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## G = D4.D9order 144 = 24·32

### The non-split extension by D4 of D9 acting via D9/C9=C2

Aliases: D4.D9, C92SD16, C4.1D18, C18.7D4, C12.1D6, Dic182C2, C36.1C22, C9⋊C81C2, (C3×D4).1S3, (D4×C9).1C2, C3.(D4.S3), C2.4(C9⋊D4), C6.14(C3⋊D4), SmallGroup(144,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — D4.D9
 Chief series C1 — C3 — C9 — C18 — C36 — Dic18 — 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=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Character table of D4.D9

 class 1 2A 2B 3 4A 4B 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 2 2 36 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 2 2 0 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 2 2 2 0 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 ρ7 2 2 0 2 -2 0 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 ρ8 2 2 -2 -1 2 0 -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 ρ9 2 2 2 -1 2 0 -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 ρ10 2 2 -2 -1 2 0 -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 ρ11 2 2 2 -1 2 0 -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 ρ12 2 2 2 -1 2 0 -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 ρ13 2 2 -2 -1 2 0 -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 ρ14 2 -2 0 2 0 0 -2 0 0 √-2 -√-2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ15 2 -2 0 2 0 0 -2 0 0 -√-2 √-2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ16 2 2 0 2 -2 0 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 2 -2 0 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 -1 -2 0 -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 ρ19 2 2 0 -1 -2 0 -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 -1 -2 0 -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 ρ21 2 2 0 -1 -2 0 -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 ρ22 2 2 0 -1 -2 0 -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 -1 -2 0 -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 4 0 0 -4 0 0 0 0 -2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ25 4 -4 0 -2 0 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 symplectic faithful, Schur index 2 ρ26 4 -4 0 -2 0 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 symplectic faithful, Schur index 2 ρ27 4 -4 0 -2 0 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 symplectic 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 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)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 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 48 14 39)(2 47 15 38)(3 46 16 37)(4 54 17 45)(5 53 18 44)(6 52 10 43)(7 51 11 42)(8 50 12 41)(9 49 13 40)(19 70 28 61)(20 69 29 60)(21 68 30 59)(22 67 31 58)(23 66 32 57)(24 65 33 56)(25 64 34 55)(26 72 35 63)(27 71 36 62)```

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

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

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

D4.D9 is a maximal subgroup of
D8⋊D9  D83D9  SD16×D9  SD16⋊D9  D366C22  D4.D18  D4.9D18  D4.D27  D12.D9  C36.D6  Dic18⋊C6  C36.17D6
D4.D9 is a maximal quotient of
C4.Dic18  C18.Q16  D4⋊Dic9  D4.D27  D12.D9  C36.D6  C36.17D6

Matrix representation of D4.D9 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 70 0 0 25 1
,
 72 0 0 0 0 1 0 0 0 0 1 0 0 0 48 72
,
 32 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 0 16 0 0 32 0 0 0 0 0 0 55 0 0 69 0
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,25,0,0,70,1],[72,0,0,0,0,1,0,0,0,0,1,48,0,0,0,72],[32,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,32,0,0,16,0,0,0,0,0,0,69,0,0,55,0] >;`

D4.D9 in GAP, Magma, Sage, TeX

`D_4.D_9`
`% in TeX`

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

`G:=SmallGroup(144,15);`
`// by ID`

`G=gap.SmallGroup(144,15);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,218,116,50,2404,208,3461]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^9=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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