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×C6).13C24, (C2×C42).21S3, C6⋊1(C42.C2), (C2×C4).52Dic6, C4.43(C2×Dic6), (C22×C4).451D6, C2.5(C22×Dic6), (C4×C12).313C22, (C2×C12).779C23, C22.60(S3×C23), (C2×Dic3).2C23, C22.35(C2×Dic6), C22.66(C4○D12), Dic3⋊C4.94C22, C4⋊Dic3.287C22, (C22×C6).375C23, C23.321(C22×S3), (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
Subgroups: 440 in 226 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×12], C22, C22 [×6], C6, C6 [×6], C2×C4 [×10], C2×C4 [×20], C23, Dic3 [×8], C12 [×4], C12 [×4], C2×C6, C2×C6 [×6], C42 [×4], C4⋊C4 [×24], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×10], C2×C12 [×4], C22×C6, C2×C42, C2×C4⋊C4 [×6], C42.C2 [×8], Dic3⋊C4 [×16], C4⋊Dic3 [×8], C4×C12 [×4], C22×Dic3 [×4], C22×C12, C22×C12 [×2], C2×C42.C2, C12.6Q8 [×8], C2×Dic3⋊C4 [×4], C2×C4⋊Dic3 [×2], C2×C4×C12, C2×C12.6Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, Dic6 [×4], C22×S3 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C2×Dic6 [×6], C4○D12 [×4], S3×C23, C2×C42.C2, C12.6Q8 [×4], C22×Dic6, C2×C4○D12 [×2], C2×C12.6Q8
Generators and relations
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 >
►Regular action on 192 points
Generators in S192
(1 83)(2 84)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 156)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 151)(21 152)(22 153)(23 154)(24 155)(25 182)(26 183)(27 184)(28 185)(29 186)(30 187)(31 188)(32 189)(33 190)(34 191)(35 192)(36 181)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 61)(46 62)(47 63)(48 64)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 97)(58 98)(59 99)(60 100)(85 120)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(121 143)(122 144)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(131 141)(132 142)(157 179)(158 180)(159 169)(160 170)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)
(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)
(1 186 47 168)(2 187 48 157)(3 188 37 158)(4 189 38 159)(5 190 39 160)(6 191 40 161)(7 192 41 162)(8 181 42 163)(9 182 43 164)(10 183 44 165)(11 184 45 166)(12 185 46 167)(13 57 141 117)(14 58 142 118)(15 59 143 119)(16 60 144 120)(17 49 133 109)(18 50 134 110)(19 51 135 111)(20 52 136 112)(21 53 137 113)(22 54 138 114)(23 55 139 115)(24 56 140 116)(25 71 174 79)(26 72 175 80)(27 61 176 81)(28 62 177 82)(29 63 178 83)(30 64 179 84)(31 65 180 73)(32 66 169 74)(33 67 170 75)(34 68 171 76)(35 69 172 77)(36 70 173 78)(85 147 100 122)(86 148 101 123)(87 149 102 124)(88 150 103 125)(89 151 104 126)(90 152 105 127)(91 153 106 128)(92 154 107 129)(93 155 108 130)(94 156 97 131)(95 145 98 132)(96 146 99 121)
(1 114 41 60)(2 113 42 59)(3 112 43 58)(4 111 44 57)(5 110 45 56)(6 109 46 55)(7 120 47 54)(8 119 48 53)(9 118 37 52)(10 117 38 51)(11 116 39 50)(12 115 40 49)(13 183 135 159)(14 182 136 158)(15 181 137 157)(16 192 138 168)(17 191 139 167)(18 190 140 166)(19 189 141 165)(20 188 142 164)(21 187 143 163)(22 186 144 162)(23 185 133 161)(24 184 134 160)(25 126 180 145)(26 125 169 156)(27 124 170 155)(28 123 171 154)(29 122 172 153)(30 121 173 152)(31 132 174 151)(32 131 175 150)(33 130 176 149)(34 129 177 148)(35 128 178 147)(36 127 179 146)(61 108 75 87)(62 107 76 86)(63 106 77 85)(64 105 78 96)(65 104 79 95)(66 103 80 94)(67 102 81 93)(68 101 82 92)(69 100 83 91)(70 99 84 90)(71 98 73 89)(72 97 74 88)
G:=sub<Sym(192)| (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,156)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,154)(24,155)(25,182)(26,183)(27,184)(28,185)(29,186)(30,187)(31,188)(32,189)(33,190)(34,191)(35,192)(36,181)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,61)(46,62)(47,63)(48,64)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(85,120)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(121,143)(122,144)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142)(157,179)(158,180)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (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), (1,186,47,168)(2,187,48,157)(3,188,37,158)(4,189,38,159)(5,190,39,160)(6,191,40,161)(7,192,41,162)(8,181,42,163)(9,182,43,164)(10,183,44,165)(11,184,45,166)(12,185,46,167)(13,57,141,117)(14,58,142,118)(15,59,143,119)(16,60,144,120)(17,49,133,109)(18,50,134,110)(19,51,135,111)(20,52,136,112)(21,53,137,113)(22,54,138,114)(23,55,139,115)(24,56,140,116)(25,71,174,79)(26,72,175,80)(27,61,176,81)(28,62,177,82)(29,63,178,83)(30,64,179,84)(31,65,180,73)(32,66,169,74)(33,67,170,75)(34,68,171,76)(35,69,172,77)(36,70,173,78)(85,147,100,122)(86,148,101,123)(87,149,102,124)(88,150,103,125)(89,151,104,126)(90,152,105,127)(91,153,106,128)(92,154,107,129)(93,155,108,130)(94,156,97,131)(95,145,98,132)(96,146,99,121), (1,114,41,60)(2,113,42,59)(3,112,43,58)(4,111,44,57)(5,110,45,56)(6,109,46,55)(7,120,47,54)(8,119,48,53)(9,118,37,52)(10,117,38,51)(11,116,39,50)(12,115,40,49)(13,183,135,159)(14,182,136,158)(15,181,137,157)(16,192,138,168)(17,191,139,167)(18,190,140,166)(19,189,141,165)(20,188,142,164)(21,187,143,163)(22,186,144,162)(23,185,133,161)(24,184,134,160)(25,126,180,145)(26,125,169,156)(27,124,170,155)(28,123,171,154)(29,122,172,153)(30,121,173,152)(31,132,174,151)(32,131,175,150)(33,130,176,149)(34,129,177,148)(35,128,178,147)(36,127,179,146)(61,108,75,87)(62,107,76,86)(63,106,77,85)(64,105,78,96)(65,104,79,95)(66,103,80,94)(67,102,81,93)(68,101,82,92)(69,100,83,91)(70,99,84,90)(71,98,73,89)(72,97,74,88)>;
G:=Group( (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,156)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,154)(24,155)(25,182)(26,183)(27,184)(28,185)(29,186)(30,187)(31,188)(32,189)(33,190)(34,191)(35,192)(36,181)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,61)(46,62)(47,63)(48,64)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(85,120)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(121,143)(122,144)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142)(157,179)(158,180)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (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), (1,186,47,168)(2,187,48,157)(3,188,37,158)(4,189,38,159)(5,190,39,160)(6,191,40,161)(7,192,41,162)(8,181,42,163)(9,182,43,164)(10,183,44,165)(11,184,45,166)(12,185,46,167)(13,57,141,117)(14,58,142,118)(15,59,143,119)(16,60,144,120)(17,49,133,109)(18,50,134,110)(19,51,135,111)(20,52,136,112)(21,53,137,113)(22,54,138,114)(23,55,139,115)(24,56,140,116)(25,71,174,79)(26,72,175,80)(27,61,176,81)(28,62,177,82)(29,63,178,83)(30,64,179,84)(31,65,180,73)(32,66,169,74)(33,67,170,75)(34,68,171,76)(35,69,172,77)(36,70,173,78)(85,147,100,122)(86,148,101,123)(87,149,102,124)(88,150,103,125)(89,151,104,126)(90,152,105,127)(91,153,106,128)(92,154,107,129)(93,155,108,130)(94,156,97,131)(95,145,98,132)(96,146,99,121), (1,114,41,60)(2,113,42,59)(3,112,43,58)(4,111,44,57)(5,110,45,56)(6,109,46,55)(7,120,47,54)(8,119,48,53)(9,118,37,52)(10,117,38,51)(11,116,39,50)(12,115,40,49)(13,183,135,159)(14,182,136,158)(15,181,137,157)(16,192,138,168)(17,191,139,167)(18,190,140,166)(19,189,141,165)(20,188,142,164)(21,187,143,163)(22,186,144,162)(23,185,133,161)(24,184,134,160)(25,126,180,145)(26,125,169,156)(27,124,170,155)(28,123,171,154)(29,122,172,153)(30,121,173,152)(31,132,174,151)(32,131,175,150)(33,130,176,149)(34,129,177,148)(35,128,178,147)(36,127,179,146)(61,108,75,87)(62,107,76,86)(63,106,77,85)(64,105,78,96)(65,104,79,95)(66,103,80,94)(67,102,81,93)(68,101,82,92)(69,100,83,91)(70,99,84,90)(71,98,73,89)(72,97,74,88) );
G=PermutationGroup([(1,83),(2,84),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,156),(14,145),(15,146),(16,147),(17,148),(18,149),(19,150),(20,151),(21,152),(22,153),(23,154),(24,155),(25,182),(26,183),(27,184),(28,185),(29,186),(30,187),(31,188),(32,189),(33,190),(34,191),(35,192),(36,181),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,61),(46,62),(47,63),(48,64),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,97),(58,98),(59,99),(60,100),(85,120),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(121,143),(122,144),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(131,141),(132,142),(157,179),(158,180),(159,169),(160,170),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178)], [(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)], [(1,186,47,168),(2,187,48,157),(3,188,37,158),(4,189,38,159),(5,190,39,160),(6,191,40,161),(7,192,41,162),(8,181,42,163),(9,182,43,164),(10,183,44,165),(11,184,45,166),(12,185,46,167),(13,57,141,117),(14,58,142,118),(15,59,143,119),(16,60,144,120),(17,49,133,109),(18,50,134,110),(19,51,135,111),(20,52,136,112),(21,53,137,113),(22,54,138,114),(23,55,139,115),(24,56,140,116),(25,71,174,79),(26,72,175,80),(27,61,176,81),(28,62,177,82),(29,63,178,83),(30,64,179,84),(31,65,180,73),(32,66,169,74),(33,67,170,75),(34,68,171,76),(35,69,172,77),(36,70,173,78),(85,147,100,122),(86,148,101,123),(87,149,102,124),(88,150,103,125),(89,151,104,126),(90,152,105,127),(91,153,106,128),(92,154,107,129),(93,155,108,130),(94,156,97,131),(95,145,98,132),(96,146,99,121)], [(1,114,41,60),(2,113,42,59),(3,112,43,58),(4,111,44,57),(5,110,45,56),(6,109,46,55),(7,120,47,54),(8,119,48,53),(9,118,37,52),(10,117,38,51),(11,116,39,50),(12,115,40,49),(13,183,135,159),(14,182,136,158),(15,181,137,157),(16,192,138,168),(17,191,139,167),(18,190,140,166),(19,189,141,165),(20,188,142,164),(21,187,143,163),(22,186,144,162),(23,185,133,161),(24,184,134,160),(25,126,180,145),(26,125,169,156),(27,124,170,155),(28,123,171,154),(29,122,172,153),(30,121,173,152),(31,132,174,151),(32,131,175,150),(33,130,176,149),(34,129,177,148),(35,128,178,147),(36,127,179,146),(61,108,75,87),(62,107,76,86),(63,106,77,85),(64,105,78,96),(65,104,79,95),(66,103,80,94),(67,102,81,93),(68,101,82,92),(69,100,83,91),(70,99,84,90),(71,98,73,89),(72,97,74,88)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀