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## G = C12.Q16order 192 = 26·3

### 11st 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.Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12⋊2Q8 — C12.Q16
 Lower central C3 — C6 — C2×C12 — C12.Q16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C12.Q16
G = < a,b,c | a12=b8=1, c2=a6b4, 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 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×Q8 [×3], C3⋊C8 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C4×C8, Q8⋊C4 [×4], C4⋊Q8, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], C6×Q8, C4.SD16, C4×C3⋊C8, C6.SD16 [×4], C122Q8, C3×C4⋊Q8, C12.Q16
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, D4.S3 [×2], C3⋊Q16 [×2], Q83S3 [×2], C2×C3⋊D4, C4.SD16, C2×D4.S3, C2×C3⋊Q16, C12.23D4, C12.Q16

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

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

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

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

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 D4.S3 C3⋊Q16 Q8⋊3S3 kernel C12.Q16 C4×C3⋊C8 C6.SD16 C12⋊2Q8 C3×C4⋊Q8 C4⋊Q8 C2×C12 C42 C4⋊C4 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.Q16 in GL6(𝔽73)

 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 3 0 0 0 0 48 72 0 0 0 0 0 0 72 2 0 0 0 0 72 1
,
 5 59 0 0 0 0 54 68 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 12 0 0 0 0 67 12
,
 30 60 0 0 0 0 13 43 0 0 0 0 0 0 46 0 0 0 0 0 18 27 0 0 0 0 0 0 52 48 0 0 0 0 3 21

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,48,0,0,0,0,3,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[5,54,0,0,0,0,59,68,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,67,0,0,0,0,12,12],[30,13,0,0,0,0,60,43,0,0,0,0,0,0,46,18,0,0,0,0,0,27,0,0,0,0,0,0,52,3,0,0,0,0,48,21] >;`

C12.Q16 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(192,652);`
`// by ID`

`G=gap.SmallGroup(192,652);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,219,100,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=a^6*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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