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## G = C12.9Q16order 192 = 26·3

### 9th non-split extension by C12 of Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.9Q16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C4×C3⋊C8 — C12.9Q16
 Lower central C3 — C6 — C2×C12 — C12.9Q16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C12.9Q16
G = < a,b,c | a12=b8=1, c2=b4, bab-1=a5, cac-1=a7, cbc-1=a6b-1 >

Subgroups: 240 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C3, C4 [×6], C4 [×4], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×6], Dic3 [×2], C12 [×6], C12 [×2], C2×C6, C42, C4⋊C4 [×5], C2×C8 [×2], C2×Q8 [×2], C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×4], C4×C8, Q8⋊C4 [×4], C4⋊Q8, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C2×Dic6, C6×Q8 [×2], C4.SD16, C4×C3⋊C8, Q82Dic3 [×4], C122Q8, C3×C4⋊Q8, C12.9Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], Q16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C2×SD16, C2×Q16, Q82S3 [×2], C3⋊Q16 [×2], D42S3 [×2], C2×C3⋊D4, C4.SD16, C23.12D6, C2×Q82S3, C2×C3⋊Q16, C12.9Q16

Smallest permutation representation of C12.9Q16
Regular action on 192 points
Generators in S192
```(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)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 98 130 113 89 39 183 75)(2 103 131 118 90 44 184 80)(3 108 132 111 91 37 185 73)(4 101 121 116 92 42 186 78)(5 106 122 109 93 47 187 83)(6 99 123 114 94 40 188 76)(7 104 124 119 95 45 189 81)(8 97 125 112 96 38 190 74)(9 102 126 117 85 43 191 79)(10 107 127 110 86 48 192 84)(11 100 128 115 87 41 181 77)(12 105 129 120 88 46 182 82)(13 167 28 176 154 139 54 61)(14 160 29 169 155 144 55 66)(15 165 30 174 156 137 56 71)(16 158 31 179 145 142 57 64)(17 163 32 172 146 135 58 69)(18 168 33 177 147 140 59 62)(19 161 34 170 148 133 60 67)(20 166 35 175 149 138 49 72)(21 159 36 180 150 143 50 65)(22 164 25 173 151 136 51 70)(23 157 26 178 152 141 52 63)(24 162 27 171 153 134 53 68)
(1 66 89 169)(2 61 90 176)(3 68 91 171)(4 63 92 178)(5 70 93 173)(6 65 94 180)(7 72 95 175)(8 67 96 170)(9 62 85 177)(10 69 86 172)(11 64 87 179)(12 71 88 174)(13 112 154 74)(14 119 155 81)(15 114 156 76)(16 109 145 83)(17 116 146 78)(18 111 147 73)(19 118 148 80)(20 113 149 75)(21 120 150 82)(22 115 151 77)(23 110 152 84)(24 117 153 79)(25 100 51 41)(26 107 52 48)(27 102 53 43)(28 97 54 38)(29 104 55 45)(30 99 56 40)(31 106 57 47)(32 101 58 42)(33 108 59 37)(34 103 60 44)(35 98 49 39)(36 105 50 46)(121 141 186 157)(122 136 187 164)(123 143 188 159)(124 138 189 166)(125 133 190 161)(126 140 191 168)(127 135 192 163)(128 142 181 158)(129 137 182 165)(130 144 183 160)(131 139 184 167)(132 134 185 162)```

`G:=sub<Sym(192)| (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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,98,130,113,89,39,183,75)(2,103,131,118,90,44,184,80)(3,108,132,111,91,37,185,73)(4,101,121,116,92,42,186,78)(5,106,122,109,93,47,187,83)(6,99,123,114,94,40,188,76)(7,104,124,119,95,45,189,81)(8,97,125,112,96,38,190,74)(9,102,126,117,85,43,191,79)(10,107,127,110,86,48,192,84)(11,100,128,115,87,41,181,77)(12,105,129,120,88,46,182,82)(13,167,28,176,154,139,54,61)(14,160,29,169,155,144,55,66)(15,165,30,174,156,137,56,71)(16,158,31,179,145,142,57,64)(17,163,32,172,146,135,58,69)(18,168,33,177,147,140,59,62)(19,161,34,170,148,133,60,67)(20,166,35,175,149,138,49,72)(21,159,36,180,150,143,50,65)(22,164,25,173,151,136,51,70)(23,157,26,178,152,141,52,63)(24,162,27,171,153,134,53,68), (1,66,89,169)(2,61,90,176)(3,68,91,171)(4,63,92,178)(5,70,93,173)(6,65,94,180)(7,72,95,175)(8,67,96,170)(9,62,85,177)(10,69,86,172)(11,64,87,179)(12,71,88,174)(13,112,154,74)(14,119,155,81)(15,114,156,76)(16,109,145,83)(17,116,146,78)(18,111,147,73)(19,118,148,80)(20,113,149,75)(21,120,150,82)(22,115,151,77)(23,110,152,84)(24,117,153,79)(25,100,51,41)(26,107,52,48)(27,102,53,43)(28,97,54,38)(29,104,55,45)(30,99,56,40)(31,106,57,47)(32,101,58,42)(33,108,59,37)(34,103,60,44)(35,98,49,39)(36,105,50,46)(121,141,186,157)(122,136,187,164)(123,143,188,159)(124,138,189,166)(125,133,190,161)(126,140,191,168)(127,135,192,163)(128,142,181,158)(129,137,182,165)(130,144,183,160)(131,139,184,167)(132,134,185,162)>;`

`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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,98,130,113,89,39,183,75)(2,103,131,118,90,44,184,80)(3,108,132,111,91,37,185,73)(4,101,121,116,92,42,186,78)(5,106,122,109,93,47,187,83)(6,99,123,114,94,40,188,76)(7,104,124,119,95,45,189,81)(8,97,125,112,96,38,190,74)(9,102,126,117,85,43,191,79)(10,107,127,110,86,48,192,84)(11,100,128,115,87,41,181,77)(12,105,129,120,88,46,182,82)(13,167,28,176,154,139,54,61)(14,160,29,169,155,144,55,66)(15,165,30,174,156,137,56,71)(16,158,31,179,145,142,57,64)(17,163,32,172,146,135,58,69)(18,168,33,177,147,140,59,62)(19,161,34,170,148,133,60,67)(20,166,35,175,149,138,49,72)(21,159,36,180,150,143,50,65)(22,164,25,173,151,136,51,70)(23,157,26,178,152,141,52,63)(24,162,27,171,153,134,53,68), (1,66,89,169)(2,61,90,176)(3,68,91,171)(4,63,92,178)(5,70,93,173)(6,65,94,180)(7,72,95,175)(8,67,96,170)(9,62,85,177)(10,69,86,172)(11,64,87,179)(12,71,88,174)(13,112,154,74)(14,119,155,81)(15,114,156,76)(16,109,145,83)(17,116,146,78)(18,111,147,73)(19,118,148,80)(20,113,149,75)(21,120,150,82)(22,115,151,77)(23,110,152,84)(24,117,153,79)(25,100,51,41)(26,107,52,48)(27,102,53,43)(28,97,54,38)(29,104,55,45)(30,99,56,40)(31,106,57,47)(32,101,58,42)(33,108,59,37)(34,103,60,44)(35,98,49,39)(36,105,50,46)(121,141,186,157)(122,136,187,164)(123,143,188,159)(124,138,189,166)(125,133,190,161)(126,140,191,168)(127,135,192,163)(128,142,181,158)(129,137,182,165)(130,144,183,160)(131,139,184,167)(132,134,185,162) );`

`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),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,98,130,113,89,39,183,75),(2,103,131,118,90,44,184,80),(3,108,132,111,91,37,185,73),(4,101,121,116,92,42,186,78),(5,106,122,109,93,47,187,83),(6,99,123,114,94,40,188,76),(7,104,124,119,95,45,189,81),(8,97,125,112,96,38,190,74),(9,102,126,117,85,43,191,79),(10,107,127,110,86,48,192,84),(11,100,128,115,87,41,181,77),(12,105,129,120,88,46,182,82),(13,167,28,176,154,139,54,61),(14,160,29,169,155,144,55,66),(15,165,30,174,156,137,56,71),(16,158,31,179,145,142,57,64),(17,163,32,172,146,135,58,69),(18,168,33,177,147,140,59,62),(19,161,34,170,148,133,60,67),(20,166,35,175,149,138,49,72),(21,159,36,180,150,143,50,65),(22,164,25,173,151,136,51,70),(23,157,26,178,152,141,52,63),(24,162,27,171,153,134,53,68)], [(1,66,89,169),(2,61,90,176),(3,68,91,171),(4,63,92,178),(5,70,93,173),(6,65,94,180),(7,72,95,175),(8,67,96,170),(9,62,85,177),(10,69,86,172),(11,64,87,179),(12,71,88,174),(13,112,154,74),(14,119,155,81),(15,114,156,76),(16,109,145,83),(17,116,146,78),(18,111,147,73),(19,118,148,80),(20,113,149,75),(21,120,150,82),(22,115,151,77),(23,110,152,84),(24,117,153,79),(25,100,51,41),(26,107,52,48),(27,102,53,43),(28,97,54,38),(29,104,55,45),(30,99,56,40),(31,106,57,47),(32,101,58,42),(33,108,59,37),(34,103,60,44),(35,98,49,39),(36,105,50,46),(121,141,186,157),(122,136,187,164),(123,143,188,159),(124,138,189,166),(125,133,190,161),(126,140,191,168),(127,135,192,163),(128,142,181,158),(129,137,182,165),(130,144,183,160),(131,139,184,167),(132,134,185,162)])`

36 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 6A 6B 6C 8A ··· 8H 12A ··· 12F 12G 12H 12I 12J order 1 2 2 2 3 4 ··· 4 4 4 4 4 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 ··· 2 8 8 24 24 2 2 2 6 ··· 6 4 ··· 4 8 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + - - image C1 C2 C2 C2 C2 S3 D4 D6 D6 SD16 Q16 C4○D4 C3⋊D4 Q8⋊2S3 C3⋊Q16 D4⋊2S3 kernel C12.9Q16 C4×C3⋊C8 Q8⋊2Dic3 C12⋊2Q8 C3×C4⋊Q8 C4⋊Q8 C2×C12 C42 C2×Q8 C12 C12 C12 C2×C4 C4 C4 C4 # reps 1 1 4 1 1 1 2 1 2 4 4 4 4 2 2 2

Matrix representation of C12.9Q16 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 20 28 0 0 0 0 30 53
,
 67 6 0 0 0 0 67 67 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27
,
 0 46 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 27 2 0 0 0 0 1 46

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,20,30,0,0,0,0,28,53],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,1,0,0,0,0,2,46] >;`

C12.9Q16 in GAP, Magma, Sage, TeX

`C_{12}._9Q_{16}`
`% in TeX`

`G:=Group("C12.9Q16");`
`// GroupNames label`

`G:=SmallGroup(192,638);`
`// by ID`

`G=gap.SmallGroup(192,638);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,135,184,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=a^6*b^-1>;`
`// generators/relations`

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