direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22xD9, C9:C23, C18:C22, C6.11D6, (C2xC18):3C2, (C2xC6).4S3, C3.(C22xS3), SmallGroup(72,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C22xD9 |
Generators and relations for C22xD9
G = < a,b,c,d | a2=b2=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Quotients: C1, C2, C22, S3, C23, D6, D9, C22xS3, D18, C22xD9
Character table of C22xD9
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 9A | 9B | 9C | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 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 | 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 | 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 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
►On 36 points
Generators in S36
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 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 31)(2 30)(3 29)(4 28)(5 36)(6 35)(7 34)(8 33)(9 32)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(18 27)
G:=sub<Sym(36)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (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), (1,31)(2,30)(3,29)(4,28)(5,36)(6,35)(7,34)(8,33)(9,32)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(18,27)>;
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (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), (1,31)(2,30)(3,29)(4,28)(5,36)(6,35)(7,34)(8,33)(9,32)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(18,27) );
G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,36),(6,35),(7,34),(8,33),(9,32),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(18,27)]])
C22xD9 is a maximal subgroup of
D18:C4 D9:A4
C22xD9 is a maximal quotient of D36:5C2 D4:2D9 Q8:3D9
Matrix representation of C22xD9 ►in GL3(F19) generated by
| 18 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 0 | 0 |
| 0 | 7 | 14 |
| 0 | 5 | 2 |
| 1 | 0 | 0 |
| 0 | 14 | 17 |
| 0 | 12 | 5 |
G:=sub<GL(3,GF(19))| [18,0,0,0,1,0,0,0,1],[1,0,0,0,18,0,0,0,18],[1,0,0,0,7,5,0,14,2],[1,0,0,0,14,12,0,17,5] >;
C22xD9 in GAP, Magma, Sage, TeX
C_2^2\times D_9
% in TeX
G:=Group("C2^2xD9"); // GroupNames label
G:=SmallGroup(72,17);
// by ID
G=gap.SmallGroup(72,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,803,138,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C22xD9 in TeX
Character table of C22xD9 in TeX