metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic18, C9⋊Q8, C4.D9, C6.6D6, C3.Dic6, C36.1C2, C12.1S3, C2.3D18, Dic9.C2, C18.1C22, SmallGroup(72,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic18
G = < a,b | a36=1, b2=a18, bab-1=a-1 >
Character table of Dic18
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 2 | 2 | 18 | 18 | 2 | 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 | trivial |
| ρ2 | 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 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 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 |
| ρ5 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ9 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ11 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ13 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | √3 | -√3 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -√3 | √3 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -√3 | √3 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ43ζ98-ζ43ζ9 | ζ4ζ95-ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | symplectic faithful, Schur index 2 |
| ρ17 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -√3 | √3 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ4ζ95+ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ97-ζ4ζ92 | symplectic faithful, Schur index 2 |
| ρ18 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | √3 | -√3 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ4ζ97+ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | symplectic faithful, Schur index 2 |
| ρ19 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | √3 | -√3 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ95-ζ4ζ94 | symplectic faithful, Schur index 2 |
| ρ20 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -√3 | √3 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ4ζ97-ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ95-ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ43ζ98-ζ43ζ9 | symplectic faithful, Schur index 2 |
| ρ21 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | √3 | -√3 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ4ζ95-ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | symplectic faithful, Schur index 2 |
►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 71 19 53)(2 70 20 52)(3 69 21 51)(4 68 22 50)(5 67 23 49)(6 66 24 48)(7 65 25 47)(8 64 26 46)(9 63 27 45)(10 62 28 44)(11 61 29 43)(12 60 30 42)(13 59 31 41)(14 58 32 40)(15 57 33 39)(16 56 34 38)(17 55 35 37)(18 54 36 72)
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), (1,71,19,53)(2,70,20,52)(3,69,21,51)(4,68,22,50)(5,67,23,49)(6,66,24,48)(7,65,25,47)(8,64,26,46)(9,63,27,45)(10,62,28,44)(11,61,29,43)(12,60,30,42)(13,59,31,41)(14,58,32,40)(15,57,33,39)(16,56,34,38)(17,55,35,37)(18,54,36,72)>;
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), (1,71,19,53)(2,70,20,52)(3,69,21,51)(4,68,22,50)(5,67,23,49)(6,66,24,48)(7,65,25,47)(8,64,26,46)(9,63,27,45)(10,62,28,44)(11,61,29,43)(12,60,30,42)(13,59,31,41)(14,58,32,40)(15,57,33,39)(16,56,34,38)(17,55,35,37)(18,54,36,72) );
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)], [(1,71,19,53),(2,70,20,52),(3,69,21,51),(4,68,22,50),(5,67,23,49),(6,66,24,48),(7,65,25,47),(8,64,26,46),(9,63,27,45),(10,62,28,44),(11,61,29,43),(12,60,30,42),(13,59,31,41),(14,58,32,40),(15,57,33,39),(16,56,34,38),(17,55,35,37),(18,54,36,72)]])
Dic18 is a maximal subgroup of
Dic36 C72⋊C2 D4.D9 C9⋊Q16 D36⋊5C2 D4⋊2D9 Q8×D9 Dic54 C9⋊Dic6 C36.C6 C12.D9 C12.1S4 C12.3S4 C45⋊Q8 Dic90
Dic18 is a maximal quotient of
Dic9⋊C4 C4⋊Dic9 Dic54 C9⋊Dic6 C12.D9 C12.1S4 C45⋊Q8 Dic90
Matrix representation of Dic18 ►in GL2(𝔽37) generated by
| 17 | 6 |
| 6 | 0 |
| 15 | 25 |
| 25 | 22 |
G:=sub<GL(2,GF(37))| [17,6,6,0],[15,25,25,22] >;
Dic18 in GAP, Magma, Sage, TeX
{\rm Dic}_{18} % in TeX
G:=Group("Dic18"); // GroupNames label
G:=SmallGroup(72,4);
// by ID
G=gap.SmallGroup(72,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,20,61,26,803,138,1204]);
// Polycyclic
G:=Group<a,b|a^36=1,b^2=a^18,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic18 in TeX
Character table of Dic18 in TeX