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## G = D4⋊2D9order 144 = 24·32

### The semidirect product of D4 and D9 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4⋊2D9
 Chief series C1 — C3 — C9 — C18 — D18 — C4×D9 — D4⋊2D9
 Lower central C9 — C18 — D4⋊2D9
 Upper central C1 — C2 — D4

Generators and relations for D42D9
G = < a,b,c,d | a4=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 199 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C4○D4, D9, C18, C18, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, Dic9, Dic9, C36, D18, C2×C18, D42S3, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, D42D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, D42S3, C22×D9, D42D9

Character table of D42D9

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 9A 9B 9C 12 18A 18B 18C 18D 18E 18F 18G 18H 18I 36A 36B 36C size 1 1 2 2 18 2 2 9 9 18 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 -2 2 0 2 -2 0 0 0 0 2 -2 2 -1 -1 -1 -2 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 0 2 2 0 0 0 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 2 2 0 0 0 0 2 2 2 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 -2 0 2 -2 0 0 0 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 2 2 0 -1 2 0 0 0 0 -1 -1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ14 2 2 2 2 0 -1 2 0 0 0 0 -1 -1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ15 2 2 -2 -2 0 -1 2 0 0 0 0 -1 1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D18 ρ16 2 2 -2 2 0 -1 -2 0 0 0 0 -1 1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ17 2 2 2 -2 0 -1 -2 0 0 0 0 -1 -1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ18 2 2 2 -2 0 -1 -2 0 0 0 0 -1 -1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ19 2 2 -2 -2 0 -1 2 0 0 0 0 -1 1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D18 ρ20 2 2 -2 2 0 -1 -2 0 0 0 0 -1 1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ21 2 2 2 2 0 -1 2 0 0 0 0 -1 -1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ22 2 2 -2 2 0 -1 -2 0 0 0 0 -1 1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ23 2 2 2 -2 0 -1 -2 0 0 0 0 -1 -1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ24 2 2 -2 -2 0 -1 2 0 0 0 0 -1 1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D18 ρ25 2 -2 0 0 0 2 0 2i -2i 0 0 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 4 -4 0 0 0 4 0 0 0 0 0 -4 0 0 -2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ28 4 -4 0 0 0 -2 0 0 0 0 0 2 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 0 -2ζ97-2ζ92 -2ζ98-2ζ9 -2ζ95-2ζ94 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ29 4 -4 0 0 0 -2 0 0 0 0 0 2 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 0 -2ζ95-2ζ94 -2ζ97-2ζ92 -2ζ98-2ζ9 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ30 4 -4 0 0 0 -2 0 0 0 0 0 2 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 0 -2ζ98-2ζ9 -2ζ95-2ζ94 -2ζ97-2ζ92 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D42D9
On 72 points
Generators in S72
```(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(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 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)```

`G:=sub<Sym(72)| (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)>;`

`G:=Group( (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72) );`

`G=PermutationGroup([[(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,55),(7,56),(8,57),(9,58),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(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),(37,44),(38,43),(39,42),(40,41),(46,53),(47,52),(48,51),(49,50),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72)]])`

Matrix representation of D42D9 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 31 0 0 0 0 6
,
 36 0 0 0 0 36 0 0 0 0 0 6 0 0 31 0
,
 20 6 0 0 31 26 0 0 0 0 1 0 0 0 0 1
,
 17 26 0 0 6 20 0 0 0 0 1 0 0 0 0 36
`G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,31,0,0,0,0,6],[36,0,0,0,0,36,0,0,0,0,0,31,0,0,6,0],[20,31,0,0,6,26,0,0,0,0,1,0,0,0,0,1],[17,6,0,0,26,20,0,0,0,0,1,0,0,0,0,36] >;`

D42D9 in GAP, Magma, Sage, TeX

`D_4\rtimes_2D_9`
`% in TeX`

`G:=Group("D4:2D9");`
`// GroupNames label`

`G:=SmallGroup(144,42);`
`// by ID`

`G=gap.SmallGroup(144,42);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,2404,208,3461]);`
`// 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,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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