metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9⋊C8, C18.C4, C4.2D9, C2.Dic9, C36.2C2, C12.4S3, C6.1Dic3, C3.(C3⋊C8), SmallGroup(72,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C8 |
Generators and relations for C9⋊C8
G = < a,b | a9=b8=1, bab-1=a-1 >
Character table of C9⋊C8
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | 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 | -i | i | -i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | -i | i | -1 | ζ83 | ζ8 | ζ87 | ζ85 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -i | -i | i | i | i | -i | linear of order 8 |
| ρ6 | 1 | -1 | 1 | i | -i | -1 | ζ8 | ζ83 | ζ85 | ζ87 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | i | i | -i | -i | -i | i | linear of order 8 |
| ρ7 | 1 | -1 | 1 | -i | i | -1 | ζ87 | ζ85 | ζ83 | ζ8 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -i | -i | i | i | i | -i | linear of order 8 |
| ρ8 | 1 | -1 | 1 | i | -i | -1 | ζ85 | ζ87 | ζ8 | ζ83 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | i | i | -i | -i | -i | i | linear of order 8 |
| ρ9 | 2 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ13 | 2 | 2 | -1 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ14 | 2 | 2 | -1 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | symplectic lifted from Dic9, Schur index 2 |
| ρ15 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | 2 | -1 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ17 | 2 | -2 | 2 | 2i | -2i | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2i | -2i | 1 | 1 | 1 | -i | -i | i | i | i | -i | complex lifted from C3⋊C8 |
| ρ18 | 2 | -2 | 2 | -2i | 2i | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2i | 2i | 1 | 1 | 1 | i | i | -i | -i | -i | i | complex lifted from C3⋊C8 |
| ρ19 | 2 | -2 | -1 | -2i | 2i | 1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | i | -i | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | ζ43ζ97+ζ43ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | ζ4ζ98+ζ4ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ98+ζ43ζ9 | complex faithful |
| ρ20 | 2 | -2 | -1 | 2i | -2i | 1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -i | i | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | ζ4ζ98+ζ4ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | ζ43ζ95+ζ43ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ95+ζ4ζ94 | complex faithful |
| ρ21 | 2 | -2 | -1 | 2i | -2i | 1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -i | i | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | ζ4ζ95+ζ4ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | ζ43ζ97+ζ43ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ97+ζ4ζ92 | complex faithful |
| ρ22 | 2 | -2 | -1 | -2i | 2i | 1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | i | -i | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | ζ43ζ95+ζ43ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | ζ4ζ97+ζ4ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ97+ζ43ζ92 | complex faithful |
| ρ23 | 2 | -2 | -1 | -2i | 2i | 1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | i | -i | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | ζ43ζ98+ζ43ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | ζ4ζ95+ζ4ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ95+ζ43ζ94 | complex faithful |
| ρ24 | 2 | -2 | -1 | 2i | -2i | 1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -i | i | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | ζ4ζ97+ζ4ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | ζ43ζ98+ζ43ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ98+ζ4ζ9 | complex faithful |
►Regular action on 72 points
Generators in S72
(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 70 29 52 11 61 20 43)(2 69 30 51 12 60 21 42)(3 68 31 50 13 59 22 41)(4 67 32 49 14 58 23 40)(5 66 33 48 15 57 24 39)(6 65 34 47 16 56 25 38)(7 64 35 46 17 55 26 37)(8 72 36 54 18 63 27 45)(9 71 28 53 10 62 19 44)
G:=sub<Sym(72)| (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,70,29,52,11,61,20,43)(2,69,30,51,12,60,21,42)(3,68,31,50,13,59,22,41)(4,67,32,49,14,58,23,40)(5,66,33,48,15,57,24,39)(6,65,34,47,16,56,25,38)(7,64,35,46,17,55,26,37)(8,72,36,54,18,63,27,45)(9,71,28,53,10,62,19,44)>;
G:=Group( (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,70,29,52,11,61,20,43)(2,69,30,51,12,60,21,42)(3,68,31,50,13,59,22,41)(4,67,32,49,14,58,23,40)(5,66,33,48,15,57,24,39)(6,65,34,47,16,56,25,38)(7,64,35,46,17,55,26,37)(8,72,36,54,18,63,27,45)(9,71,28,53,10,62,19,44) );
G=PermutationGroup([[(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,70,29,52,11,61,20,43),(2,69,30,51,12,60,21,42),(3,68,31,50,13,59,22,41),(4,67,32,49,14,58,23,40),(5,66,33,48,15,57,24,39),(6,65,34,47,16,56,25,38),(7,64,35,46,17,55,26,37),(8,72,36,54,18,63,27,45),(9,71,28,53,10,62,19,44)]])
C9⋊C8 is a maximal subgroup of
C8×D9 C8⋊D9 C4.Dic9 D4.D9 D4⋊D9 C9⋊Q16 Q8⋊2D9 C27⋊C8 C9⋊C24 C36.S3 C12.S4 C12.9S4 C45⋊3C8 C45⋊C8
C9⋊C8 is a maximal quotient of
C9⋊C16 C27⋊C8 C36.S3 C12.S4 C45⋊3C8 C45⋊C8
Matrix representation of C9⋊C8 ►in GL2(𝔽17) generated by
| 6 | 16 |
| 4 | 8 |
| 8 | 4 |
| 0 | 9 |
G:=sub<GL(2,GF(17))| [6,4,16,8],[8,0,4,9] >;
C9⋊C8 in GAP, Magma, Sage, TeX
C_9\rtimes C_8
% in TeX
G:=Group("C9:C8"); // GroupNames label
G:=SmallGroup(72,1);
// by ID
G=gap.SmallGroup(72,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,10,26,803,138,1204]);
// Polycyclic
G:=Group<a,b|a^9=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C9⋊C8 in TeX
Character table of C9⋊C8 in TeX