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## G = C2×C6.Q16order 192 = 26·3

### Direct product of C2 and C6.Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C6.Q16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C2×C6.Q16
 Lower central C3 — C6 — C12 — C2×C6.Q16
 Upper central C1 — C23 — C22×C4 — C2×C4⋊C4

Generators and relations for C2×C6.Q16
G = < a,b,c,d | a2=b12=c4=1, d2=b9c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b9c-1 >

Subgroups: 280 in 130 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×6], C22×C4, C22×C4 [×2], C3⋊C8 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×C6, C2.D8 [×4], C2×C4⋊C4, C2×C4⋊C4, C22×C8, C2×C3⋊C8 [×6], C4⋊Dic3 [×2], C4⋊Dic3, C3×C4⋊C4 [×2], C3×C4⋊C4, C22×Dic3, C22×C12, C22×C12, C2×C2.D8, C6.Q16 [×4], C22×C3⋊C8, C2×C4⋊Dic3, C6×C4⋊C4, C2×C6.Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2.D8 [×4], C2×C4⋊C4, C2×D8, C2×Q16, Dic3⋊C4 [×4], D4⋊S3 [×2], C3⋊Q16 [×2], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×C2.D8, C6.Q16 [×4], C2×Dic3⋊C4, C2×D4⋊S3, C2×C3⋊Q16, C2×C6.Q16

Smallest permutation representation of C2×C6.Q16
Regular action on 192 points
Generators in S192
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 133)(37 170)(38 171)(39 172)(40 173)(41 174)(42 175)(43 176)(44 177)(45 178)(46 179)(47 180)(48 169)(49 192)(50 181)(51 182)(52 183)(53 184)(54 185)(55 186)(56 187)(57 188)(58 189)(59 190)(60 191)(61 157)(62 158)(63 159)(64 160)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 167)(72 168)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 109)(82 110)(83 111)(84 112)(85 108)(86 97)(87 98)(88 99)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(121 152)(122 153)(123 154)(124 155)(125 156)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)
(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 59 64 149)(2 54 65 156)(3 49 66 151)(4 56 67 146)(5 51 68 153)(6 58 69 148)(7 53 70 155)(8 60 71 150)(9 55 72 145)(10 50 61 152)(11 57 62 147)(12 52 63 154)(13 103 38 111)(14 98 39 118)(15 105 40 113)(16 100 41 120)(17 107 42 115)(18 102 43 110)(19 97 44 117)(20 104 45 112)(21 99 46 119)(22 106 47 114)(23 101 48 109)(24 108 37 116)(25 182 164 122)(26 189 165 129)(27 184 166 124)(28 191 167 131)(29 186 168 126)(30 181 157 121)(31 188 158 128)(32 183 159 123)(33 190 160 130)(34 185 161 125)(35 192 162 132)(36 187 163 127)(73 136 94 173)(74 143 95 180)(75 138 96 175)(76 133 85 170)(77 140 86 177)(78 135 87 172)(79 142 88 179)(80 137 89 174)(81 144 90 169)(82 139 91 176)(83 134 92 171)(84 141 93 178)
(1 171 61 143 7 177 67 137)(2 176 62 136 8 170 68 142)(3 169 63 141 9 175 69 135)(4 174 64 134 10 180 70 140)(5 179 65 139 11 173 71 133)(6 172 66 144 12 178 72 138)(13 30 47 166 19 36 41 160)(14 35 48 159 20 29 42 165)(15 28 37 164 21 34 43 158)(16 33 38 157 22 27 44 163)(17 26 39 162 23 32 45 168)(18 31 40 167 24 25 46 161)(49 87 148 75 55 93 154 81)(50 92 149 80 56 86 155 74)(51 85 150 73 57 91 156 79)(52 90 151 78 58 96 145 84)(53 95 152 83 59 89 146 77)(54 88 153 76 60 94 147 82)(97 124 114 181 103 130 120 187)(98 129 115 186 104 123 109 192)(99 122 116 191 105 128 110 185)(100 127 117 184 106 121 111 190)(101 132 118 189 107 126 112 183)(102 125 119 182 108 131 113 188)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6G 8A ··· 8H 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 12 12 12 12 2 ··· 2 6 ··· 6 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D4 D6 D6 D8 Q16 Dic6 C4×S3 C3⋊D4 C3⋊D4 D4⋊S3 C3⋊Q16 kernel C2×C6.Q16 C6.Q16 C22×C3⋊C8 C2×C4⋊Dic3 C6×C4⋊C4 C2×C3⋊C8 C2×C4⋊C4 C2×C12 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 1 2 1 2 1 4 4 4 4 2 2 2 2

Matrix representation of C2×C6.Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 9 0 0 0 0 0 2 65
,
 17 71 0 0 0 0 71 56 0 0 0 0 0 0 5 4 0 0 0 0 67 68 0 0 0 0 0 0 46 0 0 0 0 0 0 46
,
 16 16 0 0 0 0 57 16 0 0 0 0 0 0 67 61 0 0 0 0 70 6 0 0 0 0 0 0 6 22 0 0 0 0 15 67

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,9,2,0,0,0,0,0,65],[17,71,0,0,0,0,71,56,0,0,0,0,0,0,5,67,0,0,0,0,4,68,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[16,57,0,0,0,0,16,16,0,0,0,0,0,0,67,70,0,0,0,0,61,6,0,0,0,0,0,0,6,15,0,0,0,0,22,67] >;

C2×C6.Q16 in GAP, Magma, Sage, TeX

C_2\times C_6.Q_{16}
% in TeX

G:=Group("C2xC6.Q16");
// GroupNames label

G:=SmallGroup(192,521);
// by ID

G=gap.SmallGroup(192,521);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,58,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=b^9*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^9*c^-1>;
// generators/relations

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