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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, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×C6, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C22×C12, C2×C2.D8, C6.Q16, C22×C3⋊C8, C2×C4⋊Dic3, C6×C4⋊C4, C2×C6.Q16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, Dic3⋊C4, D4⋊S3, C3⋊Q16, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×C2.D8, C6.Q16, 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
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 183)(26 184)(27 185)(28 186)(29 187)(30 188)(31 189)(32 190)(33 191)(34 192)(35 181)(36 182)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 153)(68 154)(69 155)(70 156)(71 145)(72 146)(73 128)(74 129)(75 130)(76 131)(77 132)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(97 110)(98 111)(99 112)(100 113)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 109)(133 169)(134 170)(135 171)(136 172)(137 173)(138 174)(139 175)(140 176)(141 177)(142 178)(143 179)(144 180)
(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)
(1 150 108 46)(2 145 97 41)(3 152 98 48)(4 147 99 43)(5 154 100 38)(6 149 101 45)(7 156 102 40)(8 151 103 47)(9 146 104 42)(10 153 105 37)(11 148 106 44)(12 155 107 39)(13 73 177 33)(14 80 178 28)(15 75 179 35)(16 82 180 30)(17 77 169 25)(18 84 170 32)(19 79 171 27)(20 74 172 34)(21 81 173 29)(22 76 174 36)(23 83 175 31)(24 78 176 26)(49 121 140 184)(50 128 141 191)(51 123 142 186)(52 130 143 181)(53 125 144 188)(54 132 133 183)(55 127 134 190)(56 122 135 185)(57 129 136 192)(58 124 137 187)(59 131 138 182)(60 126 139 189)(61 112 91 160)(62 119 92 167)(63 114 93 162)(64 109 94 157)(65 116 95 164)(66 111 96 159)(67 118 85 166)(68 113 86 161)(69 120 87 168)(70 115 88 163)(71 110 89 158)(72 117 90 165)
(1 80 105 25 7 74 99 31)(2 73 106 30 8 79 100 36)(3 78 107 35 9 84 101 29)(4 83 108 28 10 77 102 34)(5 76 97 33 11 82 103 27)(6 81 98 26 12 75 104 32)(13 41 174 154 19 47 180 148)(14 46 175 147 20 40 169 153)(15 39 176 152 21 45 170 146)(16 44 177 145 22 38 171 151)(17 37 178 150 23 43 172 156)(18 42 179 155 24 48 173 149)(49 96 137 63 55 90 143 69)(50 89 138 68 56 95 144 62)(51 94 139 61 57 88 133 67)(52 87 140 66 58 93 134 72)(53 92 141 71 59 86 135 65)(54 85 142 64 60 91 136 70)(109 186 166 132 115 192 160 126)(110 191 167 125 116 185 161 131)(111 184 168 130 117 190 162 124)(112 189 157 123 118 183 163 129)(113 182 158 128 119 188 164 122)(114 187 159 121 120 181 165 127)

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

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

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

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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