direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.Q8, C6⋊1(C4.Q8), C4⋊C4.224D6, C12.15(C4⋊C4), C12.62(C2×Q8), (C2×C12).15Q8, (C2×C12).131D4, (C2×C6).39SD16, C4.27(C2×Dic6), (C2×C4).27Dic6, C6.65(C2×SD16), C12.57(C22×C4), (C22×C6).181D4, (C22×C4).344D6, C4.15(Dic3⋊C4), (C2×C12).316C23, C23.103(C3⋊D4), C4⋊Dic3.322C22, C22.10(D4.S3), C22.24(Dic3⋊C4), (C22×C12).131C22, C22.10(Q8⋊2S3), (C2×C3⋊C8)⋊6C4, C3⋊2(C2×C4.Q8), C3⋊C8⋊24(C2×C4), C4.85(S3×C2×C4), (C6×C4⋊C4).4C2, C6.33(C2×C4⋊C4), (C2×C4⋊C4).5S3, (C22×C3⋊C8).4C2, C2.1(C2×D4.S3), (C2×C6).45(C4⋊C4), (C2×C12).75(C2×C4), (C2×C4).150(C4×S3), (C2×C6).436(C2×D4), C2.8(C2×Dic3⋊C4), C2.1(C2×Q8⋊2S3), (C2×C3⋊C8).240C22, (C2×C4⋊Dic3).30C2, C22.55(C2×C3⋊D4), (C2×C4).123(C3⋊D4), (C3×C4⋊C4).255C22, (C2×C4).416(C22×S3), SmallGroup(192,522)
Series: Derived ►Chief ►Lower central ►Upper central
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, 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, C4.Q8, 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×C4.Q8, C12.Q8, C22×C3⋊C8, C2×C4⋊Dic3, C6×C4⋊C4, C2×C12.Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C4.Q8, C2×C4⋊C4, C2×SD16, Dic3⋊C4, D4.S3, Q8⋊2S3, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×C4.Q8, C12.Q8, C2×Dic3⋊C4, C2×D4.S3, C2×Q8⋊2S3, C2×C12.Q8
►Regular action on 192 points
Generators in S192
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 166)(14 167)(15 168)(16 157)(17 158)(18 159)(19 160)(20 161)(21 162)(22 163)(23 164)(24 165)(25 104)(26 105)(27 106)(28 107)(29 108)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 155)(38 156)(39 145)(40 146)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 142)(50 143)(51 144)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(73 131)(74 132)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 119)(86 120)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(169 182)(170 183)(171 184)(172 185)(173 186)(174 187)(175 188)(176 189)(177 190)(178 191)(179 192)(180 181)
(1 82 39 114)(2 115 40 83)(3 84 41 116)(4 117 42 73)(5 74 43 118)(6 119 44 75)(7 76 45 120)(8 109 46 77)(9 78 47 110)(10 111 48 79)(11 80 37 112)(12 113 38 81)(13 25 183 140)(14 141 184 26)(15 27 185 142)(16 143 186 28)(17 29 187 144)(18 133 188 30)(19 31 189 134)(20 135 190 32)(21 33 191 136)(22 137 192 34)(23 35 181 138)(24 139 182 36)(49 168 106 172)(50 173 107 157)(51 158 108 174)(52 175 97 159)(53 160 98 176)(54 177 99 161)(55 162 100 178)(56 179 101 163)(57 164 102 180)(58 169 103 165)(59 166 104 170)(60 171 105 167)(61 128 145 92)(62 93 146 129)(63 130 147 94)(64 95 148 131)(65 132 149 96)(66 85 150 121)(67 122 151 86)(68 87 152 123)(69 124 153 88)(70 89 154 125)(71 126 155 90)(72 91 156 127)
(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)
(1 183 76 134 39 13 120 31)(2 139 109 188 40 36 77 18)(3 181 78 144 41 23 110 29)(4 137 111 186 42 34 79 16)(5 191 80 142 43 21 112 27)(6 135 113 184 44 32 81 14)(7 189 82 140 45 19 114 25)(8 133 115 182 46 30 83 24)(9 187 84 138 47 17 116 35)(10 143 117 192 48 28 73 22)(11 185 74 136 37 15 118 33)(12 141 119 190 38 26 75 20)(49 149 162 90 106 65 178 126)(50 95 179 154 107 131 163 70)(51 147 164 88 108 63 180 124)(52 93 169 152 97 129 165 68)(53 145 166 86 98 61 170 122)(54 91 171 150 99 127 167 66)(55 155 168 96 100 71 172 132)(56 89 173 148 101 125 157 64)(57 153 158 94 102 69 174 130)(58 87 175 146 103 123 159 62)(59 151 160 92 104 67 176 128)(60 85 177 156 105 121 161 72)
G:=sub<Sym(192)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,166)(14,167)(15,168)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,104)(26,105)(27,106)(28,107)(29,108)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,155)(38,156)(39,145)(40,146)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,142)(50,143)(51,144)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(73,131)(74,132)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,119)(86,120)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(169,182)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,181), (1,82,39,114)(2,115,40,83)(3,84,41,116)(4,117,42,73)(5,74,43,118)(6,119,44,75)(7,76,45,120)(8,109,46,77)(9,78,47,110)(10,111,48,79)(11,80,37,112)(12,113,38,81)(13,25,183,140)(14,141,184,26)(15,27,185,142)(16,143,186,28)(17,29,187,144)(18,133,188,30)(19,31,189,134)(20,135,190,32)(21,33,191,136)(22,137,192,34)(23,35,181,138)(24,139,182,36)(49,168,106,172)(50,173,107,157)(51,158,108,174)(52,175,97,159)(53,160,98,176)(54,177,99,161)(55,162,100,178)(56,179,101,163)(57,164,102,180)(58,169,103,165)(59,166,104,170)(60,171,105,167)(61,128,145,92)(62,93,146,129)(63,130,147,94)(64,95,148,131)(65,132,149,96)(66,85,150,121)(67,122,151,86)(68,87,152,123)(69,124,153,88)(70,89,154,125)(71,126,155,90)(72,91,156,127), (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), (1,183,76,134,39,13,120,31)(2,139,109,188,40,36,77,18)(3,181,78,144,41,23,110,29)(4,137,111,186,42,34,79,16)(5,191,80,142,43,21,112,27)(6,135,113,184,44,32,81,14)(7,189,82,140,45,19,114,25)(8,133,115,182,46,30,83,24)(9,187,84,138,47,17,116,35)(10,143,117,192,48,28,73,22)(11,185,74,136,37,15,118,33)(12,141,119,190,38,26,75,20)(49,149,162,90,106,65,178,126)(50,95,179,154,107,131,163,70)(51,147,164,88,108,63,180,124)(52,93,169,152,97,129,165,68)(53,145,166,86,98,61,170,122)(54,91,171,150,99,127,167,66)(55,155,168,96,100,71,172,132)(56,89,173,148,101,125,157,64)(57,153,158,94,102,69,174,130)(58,87,175,146,103,123,159,62)(59,151,160,92,104,67,176,128)(60,85,177,156,105,121,161,72)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,166)(14,167)(15,168)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,104)(26,105)(27,106)(28,107)(29,108)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,155)(38,156)(39,145)(40,146)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,142)(50,143)(51,144)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(73,131)(74,132)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,119)(86,120)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(169,182)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,181), (1,82,39,114)(2,115,40,83)(3,84,41,116)(4,117,42,73)(5,74,43,118)(6,119,44,75)(7,76,45,120)(8,109,46,77)(9,78,47,110)(10,111,48,79)(11,80,37,112)(12,113,38,81)(13,25,183,140)(14,141,184,26)(15,27,185,142)(16,143,186,28)(17,29,187,144)(18,133,188,30)(19,31,189,134)(20,135,190,32)(21,33,191,136)(22,137,192,34)(23,35,181,138)(24,139,182,36)(49,168,106,172)(50,173,107,157)(51,158,108,174)(52,175,97,159)(53,160,98,176)(54,177,99,161)(55,162,100,178)(56,179,101,163)(57,164,102,180)(58,169,103,165)(59,166,104,170)(60,171,105,167)(61,128,145,92)(62,93,146,129)(63,130,147,94)(64,95,148,131)(65,132,149,96)(66,85,150,121)(67,122,151,86)(68,87,152,123)(69,124,153,88)(70,89,154,125)(71,126,155,90)(72,91,156,127), (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), (1,183,76,134,39,13,120,31)(2,139,109,188,40,36,77,18)(3,181,78,144,41,23,110,29)(4,137,111,186,42,34,79,16)(5,191,80,142,43,21,112,27)(6,135,113,184,44,32,81,14)(7,189,82,140,45,19,114,25)(8,133,115,182,46,30,83,24)(9,187,84,138,47,17,116,35)(10,143,117,192,48,28,73,22)(11,185,74,136,37,15,118,33)(12,141,119,190,38,26,75,20)(49,149,162,90,106,65,178,126)(50,95,179,154,107,131,163,70)(51,147,164,88,108,63,180,124)(52,93,169,152,97,129,165,68)(53,145,166,86,98,61,170,122)(54,91,171,150,99,127,167,66)(55,155,168,96,100,71,172,132)(56,89,173,148,101,125,157,64)(57,153,158,94,102,69,174,130)(58,87,175,146,103,123,159,62)(59,151,160,92,104,67,176,128)(60,85,177,156,105,121,161,72) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,166),(14,167),(15,168),(16,157),(17,158),(18,159),(19,160),(20,161),(21,162),(22,163),(23,164),(24,165),(25,104),(26,105),(27,106),(28,107),(29,108),(30,97),(31,98),(32,99),(33,100),(34,101),(35,102),(36,103),(37,155),(38,156),(39,145),(40,146),(41,147),(42,148),(43,149),(44,150),(45,151),(46,152),(47,153),(48,154),(49,142),(50,143),(51,144),(52,133),(53,134),(54,135),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(73,131),(74,132),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,127),(82,128),(83,129),(84,130),(85,119),(86,120),(87,109),(88,110),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(169,182),(170,183),(171,184),(172,185),(173,186),(174,187),(175,188),(176,189),(177,190),(178,191),(179,192),(180,181)], [(1,82,39,114),(2,115,40,83),(3,84,41,116),(4,117,42,73),(5,74,43,118),(6,119,44,75),(7,76,45,120),(8,109,46,77),(9,78,47,110),(10,111,48,79),(11,80,37,112),(12,113,38,81),(13,25,183,140),(14,141,184,26),(15,27,185,142),(16,143,186,28),(17,29,187,144),(18,133,188,30),(19,31,189,134),(20,135,190,32),(21,33,191,136),(22,137,192,34),(23,35,181,138),(24,139,182,36),(49,168,106,172),(50,173,107,157),(51,158,108,174),(52,175,97,159),(53,160,98,176),(54,177,99,161),(55,162,100,178),(56,179,101,163),(57,164,102,180),(58,169,103,165),(59,166,104,170),(60,171,105,167),(61,128,145,92),(62,93,146,129),(63,130,147,94),(64,95,148,131),(65,132,149,96),(66,85,150,121),(67,122,151,86),(68,87,152,123),(69,124,153,88),(70,89,154,125),(71,126,155,90),(72,91,156,127)], [(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)], [(1,183,76,134,39,13,120,31),(2,139,109,188,40,36,77,18),(3,181,78,144,41,23,110,29),(4,137,111,186,42,34,79,16),(5,191,80,142,43,21,112,27),(6,135,113,184,44,32,81,14),(7,189,82,140,45,19,114,25),(8,133,115,182,46,30,83,24),(9,187,84,138,47,17,116,35),(10,143,117,192,48,28,73,22),(11,185,74,136,37,15,118,33),(12,141,119,190,38,26,75,20),(49,149,162,90,106,65,178,126),(50,95,179,154,107,131,163,70),(51,147,164,88,108,63,180,124),(52,93,169,152,97,129,165,68),(53,145,166,86,98,61,170,122),(54,91,171,150,99,127,167,66),(55,155,168,96,100,71,172,132),(56,89,173,148,101,125,157,64),(57,153,158,94,102,69,174,130),(58,87,175,146,103,123,159,62),(59,151,160,92,104,67,176,128),(60,85,177,156,105,121,161,72)]])
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