4i000000000000−2i2i2i−2i-22-22000000000000000000−2i2i2i−2i000000000000000000000000000000000000000000
χ1304-4-44-244002-2-44-44-4-420000−4i4i4i−4i-24-22-22-422-2-44000000000000000000−4i4i4i−4i0000000000002i−2i−2i2i-22-220000000000000000002i−2i−2i2i000000000000000000000000000000000000000000
χ1314-4-44-24-2002-2-442-22-4223−23−23234i−4i−4i4i1-2-22-2-122-112-20000−333−323−23−2323000000−2i2i2i−2i2i3−2i3−2i32i300000000−2i2i2i−2i1-11-100−333−3000000000000i−i−ii−i3i3i3−i300000000000000000000000000000000000000
χ1324-4-44-24-2002-2-442-22-4223−23−2323−4i4i4i−4i1-2-22-2-122-112-20000−333−323−23−23230000002i−2i−2i2i−2i32i32i3−2i3000000002i−2i−2i2i1-11-100−333−3000000000000−iii−ii3−i3−i3i300000000000000000000000000000000000000
χ1334-4-44-24-2002-2-442-22-42−232323−234i−4i−4i4i1-2-22-2-122-112-200003−3−33−232323−23000000−2i2i2i−2i−2i32i32i3−2i300000000−2i2i2i−2i1-11-1003−3−33000000000000i−i−iii3−i3−i3i300000000000000000000000000000000000000
χ1344-4-44-24-2002-2-442-22-42−232323−23−4i4i4i−4i1-2-22-2-122-112-200003−3−33−232323−230000002i−2i−2i2i2i3−2i3−2i32i3000000002i−2i−2i2i1-11-1003−3−33000000000000−iii−i−i3i3i3−i300000000000000000000000000000000000000
χ1358888-4-4-488-4-4-4-4-4-4-4-4-4-4-4-4-40000222222222222-4-4-4-42222222200000000000000000000000000-1-1-1-122-1-1-1-10000000000000000000000000000000000000000000000000000000000
χ1368888-4-4-4-8-8-4-4-4-4-4-4-4-4-4444400002222222222224444-2-2-2-2-2-2-2-200000000000000000000000000-1-1-1-1-2-211110000000000000000000000000000000000000000000000000000000000
χ13788-8-8-4-480044-448-8-84-400000000-4-422-244-2-44-440000000000000000000000000000000000000022-2-20000000000000000000000000000000000000000000000000000000000000000
χ13888-8-8-4-4-40044-44-4444-44343−43−4300002222-2-2-2-22-22-20000−23−232323−23−23232300000000000000000000000000-1-1110033−3−30000000000000000000000000000000000000000000000000000000000
χ13988-8-8-4-4-40044-44-4444-4−43−43434300002222-2-2-2-22-22-200002323−23−232323−23−2300000000000000000000000000-1-11100−3−3330000000000000000000000000000000000000000000000000000000000
χ1408-88-8-4-4-48-8-444444-4-44-44-440000222-2-2222-2-2-2-2-44-442-22-22-22-200000000000000000000000000-111-12-2-11-110000000000000000000000000000000000000000000000000000000000
χ1418-88-8-4-4-4-88-444444-4-444-44-40000222-2-2222-2-2-2-24-44-4-22-22-22-2200000000000000000000000000-111-1-221-11-10000000000000000000000000000000000000000000000000000000000
χ1428-8-88-4-48004-44-4-88-84400000000-4-42-2244-24-44-4000000000000000000000000000000000000002-22-20000000000000000000000000000000000000000000000000000000000000000
χ1438-8-88-4-4-4004-44-44-444443−43−43430000222-22-2-2-2-22-220000−232323−23−232323−2300000000000000000000000000-11-11003−3−330000000000000000000000000000000000000000000000000000000000
χ1448-8-88-4-4-4004-44-44-4444−434343−430000222-22-2-2-2-22-22000023−23−232323−23−232300000000000000000000000000-11-1100−333−30000000000000000000000000000000000000000000000000000000000

Here i is the imaginary unit (i² = −1).