direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C8×D13, C104⋊10C22, C52.35C23, C26⋊3(C2×C8), (C2×C104)⋊8C2, C13⋊3(C22×C8), C52.47(C2×C4), (C2×C4).97D26, C4.23(C4×D13), (C4×D13).10C4, D26.20(C2×C4), C13⋊2C8⋊13C22, C26.25(C22×C4), C22.13(C4×D13), C4.35(C22×D13), (C2×C52).110C22, (C2×Dic13).16C4, Dic13.22(C2×C4), (C4×D13).37C22, (C22×D13).10C4, C2.2(C2×C4×D13), (C2×C4×D13).20C2, (C2×C13⋊2C8)⋊13C2, (C2×C26).34(C2×C4), SmallGroup(416,120)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C8×D13 |
Generators and relations for C2×C8×D13
G = < a,b,c,d | a2=b8=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 400 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C13, C2×C8, C2×C8, C22×C4, D13, C26, C26, C22×C8, Dic13, C52, D26, C2×C26, C13⋊2C8, C104, C4×D13, C2×Dic13, C2×C52, C22×D13, C8×D13, C2×C13⋊2C8, C2×C104, C2×C4×D13, C2×C8×D13
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, D13, C22×C8, D26, C4×D13, C22×D13, C8×D13, C2×C4×D13, C2×C8×D13
►On 208 points
Generators in S208
(1 108)(2 109)(3 110)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 105)(12 106)(13 107)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 118)(21 119)(22 120)(23 121)(24 122)(25 123)(26 124)(27 132)(28 133)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 142)(38 143)(39 131)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 160)(54 161)(55 162)(56 163)(57 164)(58 165)(59 166)(60 167)(61 168)(62 169)(63 157)(64 158)(65 159)(66 181)(67 182)(68 170)(69 171)(70 172)(71 173)(72 174)(73 175)(74 176)(75 177)(76 178)(77 179)(78 180)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 183)(89 184)(90 185)(91 186)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)
(1 97 49 75 23 86 37 62)(2 98 50 76 24 87 38 63)(3 99 51 77 25 88 39 64)(4 100 52 78 26 89 27 65)(5 101 40 66 14 90 28 53)(6 102 41 67 15 91 29 54)(7 103 42 68 16 79 30 55)(8 104 43 69 17 80 31 56)(9 92 44 70 18 81 32 57)(10 93 45 71 19 82 33 58)(11 94 46 72 20 83 34 59)(12 95 47 73 21 84 35 60)(13 96 48 74 22 85 36 61)(105 198 150 174 118 191 139 166)(106 199 151 175 119 192 140 167)(107 200 152 176 120 193 141 168)(108 201 153 177 121 194 142 169)(109 202 154 178 122 195 143 157)(110 203 155 179 123 183 131 158)(111 204 156 180 124 184 132 159)(112 205 144 181 125 185 133 160)(113 206 145 182 126 186 134 161)(114 207 146 170 127 187 135 162)(115 208 147 171 128 188 136 163)(116 196 148 172 129 189 137 164)(117 197 149 173 130 190 138 165)
(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)(183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 120)(2 119)(3 118)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 116)(15 115)(16 114)(17 113)(18 112)(19 111)(20 110)(21 109)(22 108)(23 107)(24 106)(25 105)(26 117)(27 149)(28 148)(29 147)(30 146)(31 145)(32 144)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 150)(40 137)(41 136)(42 135)(43 134)(44 133)(45 132)(46 131)(47 143)(48 142)(49 141)(50 140)(51 139)(52 138)(53 172)(54 171)(55 170)(56 182)(57 181)(58 180)(59 179)(60 178)(61 177)(62 176)(63 175)(64 174)(65 173)(66 164)(67 163)(68 162)(69 161)(70 160)(71 159)(72 158)(73 157)(74 169)(75 168)(76 167)(77 166)(78 165)(79 207)(80 206)(81 205)(82 204)(83 203)(84 202)(85 201)(86 200)(87 199)(88 198)(89 197)(90 196)(91 208)(92 185)(93 184)(94 183)(95 195)(96 194)(97 193)(98 192)(99 191)(100 190)(101 189)(102 188)(103 187)(104 186)
G:=sub<Sym(208)| (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,105)(12,106)(13,107)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,118)(21,119)(22,120)(23,121)(24,122)(25,123)(26,124)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,131)(40,144)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,157)(64,158)(65,159)(66,181)(67,182)(68,170)(69,171)(70,172)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(85,193)(86,194)(87,195)(88,183)(89,184)(90,185)(91,186)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208), (1,97,49,75,23,86,37,62)(2,98,50,76,24,87,38,63)(3,99,51,77,25,88,39,64)(4,100,52,78,26,89,27,65)(5,101,40,66,14,90,28,53)(6,102,41,67,15,91,29,54)(7,103,42,68,16,79,30,55)(8,104,43,69,17,80,31,56)(9,92,44,70,18,81,32,57)(10,93,45,71,19,82,33,58)(11,94,46,72,20,83,34,59)(12,95,47,73,21,84,35,60)(13,96,48,74,22,85,36,61)(105,198,150,174,118,191,139,166)(106,199,151,175,119,192,140,167)(107,200,152,176,120,193,141,168)(108,201,153,177,121,194,142,169)(109,202,154,178,122,195,143,157)(110,203,155,179,123,183,131,158)(111,204,156,180,124,184,132,159)(112,205,144,181,125,185,133,160)(113,206,145,182,126,186,134,161)(114,207,146,170,127,187,135,162)(115,208,147,171,128,188,136,163)(116,196,148,172,129,189,137,164)(117,197,149,173,130,190,138,165), (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)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208), (1,120)(2,119)(3,118)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,116)(15,115)(16,114)(17,113)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,117)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,143)(48,142)(49,141)(50,140)(51,139)(52,138)(53,172)(54,171)(55,170)(56,182)(57,181)(58,180)(59,179)(60,178)(61,177)(62,176)(63,175)(64,174)(65,173)(66,164)(67,163)(68,162)(69,161)(70,160)(71,159)(72,158)(73,157)(74,169)(75,168)(76,167)(77,166)(78,165)(79,207)(80,206)(81,205)(82,204)(83,203)(84,202)(85,201)(86,200)(87,199)(88,198)(89,197)(90,196)(91,208)(92,185)(93,184)(94,183)(95,195)(96,194)(97,193)(98,192)(99,191)(100,190)(101,189)(102,188)(103,187)(104,186)>;
G:=Group( (1,108)(2,109)(3,110)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,105)(12,106)(13,107)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,118)(21,119)(22,120)(23,121)(24,122)(25,123)(26,124)(27,132)(28,133)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(39,131)(40,144)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,157)(64,158)(65,159)(66,181)(67,182)(68,170)(69,171)(70,172)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(85,193)(86,194)(87,195)(88,183)(89,184)(90,185)(91,186)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208), (1,97,49,75,23,86,37,62)(2,98,50,76,24,87,38,63)(3,99,51,77,25,88,39,64)(4,100,52,78,26,89,27,65)(5,101,40,66,14,90,28,53)(6,102,41,67,15,91,29,54)(7,103,42,68,16,79,30,55)(8,104,43,69,17,80,31,56)(9,92,44,70,18,81,32,57)(10,93,45,71,19,82,33,58)(11,94,46,72,20,83,34,59)(12,95,47,73,21,84,35,60)(13,96,48,74,22,85,36,61)(105,198,150,174,118,191,139,166)(106,199,151,175,119,192,140,167)(107,200,152,176,120,193,141,168)(108,201,153,177,121,194,142,169)(109,202,154,178,122,195,143,157)(110,203,155,179,123,183,131,158)(111,204,156,180,124,184,132,159)(112,205,144,181,125,185,133,160)(113,206,145,182,126,186,134,161)(114,207,146,170,127,187,135,162)(115,208,147,171,128,188,136,163)(116,196,148,172,129,189,137,164)(117,197,149,173,130,190,138,165), (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)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208), (1,120)(2,119)(3,118)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,116)(15,115)(16,114)(17,113)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,117)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,143)(48,142)(49,141)(50,140)(51,139)(52,138)(53,172)(54,171)(55,170)(56,182)(57,181)(58,180)(59,179)(60,178)(61,177)(62,176)(63,175)(64,174)(65,173)(66,164)(67,163)(68,162)(69,161)(70,160)(71,159)(72,158)(73,157)(74,169)(75,168)(76,167)(77,166)(78,165)(79,207)(80,206)(81,205)(82,204)(83,203)(84,202)(85,201)(86,200)(87,199)(88,198)(89,197)(90,196)(91,208)(92,185)(93,184)(94,183)(95,195)(96,194)(97,193)(98,192)(99,191)(100,190)(101,189)(102,188)(103,187)(104,186) );
G=PermutationGroup([[(1,108),(2,109),(3,110),(4,111),(5,112),(6,113),(7,114),(8,115),(9,116),(10,117),(11,105),(12,106),(13,107),(14,125),(15,126),(16,127),(17,128),(18,129),(19,130),(20,118),(21,119),(22,120),(23,121),(24,122),(25,123),(26,124),(27,132),(28,133),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,141),(37,142),(38,143),(39,131),(40,144),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,160),(54,161),(55,162),(56,163),(57,164),(58,165),(59,166),(60,167),(61,168),(62,169),(63,157),(64,158),(65,159),(66,181),(67,182),(68,170),(69,171),(70,172),(71,173),(72,174),(73,175),(74,176),(75,177),(76,178),(77,179),(78,180),(79,187),(80,188),(81,189),(82,190),(83,191),(84,192),(85,193),(86,194),(87,195),(88,183),(89,184),(90,185),(91,186),(92,196),(93,197),(94,198),(95,199),(96,200),(97,201),(98,202),(99,203),(100,204),(101,205),(102,206),(103,207),(104,208)], [(1,97,49,75,23,86,37,62),(2,98,50,76,24,87,38,63),(3,99,51,77,25,88,39,64),(4,100,52,78,26,89,27,65),(5,101,40,66,14,90,28,53),(6,102,41,67,15,91,29,54),(7,103,42,68,16,79,30,55),(8,104,43,69,17,80,31,56),(9,92,44,70,18,81,32,57),(10,93,45,71,19,82,33,58),(11,94,46,72,20,83,34,59),(12,95,47,73,21,84,35,60),(13,96,48,74,22,85,36,61),(105,198,150,174,118,191,139,166),(106,199,151,175,119,192,140,167),(107,200,152,176,120,193,141,168),(108,201,153,177,121,194,142,169),(109,202,154,178,122,195,143,157),(110,203,155,179,123,183,131,158),(111,204,156,180,124,184,132,159),(112,205,144,181,125,185,133,160),(113,206,145,182,126,186,134,161),(114,207,146,170,127,187,135,162),(115,208,147,171,128,188,136,163),(116,196,148,172,129,189,137,164),(117,197,149,173,130,190,138,165)], [(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),(183,184,185,186,187,188,189,190,191,192,193,194,195),(196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,120),(2,119),(3,118),(4,130),(5,129),(6,128),(7,127),(8,126),(9,125),(10,124),(11,123),(12,122),(13,121),(14,116),(15,115),(16,114),(17,113),(18,112),(19,111),(20,110),(21,109),(22,108),(23,107),(24,106),(25,105),(26,117),(27,149),(28,148),(29,147),(30,146),(31,145),(32,144),(33,156),(34,155),(35,154),(36,153),(37,152),(38,151),(39,150),(40,137),(41,136),(42,135),(43,134),(44,133),(45,132),(46,131),(47,143),(48,142),(49,141),(50,140),(51,139),(52,138),(53,172),(54,171),(55,170),(56,182),(57,181),(58,180),(59,179),(60,178),(61,177),(62,176),(63,175),(64,174),(65,173),(66,164),(67,163),(68,162),(69,161),(70,160),(71,159),(72,158),(73,157),(74,169),(75,168),(76,167),(77,166),(78,165),(79,207),(80,206),(81,205),(82,204),(83,203),(84,202),(85,201),(86,200),(87,199),(88,198),(89,197),(90,196),(91,208),(92,185),(93,184),(94,183),(95,195),(96,194),(97,193),(98,192),(99,191),(100,190),(101,189),(102,188),(103,187),(104,186)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | ··· | 8P | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D13 | D26 | D26 | C4×D13 | C4×D13 | C8×D13 |
| kernel | C2×C8×D13 | C8×D13 | C2×C13⋊2C8 | C2×C104 | C2×C4×D13 | C4×D13 | C2×Dic13 | C22×D13 | D26 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 16 | 6 | 12 | 6 | 12 | 12 | 48 |
Matrix representation of C2×C8×D13 ►in GL4(𝔽313) generated by
| 1 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 125 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 288 | 0 |
| 0 | 0 | 0 | 288 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 43 |
| 0 | 0 | 312 | 227 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 69 | 238 |
| 0 | 0 | 293 | 244 |
G:=sub<GL(4,GF(313))| [1,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[125,0,0,0,0,1,0,0,0,0,288,0,0,0,0,288],[1,0,0,0,0,1,0,0,0,0,66,312,0,0,43,227],[1,0,0,0,0,1,0,0,0,0,69,293,0,0,238,244] >;
C2×C8×D13 in GAP, Magma, Sage, TeX
C_2\times C_8\times D_{13} % in TeX
G:=Group("C2xC8xD13"); // GroupNames label
G:=SmallGroup(416,120);
// by ID
G=gap.SmallGroup(416,120);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,50,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations