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## G = C9⋊Q16order 144 = 24·32

### The semidirect product of C9 and Q16 acting via Q16/Q8=C2

Aliases: C92Q16, C18.9D4, C4.3D18, C12.3D6, Q8.2D9, C36.3C22, Dic18.2C2, C9⋊C8.C2, C3.(C3⋊Q16), (C3×Q8).5S3, (Q8×C9).1C2, C2.6(C9⋊D4), C6.16(C3⋊D4), SmallGroup(144,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C9⋊Q16
 Chief series C1 — C3 — C9 — C18 — C36 — Dic18 — C9⋊Q16
 Lower central C9 — C18 — C36 — C9⋊Q16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for C9⋊Q16
G = < a,b,c | a9=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

Character table of C9⋊Q16

 class 1 2 3 4A 4B 4C 6 8A 8B 9A 9B 9C 12A 12B 12C 18A 18B 18C 36A 36B 36C 36D 36E 36F 36G 36H 36I size 1 1 2 2 4 36 2 18 18 2 2 2 4 4 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 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 -2 0 2 0 0 -1 -1 -1 2 -2 -2 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ7 2 2 2 -2 0 0 2 0 0 2 2 2 -2 0 0 2 2 2 0 0 0 0 0 0 -2 -2 -2 orthogonal lifted from D4 ρ8 2 2 -1 2 -2 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 1 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D18 ρ9 2 2 -1 2 2 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ10 2 2 -1 2 -2 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 1 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D18 ρ11 2 2 -1 2 2 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ12 2 2 -1 2 2 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ13 2 2 -1 2 -2 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 1 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D18 ρ14 2 -2 2 0 0 0 -2 √2 -√2 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 2 0 0 0 -2 -√2 √2 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ16 2 2 2 -2 0 0 2 0 0 -1 -1 -1 -2 0 0 -1 -1 -1 -√-3 -√-3 -√-3 √-3 √-3 √-3 1 1 1 complex lifted from C3⋊D4 ρ17 2 2 -1 -2 0 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 1 -√-3 √-3 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98+ζ9 ζ97-ζ92 -ζ95+ζ94 -ζ97+ζ92 ζ98-ζ9 ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 complex lifted from C9⋊D4 ρ18 2 2 -1 -2 0 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 1 √-3 -√-3 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97+ζ92 ζ95-ζ94 ζ98-ζ9 -ζ95+ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 complex lifted from C9⋊D4 ρ19 2 2 -1 -2 0 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 1 -√-3 √-3 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97-ζ92 -ζ95+ζ94 -ζ98+ζ9 ζ95-ζ94 -ζ97+ζ92 ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 complex lifted from C9⋊D4 ρ20 2 2 2 -2 0 0 2 0 0 -1 -1 -1 -2 0 0 -1 -1 -1 √-3 √-3 √-3 -√-3 -√-3 -√-3 1 1 1 complex lifted from C3⋊D4 ρ21 2 2 -1 -2 0 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 1 √-3 -√-3 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95-ζ94 ζ98-ζ9 -ζ97+ζ92 -ζ98+ζ9 -ζ95+ζ94 ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 complex lifted from C9⋊D4 ρ22 2 2 -1 -2 0 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 1 √-3 -√-3 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98-ζ9 -ζ97+ζ92 ζ95-ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ95+ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 complex lifted from C9⋊D4 ρ23 2 2 -1 -2 0 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 1 -√-3 √-3 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95+ζ94 -ζ98+ζ9 ζ97-ζ92 ζ98-ζ9 ζ95-ζ94 -ζ97+ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 complex lifted from C9⋊D4 ρ24 4 -4 4 0 0 0 -4 0 0 -2 -2 -2 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2 ρ25 4 -4 -2 0 0 0 2 0 0 2ζ98+2ζ9 2ζ97+2ζ92 2ζ95+2ζ94 0 0 0 -2ζ97-2ζ92 -2ζ95-2ζ94 -2ζ98-2ζ9 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ26 4 -4 -2 0 0 0 2 0 0 2ζ97+2ζ92 2ζ95+2ζ94 2ζ98+2ζ9 0 0 0 -2ζ95-2ζ94 -2ζ98-2ζ9 -2ζ97-2ζ92 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ27 4 -4 -2 0 0 0 2 0 0 2ζ95+2ζ94 2ζ98+2ζ9 2ζ97+2ζ92 0 0 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 C9⋊Q16
Regular action on 144 points
Generators in S144
```(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144)
(1 133 20 124 11 142 29 115)(2 132 21 123 12 141 30 114)(3 131 22 122 13 140 31 113)(4 130 23 121 14 139 32 112)(5 129 24 120 15 138 33 111)(6 128 25 119 16 137 34 110)(7 127 26 118 17 136 35 109)(8 135 27 126 18 144 36 117)(9 134 19 125 10 143 28 116)(37 98 64 80 46 107 55 89)(38 97 65 79 47 106 56 88)(39 96 66 78 48 105 57 87)(40 95 67 77 49 104 58 86)(41 94 68 76 50 103 59 85)(42 93 69 75 51 102 60 84)(43 92 70 74 52 101 61 83)(44 91 71 73 53 100 62 82)(45 99 72 81 54 108 63 90)
(1 47 11 38)(2 48 12 39)(3 49 13 40)(4 50 14 41)(5 51 15 42)(6 52 16 43)(7 53 17 44)(8 54 18 45)(9 46 10 37)(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)(73 136 82 127)(74 137 83 128)(75 138 84 129)(76 139 85 130)(77 140 86 131)(78 141 87 132)(79 142 88 133)(80 143 89 134)(81 144 90 135)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)```

`G:=sub<Sym(144)| (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,133,20,124,11,142,29,115)(2,132,21,123,12,141,30,114)(3,131,22,122,13,140,31,113)(4,130,23,121,14,139,32,112)(5,129,24,120,15,138,33,111)(6,128,25,119,16,137,34,110)(7,127,26,118,17,136,35,109)(8,135,27,126,18,144,36,117)(9,134,19,125,10,143,28,116)(37,98,64,80,46,107,55,89)(38,97,65,79,47,106,56,88)(39,96,66,78,48,105,57,87)(40,95,67,77,49,104,58,86)(41,94,68,76,50,103,59,85)(42,93,69,75,51,102,60,84)(43,92,70,74,52,101,61,83)(44,91,71,73,53,100,62,82)(45,99,72,81,54,108,63,90), (1,47,11,38)(2,48,12,39)(3,49,13,40)(4,50,14,41)(5,51,15,42)(6,52,16,43)(7,53,17,44)(8,54,18,45)(9,46,10,37)(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)(73,136,82,127)(74,137,83,128)(75,138,84,129)(76,139,85,130)(77,140,86,131)(78,141,87,132)(79,142,88,133)(80,143,89,134)(81,144,90,135)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126)>;`

`G:=Group( (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,133,20,124,11,142,29,115)(2,132,21,123,12,141,30,114)(3,131,22,122,13,140,31,113)(4,130,23,121,14,139,32,112)(5,129,24,120,15,138,33,111)(6,128,25,119,16,137,34,110)(7,127,26,118,17,136,35,109)(8,135,27,126,18,144,36,117)(9,134,19,125,10,143,28,116)(37,98,64,80,46,107,55,89)(38,97,65,79,47,106,56,88)(39,96,66,78,48,105,57,87)(40,95,67,77,49,104,58,86)(41,94,68,76,50,103,59,85)(42,93,69,75,51,102,60,84)(43,92,70,74,52,101,61,83)(44,91,71,73,53,100,62,82)(45,99,72,81,54,108,63,90), (1,47,11,38)(2,48,12,39)(3,49,13,40)(4,50,14,41)(5,51,15,42)(6,52,16,43)(7,53,17,44)(8,54,18,45)(9,46,10,37)(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)(73,136,82,127)(74,137,83,128)(75,138,84,129)(76,139,85,130)(77,140,86,131)(78,141,87,132)(79,142,88,133)(80,143,89,134)(81,144,90,135)(91,109,100,118)(92,110,101,119)(93,111,102,120)(94,112,103,121)(95,113,104,122)(96,114,105,123)(97,115,106,124)(98,116,107,125)(99,117,108,126) );`

`G=PermutationGroup([(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144)], [(1,133,20,124,11,142,29,115),(2,132,21,123,12,141,30,114),(3,131,22,122,13,140,31,113),(4,130,23,121,14,139,32,112),(5,129,24,120,15,138,33,111),(6,128,25,119,16,137,34,110),(7,127,26,118,17,136,35,109),(8,135,27,126,18,144,36,117),(9,134,19,125,10,143,28,116),(37,98,64,80,46,107,55,89),(38,97,65,79,47,106,56,88),(39,96,66,78,48,105,57,87),(40,95,67,77,49,104,58,86),(41,94,68,76,50,103,59,85),(42,93,69,75,51,102,60,84),(43,92,70,74,52,101,61,83),(44,91,71,73,53,100,62,82),(45,99,72,81,54,108,63,90)], [(1,47,11,38),(2,48,12,39),(3,49,13,40),(4,50,14,41),(5,51,15,42),(6,52,16,43),(7,53,17,44),(8,54,18,45),(9,46,10,37),(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),(73,136,82,127),(74,137,83,128),(75,138,84,129),(76,139,85,130),(77,140,86,131),(78,141,87,132),(79,142,88,133),(80,143,89,134),(81,144,90,135),(91,109,100,118),(92,110,101,119),(93,111,102,120),(94,112,103,121),(95,113,104,122),(96,114,105,123),(97,115,106,124),(98,116,107,125),(99,117,108,126)])`

C9⋊Q16 is a maximal subgroup of
SD16⋊D9  SD163D9  Q16×D9  Q16⋊D9  C36.C23  D4.D18  D4.9D18  C27⋊Q16  Q8.D27  C12.D18  C9⋊Dic12  Dic18.C6  C36.19D6  C18.5S4
C9⋊Q16 is a maximal quotient of
C36.Q8  C18.Q16  Q82Dic9  C27⋊Q16  C12.D18  C9⋊Dic12  C36.19D6

Matrix representation of C9⋊Q16 in GL4(𝔽73) generated by

 42 45 0 0 28 70 0 0 0 0 1 0 0 0 0 1
,
 68 50 0 0 55 5 0 0 0 0 0 48 0 0 38 41
,
 72 0 0 0 0 72 0 0 0 0 54 33 0 0 20 19
`G:=sub<GL(4,GF(73))| [42,28,0,0,45,70,0,0,0,0,1,0,0,0,0,1],[68,55,0,0,50,5,0,0,0,0,0,38,0,0,48,41],[72,0,0,0,0,72,0,0,0,0,54,20,0,0,33,19] >;`

C9⋊Q16 in GAP, Magma, Sage, TeX

`C_9\rtimes Q_{16}`
`% in TeX`

`G:=Group("C9:Q16");`
`// GroupNames label`

`G:=SmallGroup(144,17);`
`// by ID`

`G=gap.SmallGroup(144,17);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,55,218,116,50,2404,208,3461]);`
`// Polycyclic`

`G:=Group<a,b,c|a^9=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;`
`// generators/relations`

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