direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D9, C36⋊2C2, D18.C2, C4○Dic9, C6.7D6, C12.5S3, C2.1D18, Dic9⋊2C2, C18.2C22, C9⋊1(C2×C4), C3.(C4×S3), SmallGroup(72,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C4×D9 |
Generators and relations for C4×D9
G = < a,b,c | a4=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C4×D9
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 9 | 9 | 2 | 1 | 1 | 9 | 9 | 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 | i | -i | i | -i | -1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | i | i | -i | -i | -i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -i | -i | i | i | i | -i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -i | -i | i | i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | i | i | -i | -i | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ14 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ15 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | -1 | -1 | -1 | -2i | 2i | 1 | 1 | 1 | i | i | -i | -i | -i | i | complex lifted from C4×S3 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | -1 | -1 | -1 | 2i | -2i | 1 | 1 | 1 | -i | -i | i | i | i | -i | complex lifted from C4×S3 |
| ρ19 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | i | -i | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ43ζ97+ζ43ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | ζ4ζ98+ζ4ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ98+ζ43ζ9 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | i | -i | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ43ζ98+ζ43ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | ζ4ζ95+ζ4ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ95+ζ43ζ94 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -i | i | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ4ζ95+ζ4ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | ζ43ζ97+ζ43ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ97+ζ4ζ92 | complex faithful |
| ρ22 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -i | i | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ4ζ97+ζ4ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | ζ43ζ98+ζ43ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ98+ζ4ζ9 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -i | i | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ4ζ98+ζ4ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | ζ43ζ95+ζ43ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ95+ζ4ζ94 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | i | -i | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ43ζ95+ζ43ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | ζ4ζ97+ζ4ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ97+ζ43ζ92 | complex faithful |
►On 36 points
Generators in S36
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)
(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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)
G:=sub<Sym(36)| (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22), (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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)>;
G:=Group( (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22), (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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36) );
G=PermutationGroup([[(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,18),(6,17),(7,16),(8,15),(9,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36)]])
C4×D9 is a maximal subgroup of
C8⋊D9 D36⋊5C2 D4⋊2D9 Q8⋊3D9 C18.D6 C12.11S4 D90.C2
C4×D9 is a maximal quotient of C8⋊D9 Dic9⋊C4 D18⋊C4 C18.D6 D90.C2
Matrix representation of C4×D9 ►in GL2(𝔽17) generated by
| 13 | 0 |
| 0 | 13 |
| 0 | 4 |
| 4 | 14 |
| 3 | 15 |
| 4 | 14 |
G:=sub<GL(2,GF(17))| [13,0,0,13],[0,4,4,14],[3,4,15,14] >;
C4×D9 in GAP, Magma, Sage, TeX
C_4\times D_9
% in TeX
G:=Group("C4xD9"); // GroupNames label
G:=SmallGroup(72,5);
// by ID
G=gap.SmallGroup(72,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,26,803,138,1204]);
// Polycyclic
G:=Group<a,b,c|a^4=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C4×D9 in TeX
Character table of C4×D9 in TeX