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## G = C2×Dic6⋊C4order 192 = 26·3

### Direct product of C2 and Dic6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Dic6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C2×Dic6⋊C4
 Lower central C3 — C6 — C2×Dic6⋊C4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×Dic6⋊C4
G = < a,b,c,d | a2=b12=d4=1, c2=b6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b7, cd=dc >

Subgroups: 536 in 298 conjugacy classes, 175 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C2×C4×Q8, Dic6⋊C4, C2×C4×Dic3, C2×C4×Dic3, C2×Dic3⋊C4, C6×C4⋊C4, C22×Dic6, C2×Dic6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C24, C4×S3, C22×S3, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, S3×C2×C4, D42S3, S3×Q8, S3×C23, C2×C4×Q8, Dic6⋊C4, S3×C22×C4, C2×D42S3, C2×S3×Q8, C2×Dic6⋊C4

Smallest permutation representation of C2×Dic6⋊C4
Regular action on 192 points
Generators in S192
(1 82)(2 83)(3 84)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 96)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 61)(59 62)(60 63)(121 190)(122 191)(123 192)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 172)(134 173)(135 174)(136 175)(137 176)(138 177)(139 178)(140 179)(141 180)(142 169)(143 170)(144 171)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(151 166)(152 167)(153 168)(154 157)(155 158)(156 159)
(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)
(1 145 7 151)(2 156 8 150)(3 155 9 149)(4 154 10 148)(5 153 11 147)(6 152 12 146)(13 119 19 113)(14 118 20 112)(15 117 21 111)(16 116 22 110)(17 115 23 109)(18 114 24 120)(25 88 31 94)(26 87 32 93)(27 86 33 92)(28 85 34 91)(29 96 35 90)(30 95 36 89)(37 133 43 139)(38 144 44 138)(39 143 45 137)(40 142 46 136)(41 141 47 135)(42 140 48 134)(49 123 55 129)(50 122 56 128)(51 121 57 127)(52 132 58 126)(53 131 59 125)(54 130 60 124)(61 183 67 189)(62 182 68 188)(63 181 69 187)(64 192 70 186)(65 191 71 185)(66 190 72 184)(73 157 79 163)(74 168 80 162)(75 167 81 161)(76 166 82 160)(77 165 83 159)(78 164 84 158)(97 172 103 178)(98 171 104 177)(99 170 105 176)(100 169 106 175)(101 180 107 174)(102 179 108 173)
(1 105 14 65)(2 100 15 72)(3 107 16 67)(4 102 17 62)(5 97 18 69)(6 104 19 64)(7 99 20 71)(8 106 21 66)(9 101 22 61)(10 108 23 68)(11 103 24 63)(12 98 13 70)(25 131 163 134)(26 126 164 141)(27 121 165 136)(28 128 166 143)(29 123 167 138)(30 130 168 133)(31 125 157 140)(32 132 158 135)(33 127 159 142)(34 122 160 137)(35 129 161 144)(36 124 162 139)(37 89 54 74)(38 96 55 81)(39 91 56 76)(40 86 57 83)(41 93 58 78)(42 88 59 73)(43 95 60 80)(44 90 49 75)(45 85 50 82)(46 92 51 77)(47 87 52 84)(48 94 53 79)(109 188 148 173)(110 183 149 180)(111 190 150 175)(112 185 151 170)(113 192 152 177)(114 187 153 172)(115 182 154 179)(116 189 155 174)(117 184 156 169)(118 191 145 176)(119 186 146 171)(120 181 147 178)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4L 4M ··· 4T 4U ··· 4AF 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D6 D6 C4○D4 C4×S3 D4⋊2S3 S3×Q8 kernel C2×Dic6⋊C4 Dic6⋊C4 C2×C4×Dic3 C2×Dic3⋊C4 C6×C4⋊C4 C22×Dic6 C2×Dic6 C2×C4⋊C4 C2×Dic3 C4⋊C4 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 8 3 2 1 1 16 1 4 4 3 4 8 2 2

Matrix representation of C2×Dic6⋊C4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 0 0 0 0 0 8 4 0 0 0 0 0 0 10 0 0 0 0 0 1 4 0 0 0 0 0 0 3 4 0 0 0 0 4 10
,
 9 3 0 0 0 0 8 4 0 0 0 0 0 0 5 9 0 0 0 0 6 8 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,8,0,0,0,0,0,4,0,0,0,0,0,0,10,1,0,0,0,0,0,4,0,0,0,0,0,0,3,4,0,0,0,0,4,10],[9,8,0,0,0,0,3,4,0,0,0,0,0,0,5,6,0,0,0,0,9,8,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C2×Dic6⋊C4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_6\rtimes C_4
% in TeX

G:=Group("C2xDic6:C4");
// GroupNames label

G:=SmallGroup(192,1055);
// by ID

G=gap.SmallGroup(192,1055);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,185,192,6278]);
// Polycyclic

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

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