metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.D13, C26.9D8, C8.5D26, C52.4D4, C13⋊2SD32, Dic52⋊3C2, C104.3C22, C13⋊2C16⋊2C2, (C13×D8).1C2, C2.5(D4⋊D13), C4.2(C13⋊D4), SmallGroup(416,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.D13
G = < a,b,c,d | a8=b2=c13=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Smallest permutation representation of D8.D13
►On 208 points
Generators in S208
(1 85 37 78 15 104 51 53)(2 86 38 66 16 92 52 54)(3 87 39 67 17 93 40 55)(4 88 27 68 18 94 41 56)(5 89 28 69 19 95 42 57)(6 90 29 70 20 96 43 58)(7 91 30 71 21 97 44 59)(8 79 31 72 22 98 45 60)(9 80 32 73 23 99 46 61)(10 81 33 74 24 100 47 62)(11 82 34 75 25 101 48 63)(12 83 35 76 26 102 49 64)(13 84 36 77 14 103 50 65)(105 157 144 196 118 170 131 183)(106 158 145 197 119 171 132 184)(107 159 146 198 120 172 133 185)(108 160 147 199 121 173 134 186)(109 161 148 200 122 174 135 187)(110 162 149 201 123 175 136 188)(111 163 150 202 124 176 137 189)(112 164 151 203 125 177 138 190)(113 165 152 204 126 178 139 191)(114 166 153 205 127 179 140 192)(115 167 154 206 128 180 141 193)(116 168 155 207 129 181 142 194)(117 169 156 208 130 182 143 195)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 77)(15 78)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 92)(39 93)(40 87)(41 88)(42 89)(43 90)(44 91)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(157 170)(158 171)(159 172)(160 173)(161 174)(162 175)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(169 182)
(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)
(1 130 15 117)(2 129 16 116)(3 128 17 115)(4 127 18 114)(5 126 19 113)(6 125 20 112)(7 124 21 111)(8 123 22 110)(9 122 23 109)(10 121 24 108)(11 120 25 107)(12 119 26 106)(13 118 14 105)(27 153 41 140)(28 152 42 139)(29 151 43 138)(30 150 44 137)(31 149 45 136)(32 148 46 135)(33 147 47 134)(34 146 48 133)(35 145 49 132)(36 144 50 131)(37 156 51 143)(38 155 52 142)(39 154 40 141)(53 182 78 169)(54 181 66 168)(55 180 67 167)(56 179 68 166)(57 178 69 165)(58 177 70 164)(59 176 71 163)(60 175 72 162)(61 174 73 161)(62 173 74 160)(63 172 75 159)(64 171 76 158)(65 170 77 157)(79 201 98 188)(80 200 99 187)(81 199 100 186)(82 198 101 185)(83 197 102 184)(84 196 103 183)(85 208 104 195)(86 207 92 194)(87 206 93 193)(88 205 94 192)(89 204 95 191)(90 203 96 190)(91 202 97 189)
G:=sub<Sym(208)| (1,85,37,78,15,104,51,53)(2,86,38,66,16,92,52,54)(3,87,39,67,17,93,40,55)(4,88,27,68,18,94,41,56)(5,89,28,69,19,95,42,57)(6,90,29,70,20,96,43,58)(7,91,30,71,21,97,44,59)(8,79,31,72,22,98,45,60)(9,80,32,73,23,99,46,61)(10,81,33,74,24,100,47,62)(11,82,34,75,25,101,48,63)(12,83,35,76,26,102,49,64)(13,84,36,77,14,103,50,65)(105,157,144,196,118,170,131,183)(106,158,145,197,119,171,132,184)(107,159,146,198,120,172,133,185)(108,160,147,199,121,173,134,186)(109,161,148,200,122,174,135,187)(110,162,149,201,123,175,136,188)(111,163,150,202,124,176,137,189)(112,164,151,203,125,177,138,190)(113,165,152,204,126,178,139,191)(114,166,153,205,127,179,140,192)(115,167,154,206,128,180,141,193)(116,168,155,207,129,181,142,194)(117,169,156,208,130,182,143,195), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,77)(15,78)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,92)(39,93)(40,87)(41,88)(42,89)(43,90)(44,91)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(157,170)(158,171)(159,172)(160,173)(161,174)(162,175)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,182), (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), (1,130,15,117)(2,129,16,116)(3,128,17,115)(4,127,18,114)(5,126,19,113)(6,125,20,112)(7,124,21,111)(8,123,22,110)(9,122,23,109)(10,121,24,108)(11,120,25,107)(12,119,26,106)(13,118,14,105)(27,153,41,140)(28,152,42,139)(29,151,43,138)(30,150,44,137)(31,149,45,136)(32,148,46,135)(33,147,47,134)(34,146,48,133)(35,145,49,132)(36,144,50,131)(37,156,51,143)(38,155,52,142)(39,154,40,141)(53,182,78,169)(54,181,66,168)(55,180,67,167)(56,179,68,166)(57,178,69,165)(58,177,70,164)(59,176,71,163)(60,175,72,162)(61,174,73,161)(62,173,74,160)(63,172,75,159)(64,171,76,158)(65,170,77,157)(79,201,98,188)(80,200,99,187)(81,199,100,186)(82,198,101,185)(83,197,102,184)(84,196,103,183)(85,208,104,195)(86,207,92,194)(87,206,93,193)(88,205,94,192)(89,204,95,191)(90,203,96,190)(91,202,97,189)>;
G:=Group( (1,85,37,78,15,104,51,53)(2,86,38,66,16,92,52,54)(3,87,39,67,17,93,40,55)(4,88,27,68,18,94,41,56)(5,89,28,69,19,95,42,57)(6,90,29,70,20,96,43,58)(7,91,30,71,21,97,44,59)(8,79,31,72,22,98,45,60)(9,80,32,73,23,99,46,61)(10,81,33,74,24,100,47,62)(11,82,34,75,25,101,48,63)(12,83,35,76,26,102,49,64)(13,84,36,77,14,103,50,65)(105,157,144,196,118,170,131,183)(106,158,145,197,119,171,132,184)(107,159,146,198,120,172,133,185)(108,160,147,199,121,173,134,186)(109,161,148,200,122,174,135,187)(110,162,149,201,123,175,136,188)(111,163,150,202,124,176,137,189)(112,164,151,203,125,177,138,190)(113,165,152,204,126,178,139,191)(114,166,153,205,127,179,140,192)(115,167,154,206,128,180,141,193)(116,168,155,207,129,181,142,194)(117,169,156,208,130,182,143,195), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,77)(15,78)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,92)(39,93)(40,87)(41,88)(42,89)(43,90)(44,91)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,85)(52,86)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(157,170)(158,171)(159,172)(160,173)(161,174)(162,175)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,182), (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), (1,130,15,117)(2,129,16,116)(3,128,17,115)(4,127,18,114)(5,126,19,113)(6,125,20,112)(7,124,21,111)(8,123,22,110)(9,122,23,109)(10,121,24,108)(11,120,25,107)(12,119,26,106)(13,118,14,105)(27,153,41,140)(28,152,42,139)(29,151,43,138)(30,150,44,137)(31,149,45,136)(32,148,46,135)(33,147,47,134)(34,146,48,133)(35,145,49,132)(36,144,50,131)(37,156,51,143)(38,155,52,142)(39,154,40,141)(53,182,78,169)(54,181,66,168)(55,180,67,167)(56,179,68,166)(57,178,69,165)(58,177,70,164)(59,176,71,163)(60,175,72,162)(61,174,73,161)(62,173,74,160)(63,172,75,159)(64,171,76,158)(65,170,77,157)(79,201,98,188)(80,200,99,187)(81,199,100,186)(82,198,101,185)(83,197,102,184)(84,196,103,183)(85,208,104,195)(86,207,92,194)(87,206,93,193)(88,205,94,192)(89,204,95,191)(90,203,96,190)(91,202,97,189) );
G=PermutationGroup([[(1,85,37,78,15,104,51,53),(2,86,38,66,16,92,52,54),(3,87,39,67,17,93,40,55),(4,88,27,68,18,94,41,56),(5,89,28,69,19,95,42,57),(6,90,29,70,20,96,43,58),(7,91,30,71,21,97,44,59),(8,79,31,72,22,98,45,60),(9,80,32,73,23,99,46,61),(10,81,33,74,24,100,47,62),(11,82,34,75,25,101,48,63),(12,83,35,76,26,102,49,64),(13,84,36,77,14,103,50,65),(105,157,144,196,118,170,131,183),(106,158,145,197,119,171,132,184),(107,159,146,198,120,172,133,185),(108,160,147,199,121,173,134,186),(109,161,148,200,122,174,135,187),(110,162,149,201,123,175,136,188),(111,163,150,202,124,176,137,189),(112,164,151,203,125,177,138,190),(113,165,152,204,126,178,139,191),(114,166,153,205,127,179,140,192),(115,167,154,206,128,180,141,193),(116,168,155,207,129,181,142,194),(117,169,156,208,130,182,143,195)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,77),(15,78),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,92),(39,93),(40,87),(41,88),(42,89),(43,90),(44,91),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,85),(52,86),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,153),(128,154),(129,155),(130,156),(157,170),(158,171),(159,172),(160,173),(161,174),(162,175),(163,176),(164,177),(165,178),(166,179),(167,180),(168,181),(169,182)], [(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)], [(1,130,15,117),(2,129,16,116),(3,128,17,115),(4,127,18,114),(5,126,19,113),(6,125,20,112),(7,124,21,111),(8,123,22,110),(9,122,23,109),(10,121,24,108),(11,120,25,107),(12,119,26,106),(13,118,14,105),(27,153,41,140),(28,152,42,139),(29,151,43,138),(30,150,44,137),(31,149,45,136),(32,148,46,135),(33,147,47,134),(34,146,48,133),(35,145,49,132),(36,144,50,131),(37,156,51,143),(38,155,52,142),(39,154,40,141),(53,182,78,169),(54,181,66,168),(55,180,67,167),(56,179,68,166),(57,178,69,165),(58,177,70,164),(59,176,71,163),(60,175,72,162),(61,174,73,161),(62,173,74,160),(63,172,75,159),(64,171,76,158),(65,170,77,157),(79,201,98,188),(80,200,99,187),(81,199,100,186),(82,198,101,185),(83,197,102,184),(84,196,103,183),(85,208,104,195),(86,207,92,194),(87,206,93,193),(88,205,94,192),(89,204,95,191),(90,203,96,190),(91,202,97,189)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 13A | ··· | 13F | 16A | 16B | 16C | 16D | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 8 | 2 | 104 | 2 | 2 | 2 | ··· | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D8 | D13 | SD32 | D26 | C13⋊D4 | D4⋊D13 | D8.D13 |
| kernel | D8.D13 | C13⋊2C16 | Dic52 | C13×D8 | C52 | C26 | D8 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 6 | 12 |
Matrix representation of D8.D13 ►in GL4(𝔽1249) generated by
| 1248 | 0 | 0 | 0 |
| 0 | 1248 | 0 | 0 |
| 0 | 0 | 0 | 142 |
| 0 | 0 | 818 | 50 |
| 1248 | 0 | 0 | 0 |
| 396 | 1 | 0 | 0 |
| 0 | 0 | 0 | 142 |
| 0 | 0 | 431 | 0 |
| 83 | 0 | 0 | 0 |
| 934 | 933 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 316 | 317 | 0 | 0 |
| 934 | 933 | 0 | 0 |
| 0 | 0 | 1188 | 1027 |
| 0 | 0 | 1187 | 61 |
G:=sub<GL(4,GF(1249))| [1248,0,0,0,0,1248,0,0,0,0,0,818,0,0,142,50],[1248,396,0,0,0,1,0,0,0,0,0,431,0,0,142,0],[83,934,0,0,0,933,0,0,0,0,1,0,0,0,0,1],[316,934,0,0,317,933,0,0,0,0,1188,1187,0,0,1027,61] >;
D8.D13 in GAP, Magma, Sage, TeX
D_8.D_{13} % in TeX
G:=Group("D8.D13"); // GroupNames label
G:=SmallGroup(416,34);
// by ID
G=gap.SmallGroup(416,34);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,73,218,116,122,579,297,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^13=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations
Export