metacyclic, supersoluble, monomial
Aliases: C9⋊C12, C18.C6, Dic9⋊C3, 3- 1+2⋊C4, C32.Dic3, C2.(C9⋊C6), C6.3(C3×S3), (C3×C6).2S3, C3.3(C3×Dic3), (C2×3- 1+2).C2, SmallGroup(108,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C12 |
Generators and relations for C9⋊C12
G = < a,b | a9=b12=1, bab-1=a5 >
Character table of C9⋊C12
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | 18B | 18C | |
| size | 1 | 1 | 2 | 3 | 3 | 9 | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | ζ32 | ζ3 | i | -i | -1 | ζ65 | ζ6 | ζ3 | ζ32 | 1 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ6 | ζ65 | -1 | linear of order 12 |
| ρ10 | 1 | -1 | 1 | ζ3 | ζ32 | i | -i | -1 | ζ6 | ζ65 | ζ32 | ζ3 | 1 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ65 | ζ6 | -1 | linear of order 12 |
| ρ11 | 1 | -1 | 1 | ζ32 | ζ3 | -i | i | -1 | ζ65 | ζ6 | ζ3 | ζ32 | 1 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ6 | ζ65 | -1 | linear of order 12 |
| ρ12 | 1 | -1 | 1 | ζ3 | ζ32 | -i | i | -1 | ζ6 | ζ65 | ζ32 | ζ3 | 1 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ65 | ζ6 | -1 | linear of order 12 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | -1 | complex lifted from C3×S3 |
| ρ16 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | -1 | complex lifted from C3×S3 |
| ρ17 | 2 | -2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -2 | 1+√-3 | 1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 1 | complex lifted from C3×Dic3 |
| ρ18 | 2 | -2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -2 | 1-√-3 | 1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 1 | complex lifted from C3×Dic3 |
| ρ19 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ20 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 36 points
Generators in S36
(1 29 21 11 25 13 5 33 17)(2 14 30 6 22 34 12 18 26)(3 35 15 9 31 19 7 27 23)(4 20 36 8 16 28 10 24 32)
(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)
G:=sub<Sym(36)| (1,29,21,11,25,13,5,33,17)(2,14,30,6,22,34,12,18,26)(3,35,15,9,31,19,7,27,23)(4,20,36,8,16,28,10,24,32), (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)>;
G:=Group( (1,29,21,11,25,13,5,33,17)(2,14,30,6,22,34,12,18,26)(3,35,15,9,31,19,7,27,23)(4,20,36,8,16,28,10,24,32), (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) );
G=PermutationGroup([[(1,29,21,11,25,13,5,33,17),(2,14,30,6,22,34,12,18,26),(3,35,15,9,31,19,7,27,23),(4,20,36,8,16,28,10,24,32)], [(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)]])
C9⋊C12 is a maximal subgroup of
C36.C6 C4×C9⋊C6 Dic9⋊C6 C33⋊Dic3 He3.3Dic3 3- 1+2.Dic3 C33.Dic3 He3.4Dic3 C32.CSU2(𝔽3) C62.Dic3 Dic9.A4 Dic9⋊A4
C9⋊C12 is a maximal quotient of
C9⋊C24 C32⋊Dic9 C9⋊C36 C33.Dic3 C62.Dic3 Dic9⋊A4
Matrix representation of C9⋊C12 ►in GL8(𝔽37)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 25 |
| 0 | 0 | 0 | 0 | 33 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 4 | 0 | 0 |
G:=sub<GL(8,GF(37))| [0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0],[1,19,0,0,0,0,0,0,20,36,0,0,0,0,0,0,0,0,33,8,0,0,0,0,0,0,12,4,0,0,0,0,0,0,0,0,0,0,33,8,0,0,0,0,0,0,12,4,0,0,0,0,12,4,0,0,0,0,0,0,29,25,0,0] >;
C9⋊C12 in GAP, Magma, Sage, TeX
C_9\rtimes C_{12} % in TeX
G:=Group("C9:C12"); // GroupNames label
G:=SmallGroup(108,9);
// by ID
G=gap.SmallGroup(108,9);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-3,30,1203,488,138,1804]);
// Polycyclic
G:=Group<a,b|a^9=b^12=1,b*a*b^-1=a^5>;
// generators/relations
Export
Subgroup lattice of C9⋊C12 in TeX
Character table of C9⋊C12 in TeX