direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C4⋊1D4, C52⋊6D4, C42⋊6C26, C4⋊1(D4×C13), (C4×C52)⋊13C2, (C2×D4)⋊3C26, C2.9(D4×C26), (D4×C26)⋊12C2, C26.72(C2×D4), C23.4(C2×C26), (C2×C26).82C23, (C2×C52).125C22, (C22×C26).4C22, C22.17(C22×C26), (C2×C4).23(C2×C26), SmallGroup(416,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4⋊1D4
G = < a,b,c,d | a13=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C13, C42, C2×D4, C26, C26, C4⋊1D4, C52, C2×C26, C2×C26, C2×C52, D4×C13, C22×C26, C4×C52, D4×C26, C13×C4⋊1D4
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C26, C4⋊1D4, C2×C26, D4×C13, C22×C26, D4×C26, C13×C4⋊1D4
►On 208 points
Generators in S208
(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 17 113 118)(2 18 114 119)(3 19 115 120)(4 20 116 121)(5 21 117 122)(6 22 105 123)(7 23 106 124)(8 24 107 125)(9 25 108 126)(10 26 109 127)(11 14 110 128)(12 15 111 129)(13 16 112 130)(27 71 142 203)(28 72 143 204)(29 73 131 205)(30 74 132 206)(31 75 133 207)(32 76 134 208)(33 77 135 196)(34 78 136 197)(35 66 137 198)(36 67 138 199)(37 68 139 200)(38 69 140 201)(39 70 141 202)(40 80 167 185)(41 81 168 186)(42 82 169 187)(43 83 157 188)(44 84 158 189)(45 85 159 190)(46 86 160 191)(47 87 161 192)(48 88 162 193)(49 89 163 194)(50 90 164 195)(51 91 165 183)(52 79 166 184)(53 97 156 181)(54 98 144 182)(55 99 145 170)(56 100 146 171)(57 101 147 172)(58 102 148 173)(59 103 149 174)(60 104 150 175)(61 92 151 176)(62 93 152 177)(63 94 153 178)(64 95 154 179)(65 96 155 180)
(1 137 85 59)(2 138 86 60)(3 139 87 61)(4 140 88 62)(5 141 89 63)(6 142 90 64)(7 143 91 65)(8 131 79 53)(9 132 80 54)(10 133 81 55)(11 134 82 56)(12 135 83 57)(13 136 84 58)(14 208 169 100)(15 196 157 101)(16 197 158 102)(17 198 159 103)(18 199 160 104)(19 200 161 92)(20 201 162 93)(21 202 163 94)(22 203 164 95)(23 204 165 96)(24 205 166 97)(25 206 167 98)(26 207 168 99)(27 195 154 105)(28 183 155 106)(29 184 156 107)(30 185 144 108)(31 186 145 109)(32 187 146 110)(33 188 147 111)(34 189 148 112)(35 190 149 113)(36 191 150 114)(37 192 151 115)(38 193 152 116)(39 194 153 117)(40 182 126 74)(41 170 127 75)(42 171 128 76)(43 172 129 77)(44 173 130 78)(45 174 118 66)(46 175 119 67)(47 176 120 68)(48 177 121 69)(49 178 122 70)(50 179 123 71)(51 180 124 72)(52 181 125 73)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 27)(23 28)(24 29)(25 30)(26 31)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 53)(79 181)(80 182)(81 170)(82 171)(83 172)(84 173)(85 174)(86 175)(87 176)(88 177)(89 178)(90 179)(91 180)(92 192)(93 193)(94 194)(95 195)(96 183)(97 184)(98 185)(99 186)(100 187)(101 188)(102 189)(103 190)(104 191)(105 203)(106 204)(107 205)(108 206)(109 207)(110 208)(111 196)(112 197)(113 198)(114 199)(115 200)(116 201)(117 202)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 131)(126 132)(127 133)(128 134)(129 135)(130 136)(144 167)(145 168)(146 169)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(155 165)(156 166)
G:=sub<Sym(208)| (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), (1,17,113,118)(2,18,114,119)(3,19,115,120)(4,20,116,121)(5,21,117,122)(6,22,105,123)(7,23,106,124)(8,24,107,125)(9,25,108,126)(10,26,109,127)(11,14,110,128)(12,15,111,129)(13,16,112,130)(27,71,142,203)(28,72,143,204)(29,73,131,205)(30,74,132,206)(31,75,133,207)(32,76,134,208)(33,77,135,196)(34,78,136,197)(35,66,137,198)(36,67,138,199)(37,68,139,200)(38,69,140,201)(39,70,141,202)(40,80,167,185)(41,81,168,186)(42,82,169,187)(43,83,157,188)(44,84,158,189)(45,85,159,190)(46,86,160,191)(47,87,161,192)(48,88,162,193)(49,89,163,194)(50,90,164,195)(51,91,165,183)(52,79,166,184)(53,97,156,181)(54,98,144,182)(55,99,145,170)(56,100,146,171)(57,101,147,172)(58,102,148,173)(59,103,149,174)(60,104,150,175)(61,92,151,176)(62,93,152,177)(63,94,153,178)(64,95,154,179)(65,96,155,180), (1,137,85,59)(2,138,86,60)(3,139,87,61)(4,140,88,62)(5,141,89,63)(6,142,90,64)(7,143,91,65)(8,131,79,53)(9,132,80,54)(10,133,81,55)(11,134,82,56)(12,135,83,57)(13,136,84,58)(14,208,169,100)(15,196,157,101)(16,197,158,102)(17,198,159,103)(18,199,160,104)(19,200,161,92)(20,201,162,93)(21,202,163,94)(22,203,164,95)(23,204,165,96)(24,205,166,97)(25,206,167,98)(26,207,168,99)(27,195,154,105)(28,183,155,106)(29,184,156,107)(30,185,144,108)(31,186,145,109)(32,187,146,110)(33,188,147,111)(34,189,148,112)(35,190,149,113)(36,191,150,114)(37,192,151,115)(38,193,152,116)(39,194,153,117)(40,182,126,74)(41,170,127,75)(42,171,128,76)(43,172,129,77)(44,173,130,78)(45,174,118,66)(46,175,119,67)(47,176,120,68)(48,177,121,69)(49,178,122,70)(50,179,123,71)(51,180,124,72)(52,181,125,73), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,27)(23,28)(24,29)(25,30)(26,31)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,53)(79,181)(80,182)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,192)(93,193)(94,194)(95,195)(96,183)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,131)(126,132)(127,133)(128,134)(129,135)(130,136)(144,167)(145,168)(146,169)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166)>;
G:=Group( (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), (1,17,113,118)(2,18,114,119)(3,19,115,120)(4,20,116,121)(5,21,117,122)(6,22,105,123)(7,23,106,124)(8,24,107,125)(9,25,108,126)(10,26,109,127)(11,14,110,128)(12,15,111,129)(13,16,112,130)(27,71,142,203)(28,72,143,204)(29,73,131,205)(30,74,132,206)(31,75,133,207)(32,76,134,208)(33,77,135,196)(34,78,136,197)(35,66,137,198)(36,67,138,199)(37,68,139,200)(38,69,140,201)(39,70,141,202)(40,80,167,185)(41,81,168,186)(42,82,169,187)(43,83,157,188)(44,84,158,189)(45,85,159,190)(46,86,160,191)(47,87,161,192)(48,88,162,193)(49,89,163,194)(50,90,164,195)(51,91,165,183)(52,79,166,184)(53,97,156,181)(54,98,144,182)(55,99,145,170)(56,100,146,171)(57,101,147,172)(58,102,148,173)(59,103,149,174)(60,104,150,175)(61,92,151,176)(62,93,152,177)(63,94,153,178)(64,95,154,179)(65,96,155,180), (1,137,85,59)(2,138,86,60)(3,139,87,61)(4,140,88,62)(5,141,89,63)(6,142,90,64)(7,143,91,65)(8,131,79,53)(9,132,80,54)(10,133,81,55)(11,134,82,56)(12,135,83,57)(13,136,84,58)(14,208,169,100)(15,196,157,101)(16,197,158,102)(17,198,159,103)(18,199,160,104)(19,200,161,92)(20,201,162,93)(21,202,163,94)(22,203,164,95)(23,204,165,96)(24,205,166,97)(25,206,167,98)(26,207,168,99)(27,195,154,105)(28,183,155,106)(29,184,156,107)(30,185,144,108)(31,186,145,109)(32,187,146,110)(33,188,147,111)(34,189,148,112)(35,190,149,113)(36,191,150,114)(37,192,151,115)(38,193,152,116)(39,194,153,117)(40,182,126,74)(41,170,127,75)(42,171,128,76)(43,172,129,77)(44,173,130,78)(45,174,118,66)(46,175,119,67)(47,176,120,68)(48,177,121,69)(49,178,122,70)(50,179,123,71)(51,180,124,72)(52,181,125,73), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,27)(23,28)(24,29)(25,30)(26,31)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,53)(79,181)(80,182)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,192)(93,193)(94,194)(95,195)(96,183)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,203)(106,204)(107,205)(108,206)(109,207)(110,208)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,131)(126,132)(127,133)(128,134)(129,135)(130,136)(144,167)(145,168)(146,169)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(155,165)(156,166) );
G=PermutationGroup([[(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)], [(1,17,113,118),(2,18,114,119),(3,19,115,120),(4,20,116,121),(5,21,117,122),(6,22,105,123),(7,23,106,124),(8,24,107,125),(9,25,108,126),(10,26,109,127),(11,14,110,128),(12,15,111,129),(13,16,112,130),(27,71,142,203),(28,72,143,204),(29,73,131,205),(30,74,132,206),(31,75,133,207),(32,76,134,208),(33,77,135,196),(34,78,136,197),(35,66,137,198),(36,67,138,199),(37,68,139,200),(38,69,140,201),(39,70,141,202),(40,80,167,185),(41,81,168,186),(42,82,169,187),(43,83,157,188),(44,84,158,189),(45,85,159,190),(46,86,160,191),(47,87,161,192),(48,88,162,193),(49,89,163,194),(50,90,164,195),(51,91,165,183),(52,79,166,184),(53,97,156,181),(54,98,144,182),(55,99,145,170),(56,100,146,171),(57,101,147,172),(58,102,148,173),(59,103,149,174),(60,104,150,175),(61,92,151,176),(62,93,152,177),(63,94,153,178),(64,95,154,179),(65,96,155,180)], [(1,137,85,59),(2,138,86,60),(3,139,87,61),(4,140,88,62),(5,141,89,63),(6,142,90,64),(7,143,91,65),(8,131,79,53),(9,132,80,54),(10,133,81,55),(11,134,82,56),(12,135,83,57),(13,136,84,58),(14,208,169,100),(15,196,157,101),(16,197,158,102),(17,198,159,103),(18,199,160,104),(19,200,161,92),(20,201,162,93),(21,202,163,94),(22,203,164,95),(23,204,165,96),(24,205,166,97),(25,206,167,98),(26,207,168,99),(27,195,154,105),(28,183,155,106),(29,184,156,107),(30,185,144,108),(31,186,145,109),(32,187,146,110),(33,188,147,111),(34,189,148,112),(35,190,149,113),(36,191,150,114),(37,192,151,115),(38,193,152,116),(39,194,153,117),(40,182,126,74),(41,170,127,75),(42,171,128,76),(43,172,129,77),(44,173,130,78),(45,174,118,66),(46,175,119,67),(47,176,120,68),(48,177,121,69),(49,178,122,70),(50,179,123,71),(51,180,124,72),(52,181,125,73)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,27),(23,28),(24,29),(25,30),(26,31),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,53),(79,181),(80,182),(81,170),(82,171),(83,172),(84,173),(85,174),(86,175),(87,176),(88,177),(89,178),(90,179),(91,180),(92,192),(93,193),(94,194),(95,195),(96,183),(97,184),(98,185),(99,186),(100,187),(101,188),(102,189),(103,190),(104,191),(105,203),(106,204),(107,205),(108,206),(109,207),(110,208),(111,196),(112,197),(113,198),(114,199),(115,200),(116,201),(117,202),(118,137),(119,138),(120,139),(121,140),(122,141),(123,142),(124,143),(125,131),(126,132),(127,133),(128,134),(129,135),(130,136),(144,167),(145,168),(146,169),(147,157),(148,158),(149,159),(150,160),(151,161),(152,162),(153,163),(154,164),(155,165),(156,166)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26CF | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | D4 | D4×C13 |
| kernel | C13×C4⋊1D4 | C4×C52 | D4×C26 | C4⋊1D4 | C42 | C2×D4 | C52 | C4 |
| # reps | 1 | 1 | 6 | 12 | 12 | 72 | 6 | 72 |
Matrix representation of C13×C4⋊1D4 ►in GL4(𝔽53) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 51 |
| 0 | 0 | 49 | 37 |
| 0 | 1 | 0 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 52 | 0 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 0 | 16 | 51 |
| 0 | 0 | 48 | 37 |
G:=sub<GL(4,GF(53))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,16,49,0,0,51,37],[0,52,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,52,0,0,52,0,0,0,0,0,16,48,0,0,51,37] >;
C13×C4⋊1D4 in GAP, Magma, Sage, TeX
C_{13}\times C_4\rtimes_1D_4 % in TeX
G:=Group("C13xC4:1D4"); // GroupNames label
G:=SmallGroup(416,188);
// by ID
G=gap.SmallGroup(416,188);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,631,3818,950]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations