direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.6Q8, C42.273D6, (C2×C12).57Q8, C12.77(C2×Q8), C6.3(C22×Q8), (C2×C42).21S3, (C2×C6).13C24, C6⋊1(C42.C2), C4.43(C2×Dic6), (C2×C4).52Dic6, (C22×C4).451D6, C2.5(C22×Dic6), (C2×C12).779C23, (C4×C12).313C22, C22.60(S3×C23), (C2×Dic3).2C23, C22.35(C2×Dic6), C22.66(C4○D12), Dic3⋊C4.94C22, C4⋊Dic3.287C22, C23.321(C22×S3), (C22×C6).375C23, (C22×C12).501C22, (C22×Dic3).73C22, (C2×C4×C12).14C2, C6.2(C2×C4○D4), C3⋊1(C2×C42.C2), C2.7(C2×C4○D12), (C2×C6).47(C2×Q8), (C2×C6).94(C4○D4), (C2×C4⋊Dic3).25C2, (C2×Dic3⋊C4).16C2, (C2×C4).647(C22×S3), SmallGroup(192,1028)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.6Q8
G = < a,b,c,d | a2=b12=c4=1, d2=b6c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b6c-1 >
Subgroups: 440 in 226 conjugacy classes, 127 normal (13 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, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C42.C2, Dic3⋊C4, C4⋊Dic3, C4×C12, C22×Dic3, C22×C12, C22×C12, C2×C42.C2, C12.6Q8, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C4×C12, C2×C12.6Q8
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, C4○D12, S3×C23, C2×C42.C2, C12.6Q8, C22×Dic6, C2×C4○D12, C2×C12.6Q8
►Regular action on 192 points
Generators in S192
(1 172)(2 173)(3 174)(4 175)(5 176)(6 177)(7 178)(8 179)(9 180)(10 169)(11 170)(12 171)(13 142)(14 143)(15 144)(16 133)(17 134)(18 135)(19 136)(20 137)(21 138)(22 139)(23 140)(24 141)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 49)(35 50)(36 51)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 109)(46 110)(47 111)(48 112)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 104)(74 105)(75 106)(76 107)(77 108)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 128)(86 129)(87 130)(88 131)(89 132)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(157 186)(158 187)(159 188)(160 189)(161 190)(162 191)(163 192)(164 181)(165 182)(166 183)(167 184)(168 185)
(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)
(1 151 24 99)(2 152 13 100)(3 153 14 101)(4 154 15 102)(5 155 16 103)(6 156 17 104)(7 145 18 105)(8 146 19 106)(9 147 20 107)(10 148 21 108)(11 149 22 97)(12 150 23 98)(25 116 90 188)(26 117 91 189)(27 118 92 190)(28 119 93 191)(29 120 94 192)(30 109 95 181)(31 110 96 182)(32 111 85 183)(33 112 86 184)(34 113 87 185)(35 114 88 186)(36 115 89 187)(37 130 168 49)(38 131 157 50)(39 132 158 51)(40 121 159 52)(41 122 160 53)(42 123 161 54)(43 124 162 55)(44 125 163 56)(45 126 164 57)(46 127 165 58)(47 128 166 59)(48 129 167 60)(61 141 80 172)(62 142 81 173)(63 143 82 174)(64 144 83 175)(65 133 84 176)(66 134 73 177)(67 135 74 178)(68 136 75 179)(69 137 76 180)(70 138 77 169)(71 139 78 170)(72 140 79 171)
(1 54 18 129)(2 53 19 128)(3 52 20 127)(4 51 21 126)(5 50 22 125)(6 49 23 124)(7 60 24 123)(8 59 13 122)(9 58 14 121)(10 57 15 132)(11 56 16 131)(12 55 17 130)(25 137 96 174)(26 136 85 173)(27 135 86 172)(28 134 87 171)(29 133 88 170)(30 144 89 169)(31 143 90 180)(32 142 91 179)(33 141 92 178)(34 140 93 177)(35 139 94 176)(36 138 95 175)(37 156 162 98)(38 155 163 97)(39 154 164 108)(40 153 165 107)(41 152 166 106)(42 151 167 105)(43 150 168 104)(44 149 157 103)(45 148 158 102)(46 147 159 101)(47 146 160 100)(48 145 161 99)(61 184 74 118)(62 183 75 117)(63 182 76 116)(64 181 77 115)(65 192 78 114)(66 191 79 113)(67 190 80 112)(68 189 81 111)(69 188 82 110)(70 187 83 109)(71 186 84 120)(72 185 73 119)
G:=sub<Sym(192)| (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,169)(11,170)(12,171)(13,142)(14,143)(15,144)(16,133)(17,134)(18,135)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,109)(46,110)(47,111)(48,112)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,128)(86,129)(87,130)(88,131)(89,132)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,181)(165,182)(166,183)(167,184)(168,185), (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), (1,151,24,99)(2,152,13,100)(3,153,14,101)(4,154,15,102)(5,155,16,103)(6,156,17,104)(7,145,18,105)(8,146,19,106)(9,147,20,107)(10,148,21,108)(11,149,22,97)(12,150,23,98)(25,116,90,188)(26,117,91,189)(27,118,92,190)(28,119,93,191)(29,120,94,192)(30,109,95,181)(31,110,96,182)(32,111,85,183)(33,112,86,184)(34,113,87,185)(35,114,88,186)(36,115,89,187)(37,130,168,49)(38,131,157,50)(39,132,158,51)(40,121,159,52)(41,122,160,53)(42,123,161,54)(43,124,162,55)(44,125,163,56)(45,126,164,57)(46,127,165,58)(47,128,166,59)(48,129,167,60)(61,141,80,172)(62,142,81,173)(63,143,82,174)(64,144,83,175)(65,133,84,176)(66,134,73,177)(67,135,74,178)(68,136,75,179)(69,137,76,180)(70,138,77,169)(71,139,78,170)(72,140,79,171), (1,54,18,129)(2,53,19,128)(3,52,20,127)(4,51,21,126)(5,50,22,125)(6,49,23,124)(7,60,24,123)(8,59,13,122)(9,58,14,121)(10,57,15,132)(11,56,16,131)(12,55,17,130)(25,137,96,174)(26,136,85,173)(27,135,86,172)(28,134,87,171)(29,133,88,170)(30,144,89,169)(31,143,90,180)(32,142,91,179)(33,141,92,178)(34,140,93,177)(35,139,94,176)(36,138,95,175)(37,156,162,98)(38,155,163,97)(39,154,164,108)(40,153,165,107)(41,152,166,106)(42,151,167,105)(43,150,168,104)(44,149,157,103)(45,148,158,102)(46,147,159,101)(47,146,160,100)(48,145,161,99)(61,184,74,118)(62,183,75,117)(63,182,76,116)(64,181,77,115)(65,192,78,114)(66,191,79,113)(67,190,80,112)(68,189,81,111)(69,188,82,110)(70,187,83,109)(71,186,84,120)(72,185,73,119)>;
G:=Group( (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,169)(11,170)(12,171)(13,142)(14,143)(15,144)(16,133)(17,134)(18,135)(19,136)(20,137)(21,138)(22,139)(23,140)(24,141)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,109)(46,110)(47,111)(48,112)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,128)(86,129)(87,130)(88,131)(89,132)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,181)(165,182)(166,183)(167,184)(168,185), (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), (1,151,24,99)(2,152,13,100)(3,153,14,101)(4,154,15,102)(5,155,16,103)(6,156,17,104)(7,145,18,105)(8,146,19,106)(9,147,20,107)(10,148,21,108)(11,149,22,97)(12,150,23,98)(25,116,90,188)(26,117,91,189)(27,118,92,190)(28,119,93,191)(29,120,94,192)(30,109,95,181)(31,110,96,182)(32,111,85,183)(33,112,86,184)(34,113,87,185)(35,114,88,186)(36,115,89,187)(37,130,168,49)(38,131,157,50)(39,132,158,51)(40,121,159,52)(41,122,160,53)(42,123,161,54)(43,124,162,55)(44,125,163,56)(45,126,164,57)(46,127,165,58)(47,128,166,59)(48,129,167,60)(61,141,80,172)(62,142,81,173)(63,143,82,174)(64,144,83,175)(65,133,84,176)(66,134,73,177)(67,135,74,178)(68,136,75,179)(69,137,76,180)(70,138,77,169)(71,139,78,170)(72,140,79,171), (1,54,18,129)(2,53,19,128)(3,52,20,127)(4,51,21,126)(5,50,22,125)(6,49,23,124)(7,60,24,123)(8,59,13,122)(9,58,14,121)(10,57,15,132)(11,56,16,131)(12,55,17,130)(25,137,96,174)(26,136,85,173)(27,135,86,172)(28,134,87,171)(29,133,88,170)(30,144,89,169)(31,143,90,180)(32,142,91,179)(33,141,92,178)(34,140,93,177)(35,139,94,176)(36,138,95,175)(37,156,162,98)(38,155,163,97)(39,154,164,108)(40,153,165,107)(41,152,166,106)(42,151,167,105)(43,150,168,104)(44,149,157,103)(45,148,158,102)(46,147,159,101)(47,146,160,100)(48,145,161,99)(61,184,74,118)(62,183,75,117)(63,182,76,116)(64,181,77,115)(65,192,78,114)(66,191,79,113)(67,190,80,112)(68,189,81,111)(69,188,82,110)(70,187,83,109)(71,186,84,120)(72,185,73,119) );
G=PermutationGroup([[(1,172),(2,173),(3,174),(4,175),(5,176),(6,177),(7,178),(8,179),(9,180),(10,169),(11,170),(12,171),(13,142),(14,143),(15,144),(16,133),(17,134),(18,135),(19,136),(20,137),(21,138),(22,139),(23,140),(24,141),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,49),(35,50),(36,51),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,109),(46,110),(47,111),(48,112),(61,151),(62,152),(63,153),(64,154),(65,155),(66,156),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,104),(74,105),(75,106),(76,107),(77,108),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,128),(86,129),(87,130),(88,131),(89,132),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(157,186),(158,187),(159,188),(160,189),(161,190),(162,191),(163,192),(164,181),(165,182),(166,183),(167,184),(168,185)], [(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)], [(1,151,24,99),(2,152,13,100),(3,153,14,101),(4,154,15,102),(5,155,16,103),(6,156,17,104),(7,145,18,105),(8,146,19,106),(9,147,20,107),(10,148,21,108),(11,149,22,97),(12,150,23,98),(25,116,90,188),(26,117,91,189),(27,118,92,190),(28,119,93,191),(29,120,94,192),(30,109,95,181),(31,110,96,182),(32,111,85,183),(33,112,86,184),(34,113,87,185),(35,114,88,186),(36,115,89,187),(37,130,168,49),(38,131,157,50),(39,132,158,51),(40,121,159,52),(41,122,160,53),(42,123,161,54),(43,124,162,55),(44,125,163,56),(45,126,164,57),(46,127,165,58),(47,128,166,59),(48,129,167,60),(61,141,80,172),(62,142,81,173),(63,143,82,174),(64,144,83,175),(65,133,84,176),(66,134,73,177),(67,135,74,178),(68,136,75,179),(69,137,76,180),(70,138,77,169),(71,139,78,170),(72,140,79,171)], [(1,54,18,129),(2,53,19,128),(3,52,20,127),(4,51,21,126),(5,50,22,125),(6,49,23,124),(7,60,24,123),(8,59,13,122),(9,58,14,121),(10,57,15,132),(11,56,16,131),(12,55,17,130),(25,137,96,174),(26,136,85,173),(27,135,86,172),(28,134,87,171),(29,133,88,170),(30,144,89,169),(31,143,90,180),(32,142,91,179),(33,141,92,178),(34,140,93,177),(35,139,94,176),(36,138,95,175),(37,156,162,98),(38,155,163,97),(39,154,164,108),(40,153,165,107),(41,152,166,106),(42,151,167,105),(43,150,168,104),(44,149,157,103),(45,148,158,102),(46,147,159,101),(47,146,160,100),(48,145,161,99),(61,184,74,118),(62,183,75,117),(63,182,76,116),(64,181,77,115),(65,192,78,114),(66,191,79,113),(67,190,80,112),(68,189,81,111),(69,188,82,110),(70,187,83,109),(71,186,84,120),(72,185,73,119)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | Dic6 | C4○D12 |
| kernel | C2×C12.6Q8 | C12.6Q8 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×C4×C12 | C2×C42 | C2×C12 | C42 | C22×C4 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 8 | 4 | 2 | 1 | 1 | 4 | 4 | 3 | 8 | 8 | 16 |
Matrix representation of C2×C12.6Q8 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 3 | 0 | 0 |
| 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 6 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 11 | 9 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,10,10,0,0,0,3,7,0,0,0,0,0,12,1,0,0,0,12,0],[12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,10,6,0,0,0,7,3],[1,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,4,11,0,0,0,2,9] >;
C2×C12.6Q8 in GAP, Magma, Sage, TeX
C_2\times C_{12}._6Q_8 % in TeX
G:=Group("C2xC12.6Q8"); // GroupNames label
G:=SmallGroup(192,1028);
// by ID
G=gap.SmallGroup(192,1028);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,100,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=b^6*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations