direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D9, C4⋊1D18, C36⋊C22, D36⋊3C2, C12.5D6, C22⋊2D18, D18⋊2C22, C18.5C23, Dic9⋊1C22, C9⋊2(C2×D4), C3.(S3×D4), (D4×C9)⋊2C2, (C4×D9)⋊1C2, C9⋊D4⋊1C2, (C2×C18)⋊C22, (C2×C6).2D6, (C3×D4).3S3, (C22×D9)⋊2C2, C2.6(C22×D9), C6.23(C22×S3), SmallGroup(144,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D9
G = < a,b,c,d | a4=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 355 in 81 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C9, Dic3, C12, D6, C2×C6, C2×D4, D9, D9, C18, C18, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, Dic9, C36, D18, D18, D18, C2×C18, S3×D4, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, D4×D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, D18, S3×D4, C22×D9, D4×D9
Character table of D4×D9
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | 36A | 36B | 36C | |
| size | 1 | 1 | 2 | 2 | 9 | 9 | 18 | 18 | 2 | 2 | 18 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | -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 | -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 | 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 | 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 | -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 | -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 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | -2 | -2 | -1 | -1 | -1 | 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 | 0 | 2 | -2 | 2 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 2 | -2 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | -1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ16 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | -1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ17 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | -1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ18 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | 1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ21 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | 1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | 1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ23 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | 1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | 1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ25 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | 1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | -2 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 0 | -2ζ97-2ζ92 | -2ζ98-2ζ9 | -2ζ95-2ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 2ζ97+2ζ92 | 0 | -2ζ95-2ζ94 | -2ζ97-2ζ92 | -2ζ98-2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2ζ97+2ζ92 | 2ζ98+2ζ9 | 2ζ95+2ζ94 | 0 | -2ζ98-2ζ9 | -2ζ95-2ζ94 | -2ζ97-2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal 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 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 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), (1,23)(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), (1,2,3,4,5,6,7,8,9)(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), (1,23)(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), (1,2,3,4,5,6,7,8,9)(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)], [(1,23),(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)], [(1,2,3,4,5,6,7,8,9),(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)]])
D4×D9 is a maximal subgroup of
D8⋊D9 D72⋊C2 D4⋊6D18 D4⋊8D18 C36⋊D6 D18⋊D6
D4×D9 is a maximal quotient of
C22⋊2Dic18 Dic9⋊4D4 C22⋊3D36 C23.9D18 D18⋊D4 Dic9.D4 C36⋊Q8 D36⋊C4 D18.D4 C4⋊D36 D18⋊Q8 D8⋊D9 D8⋊3D9 D72⋊C2 SD16⋊D9 SD16⋊3D9 Q16⋊D9 D72⋊5C2 C23⋊2D18 C36⋊2D4 Dic9⋊D4 C36⋊D4 C36⋊D6 D18⋊D6
Matrix representation of D4×D9 ►in GL4(𝔽37) generated by
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 20 |
| 0 | 0 | 17 | 11 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 17 | 6 |
| 0 | 0 | 26 | 20 |
G:=sub<GL(4,GF(37))| [0,36,0,0,1,0,0,0,0,0,36,0,0,0,0,36],[0,1,0,0,1,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,31,17,0,0,20,11],[36,0,0,0,0,36,0,0,0,0,17,26,0,0,6,20] >;
D4×D9 in GAP, Magma, Sage, TeX
D_4\times D_9
% in TeX
G:=Group("D4xD9"); // GroupNames label
G:=SmallGroup(144,41);
// by ID
G=gap.SmallGroup(144,41);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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