metabelian, soluble, monomial, A-group
Aliases: C32⋊3Dic9, C33.5Dic3, C3⋊S3.D9, C9⋊(C32⋊C4), (C32×C9)⋊2C4, C3.(C33⋊C4), (C3×C3⋊S3).3S3, (C9×C3⋊S3).2C2, SmallGroup(324,112)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×C9 — C32⋊3Dic9 |
Generators and relations for C32⋊3Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >
Character table of C32⋊3Dic9
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 4A | 4B | 6 | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 9M | 9N | 9O | 18A | 18B | 18C | |
| size | 1 | 9 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 81 | 81 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | |
| ρ1 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ7 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | -2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ11 | 2 | -2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | symplectic lifted from Dic9, Schur index 2 |
| ρ12 | 2 | -2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ13 | 4 | 0 | 4 | -2 | 1 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 4 | 4 | 4 | -2 | 1 | -2 | -2 | 1 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ14 | 4 | 0 | 4 | 1 | -2 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | 4 | 4 | 4 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ15 | 4 | 0 | 4 | -2 | 1 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | -2 | 1 | -1+3√-3/2 | 1 | 1 | -1-3√-3/2 | 1 | 1 | 1 | -1-3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ16 | 4 | 0 | 4 | 1 | -2 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -1+3√-3/2 | 1 | -1+3√-3/2 | -1+3√-3/2 | 1 | -1-3√-3/2 | -1-3√-3/2 | -1-3√-3/2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ17 | 4 | 0 | 4 | -2 | 1 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | -2 | 1 | -1-3√-3/2 | 1 | 1 | -1+3√-3/2 | 1 | 1 | 1 | -1+3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ18 | 4 | 0 | 4 | 1 | -2 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -1-3√-3/2 | 1 | -1-3√-3/2 | -1-3√-3/2 | 1 | -1+3√-3/2 | -1+3√-3/2 | -1+3√-3/2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ19 | 4 | 0 | -2 | -1+3√-3/2 | 1 | 1 | -2 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95-ζ94 | -ζ98-ζ9 | 2ζ98-ζ9 | -ζ97+2ζ92 | -ζ97-ζ92 | 2ζ97-ζ92 | -ζ98+2ζ9 | -ζ95+2ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | 0 | 0 | 0 | complex faithful |
| ρ20 | 4 | 0 | -2 | -1+3√-3/2 | 1 | 1 | -2 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | -ζ97+2ζ92 | -ζ95-ζ94 | 2ζ95-ζ94 | 2ζ98-ζ9 | -ζ98-ζ9 | -ζ98+2ζ9 | -ζ95+2ζ94 | 2ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | 0 | 0 | 0 | complex faithful |
| ρ21 | 4 | 0 | -2 | 1 | -1+3√-3/2 | -2 | 1 | 1 | -1-3√-3/2 | 0 | 0 | 0 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | -ζ98-ζ9 | -ζ97+2ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95+2ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | 2ζ97-ζ92 | 2ζ98-ζ9 | -ζ98+2ζ9 | 2ζ95-ζ94 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | 0 | -2 | -1+3√-3/2 | 1 | 1 | -2 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98-ζ9 | -ζ97-ζ92 | -ζ97+2ζ92 | 2ζ95-ζ94 | -ζ95-ζ94 | -ζ95+2ζ94 | 2ζ97-ζ92 | -ζ98+2ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | 0 | -2 | -1-3√-3/2 | 1 | 1 | -2 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97-ζ92 | -ζ95-ζ94 | -ζ95+2ζ94 | -ζ98+2ζ9 | -ζ98-ζ9 | 2ζ98-ζ9 | 2ζ95-ζ94 | -ζ97+2ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | 0 | -2 | 1 | -1-3√-3/2 | -2 | 1 | 1 | -1+3√-3/2 | 0 | 0 | 0 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | -ζ95-ζ94 | -ζ98+2ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97+2ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | 2ζ98-ζ9 | -ζ95+2ζ94 | 2ζ95-ζ94 | 2ζ97-ζ92 | 0 | 0 | 0 | complex faithful |
| ρ25 | 4 | 0 | -2 | -1-3√-3/2 | 1 | 1 | -2 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | -ζ98+2ζ9 | -ζ97-ζ92 | 2ζ97-ζ92 | -ζ95+2ζ94 | -ζ95-ζ94 | 2ζ95-ζ94 | -ζ97+2ζ92 | 2ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | 0 | -2 | 1 | -1-3√-3/2 | -2 | 1 | 1 | -1+3√-3/2 | 0 | 0 | 0 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | -ζ98-ζ9 | 2ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | 2ζ95-ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97+2ζ92 | -ζ98+2ζ9 | 2ζ98-ζ9 | -ζ95+2ζ94 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | 0 | -2 | -1-3√-3/2 | 1 | 1 | -2 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | -ζ95+2ζ94 | -ζ98-ζ9 | -ζ98+2ζ9 | 2ζ97-ζ92 | -ζ97-ζ92 | -ζ97+2ζ92 | 2ζ98-ζ9 | 2ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | 0 | 0 | 0 | complex faithful |
| ρ28 | 4 | 0 | -2 | 1 | -1+3√-3/2 | -2 | 1 | 1 | -1-3√-3/2 | 0 | 0 | 0 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | -ζ95-ζ94 | 2ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | 2ζ97-ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98+2ζ9 | 2ζ95-ζ94 | -ζ95+2ζ94 | -ζ97+2ζ92 | 0 | 0 | 0 | complex faithful |
| ρ29 | 4 | 0 | -2 | 1 | -1+3√-3/2 | -2 | 1 | 1 | -1-3√-3/2 | 0 | 0 | 0 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | -ζ97-ζ92 | 2ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98+2ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95+2ζ94 | -ζ97+2ζ92 | 2ζ97-ζ92 | 2ζ98-ζ9 | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | 0 | -2 | 1 | -1-3√-3/2 | -2 | 1 | 1 | -1+3√-3/2 | 0 | 0 | 0 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | -ζ97-ζ92 | -ζ95+2ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | 2ζ98-ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | 2ζ95-ζ94 | 2ζ97-ζ92 | -ζ97+2ζ92 | -ζ98+2ζ9 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)
(1 34 10 25)(2 33 11 24)(3 32 12 23)(4 31 13 22)(5 30 14 21)(6 29 15 20)(7 28 16 19)(8 27 17 36)(9 26 18 35)
G:=sub<Sym(36)| (19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18)(19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36), (1,34,10,25)(2,33,11,24)(3,32,12,23)(4,31,13,22)(5,30,14,21)(6,29,15,20)(7,28,16,19)(8,27,17,36)(9,26,18,35)>;
G:=Group( (19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18)(19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36), (1,34,10,25)(2,33,11,24)(3,32,12,23)(4,31,13,22)(5,30,14,21)(6,29,15,20)(7,28,16,19)(8,27,17,36)(9,26,18,35) );
G=PermutationGroup([[(19,25,31),(20,32,26),(21,27,33),(22,34,28),(23,29,35),(24,36,30)], [(1,13,7),(2,8,14),(3,15,9),(4,10,16),(5,17,11),(6,12,18),(19,25,31),(20,32,26),(21,27,33),(22,34,28),(23,29,35),(24,36,30)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)], [(1,34,10,25),(2,33,11,24),(3,32,12,23),(4,31,13,22),(5,30,14,21),(6,29,15,20),(7,28,16,19),(8,27,17,36),(9,26,18,35)]])
Matrix representation of C32⋊3Dic9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 26 |
| 10 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 26 |
| 0 | 34 | 0 | 0 |
| 34 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,26],[10,0,0,0,0,26,0,0,0,0,10,0,0,0,0,26],[0,34,0,0,34,0,0,0,0,0,0,12,0,0,12,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C32⋊3Dic9 in GAP, Magma, Sage, TeX
C_3^2\rtimes_3{\rm Dic}_9 % in TeX
G:=Group("C3^2:3Dic9"); // GroupNames label
G:=SmallGroup(324,112);
// by ID
G=gap.SmallGroup(324,112);
# by ID
G:=PCGroup([6,-2,-2,-3,3,-3,-3,12,362,80,387,297,5404,208,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊3Dic9 in TeX
Character table of C32⋊3Dic9 in TeX