metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12).288D4, (C2×Dic3).8Q8, C22.49(S3×Q8), C2.7(D6⋊3Q8), (C22×C4).118D6, C6.71(C22⋊Q8), C6.40(C4.4D4), C6.24(C42.C2), C6.26(C42⋊2C2), C2.7(C23.12D6), C2.14(Dic3.Q8), C6.C42.20C2, (C22×C6).349C23, C23.388(C22×S3), C22.106(C4○D12), (C22×C12).392C22, C3⋊4(C23.83C23), C22.49(Q8⋊3S3), C6.63(C22.D4), C22.102(D4⋊2S3), C2.13(C23.28D6), (C22×Dic3).56C22, (C6×C4⋊C4).32C2, (C2×C4⋊C4).22S3, (C2×C6).82(C2×Q8), (C2×C6).449(C2×D4), (C2×C4).39(C3⋊D4), C2.12(C4⋊C4⋊S3), (C2×C6).155(C4○D4), (C2×Dic3⋊C4).32C2, C22.138(C2×C3⋊D4), SmallGroup(192,544)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C12).288D4
G = < a,b,c,d | a2=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab5, dcd-1=ac-1 >
Subgroups: 328 in 134 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, 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, Dic3⋊C4, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23.83C23, C6.C42, C6.C42, C2×Dic3⋊C4, C6×C4⋊C4, (C2×C12).288D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4○D12, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C23.83C23, Dic3.Q8, C4⋊C4⋊S3, C23.28D6, C23.12D6, D6⋊3Q8, (C2×C12).288D4
►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 72)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 125)(26 126)(27 127)(28 128)(29 129)(30 130)(31 131)(32 132)(33 121)(34 122)(35 123)(36 124)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 49)(46 50)(47 51)(48 52)(85 171)(86 172)(87 173)(88 174)(89 175)(90 176)(91 177)(92 178)(93 179)(94 180)(95 169)(96 170)(97 168)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 192)(110 181)(111 182)(112 183)(113 184)(114 185)(115 186)(116 187)(117 188)(118 189)(119 190)(120 191)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)(141 156)(142 145)(143 146)(144 147)
(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)
(1 184 17 163)(2 112 18 103)(3 182 19 161)(4 110 20 101)(5 192 21 159)(6 120 22 99)(7 190 23 157)(8 118 24 97)(9 188 13 167)(10 116 14 107)(11 186 15 165)(12 114 16 105)(25 144 174 39)(26 146 175 54)(27 142 176 37)(28 156 177 52)(29 140 178 47)(30 154 179 50)(31 138 180 45)(32 152 169 60)(33 136 170 43)(34 150 171 58)(35 134 172 41)(36 148 173 56)(38 126 143 89)(40 124 133 87)(42 122 135 85)(44 132 137 95)(46 130 139 93)(48 128 141 91)(49 131 153 94)(51 129 155 92)(53 127 145 90)(55 125 147 88)(57 123 149 86)(59 121 151 96)(61 166 79 187)(62 106 80 115)(63 164 81 185)(64 104 82 113)(65 162 83 183)(66 102 84 111)(67 160 73 181)(68 100 74 109)(69 158 75 191)(70 98 76 119)(71 168 77 189)(72 108 78 117)
(1 87 7 93)(2 178 8 172)(3 85 9 91)(4 176 10 170)(5 95 11 89)(6 174 12 180)(13 128 19 122)(14 33 20 27)(15 126 21 132)(16 31 22 25)(17 124 23 130)(18 29 24 35)(26 68 32 62)(28 66 34 72)(30 64 36 70)(37 166 43 160)(38 100 44 106)(39 164 45 158)(40 98 46 104)(41 162 47 168)(42 108 48 102)(49 99 55 105)(50 163 56 157)(51 97 57 103)(52 161 58 167)(53 107 59 101)(54 159 60 165)(61 121 67 127)(63 131 69 125)(65 129 71 123)(73 90 79 96)(74 169 80 175)(75 88 81 94)(76 179 82 173)(77 86 83 92)(78 177 84 171)(109 137 115 143)(110 145 116 151)(111 135 117 141)(112 155 118 149)(113 133 119 139)(114 153 120 147)(134 183 140 189)(136 181 142 187)(138 191 144 185)(146 192 152 186)(148 190 154 184)(150 188 156 182)
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)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,121)(34,122)(35,123)(36,124)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,49)(46,50)(47,51)(48,52)(85,171)(86,172)(87,173)(88,174)(89,175)(90,176)(91,177)(92,178)(93,179)(94,180)(95,169)(96,170)(97,168)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,192)(110,181)(111,182)(112,183)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,145)(143,146)(144,147), (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), (1,184,17,163)(2,112,18,103)(3,182,19,161)(4,110,20,101)(5,192,21,159)(6,120,22,99)(7,190,23,157)(8,118,24,97)(9,188,13,167)(10,116,14,107)(11,186,15,165)(12,114,16,105)(25,144,174,39)(26,146,175,54)(27,142,176,37)(28,156,177,52)(29,140,178,47)(30,154,179,50)(31,138,180,45)(32,152,169,60)(33,136,170,43)(34,150,171,58)(35,134,172,41)(36,148,173,56)(38,126,143,89)(40,124,133,87)(42,122,135,85)(44,132,137,95)(46,130,139,93)(48,128,141,91)(49,131,153,94)(51,129,155,92)(53,127,145,90)(55,125,147,88)(57,123,149,86)(59,121,151,96)(61,166,79,187)(62,106,80,115)(63,164,81,185)(64,104,82,113)(65,162,83,183)(66,102,84,111)(67,160,73,181)(68,100,74,109)(69,158,75,191)(70,98,76,119)(71,168,77,189)(72,108,78,117), (1,87,7,93)(2,178,8,172)(3,85,9,91)(4,176,10,170)(5,95,11,89)(6,174,12,180)(13,128,19,122)(14,33,20,27)(15,126,21,132)(16,31,22,25)(17,124,23,130)(18,29,24,35)(26,68,32,62)(28,66,34,72)(30,64,36,70)(37,166,43,160)(38,100,44,106)(39,164,45,158)(40,98,46,104)(41,162,47,168)(42,108,48,102)(49,99,55,105)(50,163,56,157)(51,97,57,103)(52,161,58,167)(53,107,59,101)(54,159,60,165)(61,121,67,127)(63,131,69,125)(65,129,71,123)(73,90,79,96)(74,169,80,175)(75,88,81,94)(76,179,82,173)(77,86,83,92)(78,177,84,171)(109,137,115,143)(110,145,116,151)(111,135,117,141)(112,155,118,149)(113,133,119,139)(114,153,120,147)(134,183,140,189)(136,181,142,187)(138,191,144,185)(146,192,152,186)(148,190,154,184)(150,188,156,182)>;
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)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,121)(34,122)(35,123)(36,124)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,49)(46,50)(47,51)(48,52)(85,171)(86,172)(87,173)(88,174)(89,175)(90,176)(91,177)(92,178)(93,179)(94,180)(95,169)(96,170)(97,168)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,192)(110,181)(111,182)(112,183)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155)(141,156)(142,145)(143,146)(144,147), (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), (1,184,17,163)(2,112,18,103)(3,182,19,161)(4,110,20,101)(5,192,21,159)(6,120,22,99)(7,190,23,157)(8,118,24,97)(9,188,13,167)(10,116,14,107)(11,186,15,165)(12,114,16,105)(25,144,174,39)(26,146,175,54)(27,142,176,37)(28,156,177,52)(29,140,178,47)(30,154,179,50)(31,138,180,45)(32,152,169,60)(33,136,170,43)(34,150,171,58)(35,134,172,41)(36,148,173,56)(38,126,143,89)(40,124,133,87)(42,122,135,85)(44,132,137,95)(46,130,139,93)(48,128,141,91)(49,131,153,94)(51,129,155,92)(53,127,145,90)(55,125,147,88)(57,123,149,86)(59,121,151,96)(61,166,79,187)(62,106,80,115)(63,164,81,185)(64,104,82,113)(65,162,83,183)(66,102,84,111)(67,160,73,181)(68,100,74,109)(69,158,75,191)(70,98,76,119)(71,168,77,189)(72,108,78,117), (1,87,7,93)(2,178,8,172)(3,85,9,91)(4,176,10,170)(5,95,11,89)(6,174,12,180)(13,128,19,122)(14,33,20,27)(15,126,21,132)(16,31,22,25)(17,124,23,130)(18,29,24,35)(26,68,32,62)(28,66,34,72)(30,64,36,70)(37,166,43,160)(38,100,44,106)(39,164,45,158)(40,98,46,104)(41,162,47,168)(42,108,48,102)(49,99,55,105)(50,163,56,157)(51,97,57,103)(52,161,58,167)(53,107,59,101)(54,159,60,165)(61,121,67,127)(63,131,69,125)(65,129,71,123)(73,90,79,96)(74,169,80,175)(75,88,81,94)(76,179,82,173)(77,86,83,92)(78,177,84,171)(109,137,115,143)(110,145,116,151)(111,135,117,141)(112,155,118,149)(113,133,119,139)(114,153,120,147)(134,183,140,189)(136,181,142,187)(138,191,144,185)(146,192,152,186)(148,190,154,184)(150,188,156,182) );
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),(13,72),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,125),(26,126),(27,127),(28,128),(29,129),(30,130),(31,131),(32,132),(33,121),(34,122),(35,123),(36,124),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,49),(46,50),(47,51),(48,52),(85,171),(86,172),(87,173),(88,174),(89,175),(90,176),(91,177),(92,178),(93,179),(94,180),(95,169),(96,170),(97,168),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,192),(110,181),(111,182),(112,183),(113,184),(114,185),(115,186),(116,187),(117,188),(118,189),(119,190),(120,191),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155),(141,156),(142,145),(143,146),(144,147)], [(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)], [(1,184,17,163),(2,112,18,103),(3,182,19,161),(4,110,20,101),(5,192,21,159),(6,120,22,99),(7,190,23,157),(8,118,24,97),(9,188,13,167),(10,116,14,107),(11,186,15,165),(12,114,16,105),(25,144,174,39),(26,146,175,54),(27,142,176,37),(28,156,177,52),(29,140,178,47),(30,154,179,50),(31,138,180,45),(32,152,169,60),(33,136,170,43),(34,150,171,58),(35,134,172,41),(36,148,173,56),(38,126,143,89),(40,124,133,87),(42,122,135,85),(44,132,137,95),(46,130,139,93),(48,128,141,91),(49,131,153,94),(51,129,155,92),(53,127,145,90),(55,125,147,88),(57,123,149,86),(59,121,151,96),(61,166,79,187),(62,106,80,115),(63,164,81,185),(64,104,82,113),(65,162,83,183),(66,102,84,111),(67,160,73,181),(68,100,74,109),(69,158,75,191),(70,98,76,119),(71,168,77,189),(72,108,78,117)], [(1,87,7,93),(2,178,8,172),(3,85,9,91),(4,176,10,170),(5,95,11,89),(6,174,12,180),(13,128,19,122),(14,33,20,27),(15,126,21,132),(16,31,22,25),(17,124,23,130),(18,29,24,35),(26,68,32,62),(28,66,34,72),(30,64,36,70),(37,166,43,160),(38,100,44,106),(39,164,45,158),(40,98,46,104),(41,162,47,168),(42,108,48,102),(49,99,55,105),(50,163,56,157),(51,97,57,103),(52,161,58,167),(53,107,59,101),(54,159,60,165),(61,121,67,127),(63,131,69,125),(65,129,71,123),(73,90,79,96),(74,169,80,175),(75,88,81,94),(76,179,82,173),(77,86,83,92),(78,177,84,171),(109,137,115,143),(110,145,116,151),(111,135,117,141),(112,155,118,149),(113,133,119,139),(114,153,120,147),(134,183,140,189),(136,181,142,187),(138,191,144,185),(146,192,152,186),(148,190,154,184),(150,188,156,182)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4F | 4G | ··· | 4N | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | C4○D4 | C3⋊D4 | C4○D12 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | (C2×C12).288D4 | C6.C42 | C2×Dic3⋊C4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 5 | 1 | 1 | 1 | 2 | 2 | 3 | 10 | 4 | 8 | 2 | 1 | 1 |
Matrix representation of (C2×C12).288D4 ►in GL6(𝔽13)
| 1 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,0,0,0,0,0,0,10],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[9,10,0,0,0,0,10,4,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
(C2×C12).288D4 in GAP, Magma, Sage, TeX
(C_2\times C_{12})._{288}D_4 % in TeX
G:=Group("(C2xC12).288D4"); // GroupNames label
G:=SmallGroup(192,544);
// by ID
G=gap.SmallGroup(192,544);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,254,387,268,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^5,d*c*d^-1=a*c^-1>;
// generators/relations