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## G = C2×C12.Q8order 192 = 26·3

### Direct product of C2 and C12.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C12.Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C2×C12.Q8
 Lower central C3 — C6 — C12 — C2×C12.Q8
 Upper central C1 — C23 — C22×C4 — C2×C4⋊C4

Generators and relations for C2×C12.Q8
G = < a,b,c,d | a2=b4=c12=1, d2=bc6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c-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, C4.Q8 [×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×C4.Q8, C12.Q8 [×4], C22×C3⋊C8, C2×C4⋊Dic3, C6×C4⋊C4, C2×C12.Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4.Q8 [×4], C2×C4⋊C4, C2×SD16 [×2], Dic3⋊C4 [×4], D4.S3 [×2], Q82S3 [×2], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×C4.Q8, C12.Q8 [×4], C2×Dic3⋊C4, C2×D4.S3, C2×Q82S3, C2×C12.Q8

Smallest permutation representation of C2×C12.Q8
Regular action on 192 points
Generators in S192
(1 141)(2 142)(3 143)(4 144)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 192)(24 181)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 61)(46 62)(47 63)(48 64)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(73 128)(74 129)(75 130)(76 131)(77 132)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 175)(98 176)(99 177)(100 178)(101 179)(102 180)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)
(1 160 126 181)(2 182 127 161)(3 162 128 183)(4 184 129 163)(5 164 130 185)(6 186 131 165)(7 166 132 187)(8 188 121 167)(9 168 122 189)(10 190 123 157)(11 158 124 191)(12 192 125 159)(13 84 53 142)(14 143 54 73)(15 74 55 144)(16 133 56 75)(17 76 57 134)(18 135 58 77)(19 78 59 136)(20 137 60 79)(21 80 49 138)(22 139 50 81)(23 82 51 140)(24 141 52 83)(25 176 86 48)(26 37 87 177)(27 178 88 38)(28 39 89 179)(29 180 90 40)(30 41 91 169)(31 170 92 42)(32 43 93 171)(33 172 94 44)(34 45 95 173)(35 174 96 46)(36 47 85 175)(61 113 107 148)(62 149 108 114)(63 115 97 150)(64 151 98 116)(65 117 99 152)(66 153 100 118)(67 119 101 154)(68 155 102 120)(69 109 103 156)(70 145 104 110)(71 111 105 146)(72 147 106 112)
(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 96 166 40 126 35 187 180)(2 45 188 89 127 173 167 28)(3 94 168 38 128 33 189 178)(4 43 190 87 129 171 157 26)(5 92 158 48 130 31 191 176)(6 41 192 85 131 169 159 36)(7 90 160 46 132 29 181 174)(8 39 182 95 121 179 161 34)(9 88 162 44 122 27 183 172)(10 37 184 93 123 177 163 32)(11 86 164 42 124 25 185 170)(12 47 186 91 125 175 165 30)(13 113 78 101 53 148 136 67)(14 106 137 118 54 72 79 153)(15 111 80 99 55 146 138 65)(16 104 139 116 56 70 81 151)(17 109 82 97 57 156 140 63)(18 102 141 114 58 68 83 149)(19 119 84 107 59 154 142 61)(20 100 143 112 60 66 73 147)(21 117 74 105 49 152 144 71)(22 98 133 110 50 64 75 145)(23 115 76 103 51 150 134 69)(24 108 135 120 52 62 77 155)

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

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

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

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 4 4 type + + + + + + + - + + + - - + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D4 D6 D6 SD16 Dic6 C4×S3 C3⋊D4 C3⋊D4 D4.S3 Q8⋊2S3 kernel C2×C12.Q8 C12.Q8 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×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 1 2 1 2 1 8 4 4 2 2 2 2

Matrix representation of C2×C12.Q8 in GL5(𝔽73)

 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 2 0 0 0 72 1
,
 27 0 0 0 0 0 0 1 0 0 0 72 1 0 0 0 0 0 12 61 0 0 0 6 61
,
 72 0 0 0 0 0 31 70 0 0 0 28 42 0 0 0 0 0 0 61 0 0 0 6 61

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,2,1],[27,0,0,0,0,0,0,72,0,0,0,1,1,0,0,0,0,0,12,6,0,0,0,61,61],[72,0,0,0,0,0,31,28,0,0,0,70,42,0,0,0,0,0,0,6,0,0,0,61,61] >;

C2×C12.Q8 in GAP, Magma, Sage, TeX

C_2\times C_{12}.Q_8
% in TeX

G:=Group("C2xC12.Q8");
// GroupNames label

G:=SmallGroup(192,522);
// by ID

G=gap.SmallGroup(192,522);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1094,58,438,102,6278]);
// Polycyclic

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

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