metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D9, C9⋊2D8, D36⋊2C2, C18.8D4, C4.2D18, C12.2D6, C36.2C22, C9⋊C8⋊2C2, (D4×C9)⋊1C2, C3.(D4⋊S3), (C3×D4).2S3, C2.5(C9⋊D4), C6.15(C3⋊D4), SmallGroup(144,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D9
G = < a,b,c,d | a4=b2=c9=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Character table of D4⋊D9
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | 36A | 36B | 36C | |
| size | 1 | 1 | 4 | 36 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | √2 | -√2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ11 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | -√2 | √2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ13 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ14 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ16 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | -√-3 | -√-3 | √-3 | √-3 | √-3 | -√-3 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -1 | -1 | -1 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | √-3 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | √-3 | -√-3 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95+ζ94 | -ζ98+ζ9 | ζ95-ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex lifted from C9⋊D4 |
| ρ19 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | -√-3 | √-3 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95-ζ94 | ζ98-ζ9 | -ζ95+ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex lifted from C9⋊D4 |
| ρ20 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | -√-3 | √-3 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98-ζ9 | -ζ97+ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ97-ζ92 | ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex lifted from C9⋊D4 |
| ρ21 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | √-3 | -√-3 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97-ζ92 | -ζ95+ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex lifted from C9⋊D4 |
| ρ22 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | √-3 | -√-3 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98+ζ9 | ζ97-ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ97+ζ92 | -ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex lifted from C9⋊D4 |
| ρ23 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | -√-3 | √-3 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97+ζ92 | ζ95-ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex lifted from C9⋊D4 |
| ρ24 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 0 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 0 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | -2ζ97-2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 0 | -2ζ97-2ζ92 | -2ζ95-2ζ94 | -2ζ98-2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 72 points
Generators in S72
(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)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 64)(7 65)(8 66)(9 67)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)
G:=sub<Sym(72)| (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)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,68)(2,69)(3,70)(4,71)(5,72)(6,64)(7,65)(8,66)(9,67)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)>;
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)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,68)(2,69)(3,70)(4,71)(5,72)(6,64)(7,65)(8,66)(9,67)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72) );
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),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,64),(7,65),(8,66),(9,67),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72)]])
D4⋊D9 is a maximal subgroup of
D8×D9 D8⋊D9 D72⋊C2 SD16⋊3D9 D36⋊6C22 D4⋊D18 D4.9D18 D4⋊D27 D36⋊S3 C9⋊D24 D36⋊C6 C36.18D6
D4⋊D9 is a maximal quotient of
C36.Q8 C18.D8 C9⋊D16 D8.D9 C9⋊SD32 C9⋊Q32 D4⋊Dic9 D4⋊D27 D36⋊S3 C9⋊D24 C36.18D6
Matrix representation of D4⋊D9 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 48 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 48 |
| 0 | 0 | 35 | 0 |
| 45 | 31 | 0 | 0 |
| 42 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 42 | 3 | 0 | 0 |
| 45 | 31 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,48,0,0,3,72],[1,0,0,0,0,1,0,0,0,0,0,35,0,0,48,0],[45,42,0,0,31,3,0,0,0,0,1,0,0,0,0,1],[42,45,0,0,3,31,0,0,0,0,1,48,0,0,0,72] >;
D4⋊D9 in GAP, Magma, Sage, TeX
D_4\rtimes D_9
% in TeX
G:=Group("D4:D9"); // GroupNames label
G:=SmallGroup(144,16);
// by ID
G=gap.SmallGroup(144,16);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^9=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of D4⋊D9 in TeX
Character table of D4⋊D9 in TeX