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## G = D4⋊D9order 144 = 24·32

### The semidirect product of D4 and D9 acting via D9/C9=C2

Aliases: D4⋊D9, C92D8, D362C2, C18.8D4, C4.2D18, C12.2D6, C36.2C22, C9⋊C82C2, (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

 Derived series C1 — C36 — D4⋊D9
 Chief series C1 — C3 — C9 — C18 — C36 — D36 — D4⋊D9
 Lower central C9 — C18 — C36 — D4⋊D9
 Upper central C1 — C2 — C4 — D4

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

Smallest permutation representation of D4⋊D9
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  SD163D9  D366C22  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`

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