metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic9, C9⋊C4, C2.D9, C18.C2, C6.1S3, C3.Dic3, SmallGroup(36,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — Dic9 |
Generators and relations for Dic9
G = < a,b | a18=1, b2=a9, bab-1=a-1 >
Character table of Dic9
| class | 1 | 2 | 3 | 4A | 4B | 6 | 9A | 9B | 9C | 18A | 18B | 18C | |
| size | 1 | 1 | 2 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | i | -i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -i | i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | 0 | 0 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ7 | 2 | 2 | -1 | 0 | 0 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | -1 | 0 | 0 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | -2 | 2 | 0 | 0 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | -2 | -1 | 0 | 0 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | symplectic faithful, Schur index 2 |
| ρ11 | 2 | -2 | -1 | 0 | 0 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | symplectic faithful, Schur index 2 |
| ρ12 | 2 | -2 | -1 | 0 | 0 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | symplectic faithful, Schur index 2 |
►Regular action on 36 points
Generators in S36
(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 24 10 33)(2 23 11 32)(3 22 12 31)(4 21 13 30)(5 20 14 29)(6 19 15 28)(7 36 16 27)(8 35 17 26)(9 34 18 25)
G:=sub<Sym(36)| (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), (1,24,10,33)(2,23,11,32)(3,22,12,31)(4,21,13,30)(5,20,14,29)(6,19,15,28)(7,36,16,27)(8,35,17,26)(9,34,18,25)>;
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), (1,24,10,33)(2,23,11,32)(3,22,12,31)(4,21,13,30)(5,20,14,29)(6,19,15,28)(7,36,16,27)(8,35,17,26)(9,34,18,25) );
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)], [(1,24,10,33),(2,23,11,32),(3,22,12,31),(4,21,13,30),(5,20,14,29),(6,19,15,28),(7,36,16,27),(8,35,17,26),(9,34,18,25)]])
Dic9 is a maximal subgroup of
C4×D9 C9⋊D4 C9⋊C12 C9⋊Dic3 Q8.D9 C6.S4 C9⋊F5 C32⋊3Dic9 C13⋊Dic9
Dic9p: Dic18 Dic27 Dic45 Dic63 Dic99 Dic117 ...
Dic9 is a maximal quotient of
C6.S4 C9⋊F5 C32⋊3Dic9 C13⋊Dic9
C2p.D9: C9⋊C8 Dic27 C9⋊Dic3 Dic45 Dic63 Dic99 Dic117 ...
Matrix representation of Dic9 ►in GL2(𝔽17) generated by
| 2 | 10 |
| 10 | 8 |
| 13 | 1 |
| 0 | 4 |
G:=sub<GL(2,GF(17))| [2,10,10,8],[13,0,1,4] >;
Dic9 in GAP, Magma, Sage, TeX
{\rm Dic}_9 % in TeX
G:=Group("Dic9"); // GroupNames label
G:=SmallGroup(36,1);
// by ID
G=gap.SmallGroup(36,1);
# by ID
G:=PCGroup([4,-2,-2,-3,-3,8,242,82,387]);
// Polycyclic
G:=Group<a,b|a^18=1,b^2=a^9,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic9 in TeX
Character table of Dic9 in TeX