direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic6, C4⋊C4.260D6, C12.66(C2×Q8), (C2×C12).30Q8, C6.8(C22×Q8), (C2×C6).43C24, C6⋊3(C42.C2), (C2×C4).37Dic6, C4.31(C2×Dic6), (C22×C4).374D6, (C2×C12).134C23, C22.81(S3×C23), C2.10(C22×Dic6), C22.37(C2×Dic6), C4⋊Dic3.357C22, C23.333(C22×S3), (C22×C12).73C22, (C22×C6).392C23, C22.72(D4⋊2S3), Dic3⋊C4.104C22, C22.32(Q8⋊3S3), (C4×Dic3).248C22, (C2×Dic3).183C23, (C22×Dic3).208C22, (C6×C4⋊C4).19C2, (C2×C4⋊C4).26S3, C3⋊3(C2×C42.C2), C6.70(C2×C4○D4), (C2×C6).51(C2×Q8), C2.5(C2×Q8⋊3S3), (C2×C4×Dic3).13C2, C2.14(C2×D4⋊2S3), (C2×C4⋊Dic3).43C2, (C2×C6).170(C4○D4), (C3×C4⋊C4).292C22, (C2×Dic3⋊C4).22C2, (C2×C4).139(C22×S3), SmallGroup(192,1058)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic6
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b6c-1 >
Subgroups: 440 in 226 conjugacy classes, 127 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C2×C42.C2, C4.Dic6, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C4⋊Dic3, C6×C4⋊C4, C2×C4.Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, Dic6, C22×S3, C42.C2, C22×Q8, C2×C4○D4, C2×Dic6, D4⋊2S3, Q8⋊3S3, S3×C23, C2×C42.C2, C4.Dic6, C22×Dic6, C2×D4⋊2S3, C2×Q8⋊3S3, C2×C4.Dic6
►Regular action on 192 points
Generators in S192
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 184)(14 185)(15 186)(16 187)(17 188)(18 189)(19 190)(20 191)(21 192)(22 181)(23 182)(24 183)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 157)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 169)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 73)(85 155)(86 156)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 119)(98 120)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(121 140)(122 141)(123 142)(124 143)(125 144)(126 133)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)
(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 68 170 118)(2 63 171 113)(3 70 172 120)(4 65 173 115)(5 72 174 110)(6 67 175 117)(7 62 176 112)(8 69 177 119)(9 64 178 114)(10 71 179 109)(11 66 180 116)(12 61 169 111)(13 125 93 161)(14 132 94 168)(15 127 95 163)(16 122 96 158)(17 129 85 165)(18 124 86 160)(19 131 87 167)(20 126 88 162)(21 121 89 157)(22 128 90 164)(23 123 91 159)(24 130 92 166)(25 82 56 97)(26 77 57 104)(27 84 58 99)(28 79 59 106)(29 74 60 101)(30 81 49 108)(31 76 50 103)(32 83 51 98)(33 78 52 105)(34 73 53 100)(35 80 54 107)(36 75 55 102)(37 187 141 154)(38 182 142 149)(39 189 143 156)(40 184 144 151)(41 191 133 146)(42 186 134 153)(43 181 135 148)(44 188 136 155)(45 183 137 150)(46 190 138 145)(47 185 139 152)(48 192 140 147)
(1 166 170 130)(2 159 171 123)(3 164 172 128)(4 157 173 121)(5 162 174 126)(6 167 175 131)(7 160 176 124)(8 165 177 129)(9 158 178 122)(10 163 179 127)(11 168 180 132)(12 161 169 125)(13 67 93 117)(14 72 94 110)(15 65 95 115)(16 70 96 120)(17 63 85 113)(18 68 86 118)(19 61 87 111)(20 66 88 116)(21 71 89 109)(22 64 90 114)(23 69 91 119)(24 62 92 112)(25 44 56 136)(26 37 57 141)(27 42 58 134)(28 47 59 139)(29 40 60 144)(30 45 49 137)(31 38 50 142)(32 43 51 135)(33 48 52 140)(34 41 53 133)(35 46 54 138)(36 39 55 143)(73 152 100 185)(74 145 101 190)(75 150 102 183)(76 155 103 188)(77 148 104 181)(78 153 105 186)(79 146 106 191)(80 151 107 184)(81 156 108 189)(82 149 97 182)(83 154 98 187)(84 147 99 192)
G:=sub<Sym(192)| (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,181)(23,182)(24,183)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,157)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,169)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73)(85,155)(86,156)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,119)(98,120)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(121,140)(122,141)(123,142)(124,143)(125,144)(126,133)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139), (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), (1,68,170,118)(2,63,171,113)(3,70,172,120)(4,65,173,115)(5,72,174,110)(6,67,175,117)(7,62,176,112)(8,69,177,119)(9,64,178,114)(10,71,179,109)(11,66,180,116)(12,61,169,111)(13,125,93,161)(14,132,94,168)(15,127,95,163)(16,122,96,158)(17,129,85,165)(18,124,86,160)(19,131,87,167)(20,126,88,162)(21,121,89,157)(22,128,90,164)(23,123,91,159)(24,130,92,166)(25,82,56,97)(26,77,57,104)(27,84,58,99)(28,79,59,106)(29,74,60,101)(30,81,49,108)(31,76,50,103)(32,83,51,98)(33,78,52,105)(34,73,53,100)(35,80,54,107)(36,75,55,102)(37,187,141,154)(38,182,142,149)(39,189,143,156)(40,184,144,151)(41,191,133,146)(42,186,134,153)(43,181,135,148)(44,188,136,155)(45,183,137,150)(46,190,138,145)(47,185,139,152)(48,192,140,147), (1,166,170,130)(2,159,171,123)(3,164,172,128)(4,157,173,121)(5,162,174,126)(6,167,175,131)(7,160,176,124)(8,165,177,129)(9,158,178,122)(10,163,179,127)(11,168,180,132)(12,161,169,125)(13,67,93,117)(14,72,94,110)(15,65,95,115)(16,70,96,120)(17,63,85,113)(18,68,86,118)(19,61,87,111)(20,66,88,116)(21,71,89,109)(22,64,90,114)(23,69,91,119)(24,62,92,112)(25,44,56,136)(26,37,57,141)(27,42,58,134)(28,47,59,139)(29,40,60,144)(30,45,49,137)(31,38,50,142)(32,43,51,135)(33,48,52,140)(34,41,53,133)(35,46,54,138)(36,39,55,143)(73,152,100,185)(74,145,101,190)(75,150,102,183)(76,155,103,188)(77,148,104,181)(78,153,105,186)(79,146,106,191)(80,151,107,184)(81,156,108,189)(82,149,97,182)(83,154,98,187)(84,147,99,192)>;
G:=Group( (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,181)(23,182)(24,183)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,157)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,169)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,73)(85,155)(86,156)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,119)(98,120)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(121,140)(122,141)(123,142)(124,143)(125,144)(126,133)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139), (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), (1,68,170,118)(2,63,171,113)(3,70,172,120)(4,65,173,115)(5,72,174,110)(6,67,175,117)(7,62,176,112)(8,69,177,119)(9,64,178,114)(10,71,179,109)(11,66,180,116)(12,61,169,111)(13,125,93,161)(14,132,94,168)(15,127,95,163)(16,122,96,158)(17,129,85,165)(18,124,86,160)(19,131,87,167)(20,126,88,162)(21,121,89,157)(22,128,90,164)(23,123,91,159)(24,130,92,166)(25,82,56,97)(26,77,57,104)(27,84,58,99)(28,79,59,106)(29,74,60,101)(30,81,49,108)(31,76,50,103)(32,83,51,98)(33,78,52,105)(34,73,53,100)(35,80,54,107)(36,75,55,102)(37,187,141,154)(38,182,142,149)(39,189,143,156)(40,184,144,151)(41,191,133,146)(42,186,134,153)(43,181,135,148)(44,188,136,155)(45,183,137,150)(46,190,138,145)(47,185,139,152)(48,192,140,147), (1,166,170,130)(2,159,171,123)(3,164,172,128)(4,157,173,121)(5,162,174,126)(6,167,175,131)(7,160,176,124)(8,165,177,129)(9,158,178,122)(10,163,179,127)(11,168,180,132)(12,161,169,125)(13,67,93,117)(14,72,94,110)(15,65,95,115)(16,70,96,120)(17,63,85,113)(18,68,86,118)(19,61,87,111)(20,66,88,116)(21,71,89,109)(22,64,90,114)(23,69,91,119)(24,62,92,112)(25,44,56,136)(26,37,57,141)(27,42,58,134)(28,47,59,139)(29,40,60,144)(30,45,49,137)(31,38,50,142)(32,43,51,135)(33,48,52,140)(34,41,53,133)(35,46,54,138)(36,39,55,143)(73,152,100,185)(74,145,101,190)(75,150,102,183)(76,155,103,188)(77,148,104,181)(78,153,105,186)(79,146,106,191)(80,151,107,184)(81,156,108,189)(82,149,97,182)(83,154,98,187)(84,147,99,192) );
G=PermutationGroup([[(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,184),(14,185),(15,186),(16,187),(17,188),(18,189),(19,190),(20,191),(21,192),(22,181),(23,182),(24,183),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,157),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,177),(57,178),(58,179),(59,180),(60,169),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,73),(85,155),(86,156),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,119),(98,120),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(121,140),(122,141),(123,142),(124,143),(125,144),(126,133),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139)], [(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)], [(1,68,170,118),(2,63,171,113),(3,70,172,120),(4,65,173,115),(5,72,174,110),(6,67,175,117),(7,62,176,112),(8,69,177,119),(9,64,178,114),(10,71,179,109),(11,66,180,116),(12,61,169,111),(13,125,93,161),(14,132,94,168),(15,127,95,163),(16,122,96,158),(17,129,85,165),(18,124,86,160),(19,131,87,167),(20,126,88,162),(21,121,89,157),(22,128,90,164),(23,123,91,159),(24,130,92,166),(25,82,56,97),(26,77,57,104),(27,84,58,99),(28,79,59,106),(29,74,60,101),(30,81,49,108),(31,76,50,103),(32,83,51,98),(33,78,52,105),(34,73,53,100),(35,80,54,107),(36,75,55,102),(37,187,141,154),(38,182,142,149),(39,189,143,156),(40,184,144,151),(41,191,133,146),(42,186,134,153),(43,181,135,148),(44,188,136,155),(45,183,137,150),(46,190,138,145),(47,185,139,152),(48,192,140,147)], [(1,166,170,130),(2,159,171,123),(3,164,172,128),(4,157,173,121),(5,162,174,126),(6,167,175,131),(7,160,176,124),(8,165,177,129),(9,158,178,122),(10,163,179,127),(11,168,180,132),(12,161,169,125),(13,67,93,117),(14,72,94,110),(15,65,95,115),(16,70,96,120),(17,63,85,113),(18,68,86,118),(19,61,87,111),(20,66,88,116),(21,71,89,109),(22,64,90,114),(23,69,91,119),(24,62,92,112),(25,44,56,136),(26,37,57,141),(27,42,58,134),(28,47,59,139),(29,40,60,144),(30,45,49,137),(31,38,50,142),(32,43,51,135),(33,48,52,140),(34,41,53,133),(35,46,54,138),(36,39,55,143),(73,152,100,185),(74,145,101,190),(75,150,102,183),(76,155,103,188),(77,148,104,181),(78,153,105,186),(79,146,106,191),(80,151,107,184),(81,156,108,189),(82,149,97,182),(83,154,98,187),(84,147,99,192)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | Dic6 | D4⋊2S3 | Q8⋊3S3 |
| kernel | C2×C4.Dic6 | C4.Dic6 | C2×C4×Dic3 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C6×C4⋊C4 | C2×C4⋊C4 | C2×C12 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 3 | 1 | 1 | 4 | 4 | 3 | 8 | 8 | 2 | 2 |
Matrix representation of C2×C4.Dic6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 12 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 4 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 12 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 6 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,3,0,0,0,0,0,10,0,0,0,0,0,0,3,12,0,0,0,0,0,9,0,0,0,0,0,0,6,7,0,0,0,0,4,7],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,11,3,0,0,0,0,12,2,0,0,0,0,0,0,9,4,0,0,0,0,6,4] >;
C2×C4.Dic6 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_6 % in TeX
G:=Group("C2xC4.Dic6"); // GroupNames label
G:=SmallGroup(192,1058);
// by ID
G=gap.SmallGroup(192,1058);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,1571,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=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^6*c^-1>;
// generators/relations