metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic13⋊4D4, C23.14D26, C13⋊3(C4×D4), D26⋊3(C2×C4), C13⋊D4⋊2C4, C2.2(D4×D13), C22⋊C4⋊7D13, C26.18(C2×D4), (C2×C4).28D26, C22⋊1(C4×D13), Dic13⋊2(C2×C4), C26.D4⋊9C2, D26⋊C4⋊10C2, (C4×Dic13)⋊11C2, C26.22(C4○D4), (C2×C26).22C23, C26.20(C22×C4), (C2×C52).51C22, C2.2(D4⋊2D13), (C22×Dic13)⋊1C2, (C22×C26).11C22, C22.14(C22×D13), (C2×Dic13).61C22, (C22×D13).19C22, (C2×C4×D13)⋊9C2, C2.9(C2×C4×D13), (C2×C26)⋊5(C2×C4), (C13×C22⋊C4)⋊9C2, (C2×C13⋊D4).2C2, SmallGroup(416,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic13⋊4D4
G = < a,b,c,d | a26=c4=d2=1, b2=a13, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 592 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C26, C4×D4, Dic13, Dic13, C52, D26, D26, C2×C26, C2×C26, C2×C26, C4×D13, C2×Dic13, C2×Dic13, C13⋊D4, C2×C52, C22×D13, C22×C26, C4×Dic13, C26.D4, D26⋊C4, C13×C22⋊C4, C2×C4×D13, C22×Dic13, C2×C13⋊D4, Dic13⋊4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, D26, C4×D13, C22×D13, C2×C4×D13, D4×D13, D4⋊2D13, Dic13⋊4D4
►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)
(1 106 14 119)(2 105 15 118)(3 130 16 117)(4 129 17 116)(5 128 18 115)(6 127 19 114)(7 126 20 113)(8 125 21 112)(9 124 22 111)(10 123 23 110)(11 122 24 109)(12 121 25 108)(13 120 26 107)(27 101 40 88)(28 100 41 87)(29 99 42 86)(30 98 43 85)(31 97 44 84)(32 96 45 83)(33 95 46 82)(34 94 47 81)(35 93 48 80)(36 92 49 79)(37 91 50 104)(38 90 51 103)(39 89 52 102)(53 182 66 169)(54 181 67 168)(55 180 68 167)(56 179 69 166)(57 178 70 165)(58 177 71 164)(59 176 72 163)(60 175 73 162)(61 174 74 161)(62 173 75 160)(63 172 76 159)(64 171 77 158)(65 170 78 157)(131 206 144 193)(132 205 145 192)(133 204 146 191)(134 203 147 190)(135 202 148 189)(136 201 149 188)(137 200 150 187)(138 199 151 186)(139 198 152 185)(140 197 153 184)(141 196 154 183)(142 195 155 208)(143 194 156 207)
(1 144 97 181)(2 143 98 180)(3 142 99 179)(4 141 100 178)(5 140 101 177)(6 139 102 176)(7 138 103 175)(8 137 104 174)(9 136 79 173)(10 135 80 172)(11 134 81 171)(12 133 82 170)(13 132 83 169)(14 131 84 168)(15 156 85 167)(16 155 86 166)(17 154 87 165)(18 153 88 164)(19 152 89 163)(20 151 90 162)(21 150 91 161)(22 149 92 160)(23 148 93 159)(24 147 94 158)(25 146 95 157)(26 145 96 182)(27 58 115 184)(28 57 116 183)(29 56 117 208)(30 55 118 207)(31 54 119 206)(32 53 120 205)(33 78 121 204)(34 77 122 203)(35 76 123 202)(36 75 124 201)(37 74 125 200)(38 73 126 199)(39 72 127 198)(40 71 128 197)(41 70 129 196)(42 69 130 195)(43 68 105 194)(44 67 106 193)(45 66 107 192)(46 65 108 191)(47 64 109 190)(48 63 110 189)(49 62 111 188)(50 61 112 187)(51 60 113 186)(52 59 114 185)
(1 181)(2 182)(3 157)(4 158)(5 159)(6 160)(7 161)(8 162)(9 163)(10 164)(11 165)(12 166)(13 167)(14 168)(15 169)(16 170)(17 171)(18 172)(19 173)(20 174)(21 175)(22 176)(23 177)(24 178)(25 179)(26 180)(27 202)(28 203)(29 204)(30 205)(31 206)(32 207)(33 208)(34 183)(35 184)(36 185)(37 186)(38 187)(39 188)(40 189)(41 190)(42 191)(43 192)(44 193)(45 194)(46 195)(47 196)(48 197)(49 198)(50 199)(51 200)(52 201)(53 118)(54 119)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 152)(80 153)(81 154)(82 155)(83 156)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)
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), (1,106,14,119)(2,105,15,118)(3,130,16,117)(4,129,17,116)(5,128,18,115)(6,127,19,114)(7,126,20,113)(8,125,21,112)(9,124,22,111)(10,123,23,110)(11,122,24,109)(12,121,25,108)(13,120,26,107)(27,101,40,88)(28,100,41,87)(29,99,42,86)(30,98,43,85)(31,97,44,84)(32,96,45,83)(33,95,46,82)(34,94,47,81)(35,93,48,80)(36,92,49,79)(37,91,50,104)(38,90,51,103)(39,89,52,102)(53,182,66,169)(54,181,67,168)(55,180,68,167)(56,179,69,166)(57,178,70,165)(58,177,71,164)(59,176,72,163)(60,175,73,162)(61,174,74,161)(62,173,75,160)(63,172,76,159)(64,171,77,158)(65,170,78,157)(131,206,144,193)(132,205,145,192)(133,204,146,191)(134,203,147,190)(135,202,148,189)(136,201,149,188)(137,200,150,187)(138,199,151,186)(139,198,152,185)(140,197,153,184)(141,196,154,183)(142,195,155,208)(143,194,156,207), (1,144,97,181)(2,143,98,180)(3,142,99,179)(4,141,100,178)(5,140,101,177)(6,139,102,176)(7,138,103,175)(8,137,104,174)(9,136,79,173)(10,135,80,172)(11,134,81,171)(12,133,82,170)(13,132,83,169)(14,131,84,168)(15,156,85,167)(16,155,86,166)(17,154,87,165)(18,153,88,164)(19,152,89,163)(20,151,90,162)(21,150,91,161)(22,149,92,160)(23,148,93,159)(24,147,94,158)(25,146,95,157)(26,145,96,182)(27,58,115,184)(28,57,116,183)(29,56,117,208)(30,55,118,207)(31,54,119,206)(32,53,120,205)(33,78,121,204)(34,77,122,203)(35,76,123,202)(36,75,124,201)(37,74,125,200)(38,73,126,199)(39,72,127,198)(40,71,128,197)(41,70,129,196)(42,69,130,195)(43,68,105,194)(44,67,106,193)(45,66,107,192)(46,65,108,191)(47,64,109,190)(48,63,110,189)(49,62,111,188)(50,61,112,187)(51,60,113,186)(52,59,114,185), (1,181)(2,182)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,169)(16,170)(17,171)(18,172)(19,173)(20,174)(21,175)(22,176)(23,177)(24,178)(25,179)(26,180)(27,202)(28,203)(29,204)(30,205)(31,206)(32,207)(33,208)(34,183)(35,184)(36,185)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,193)(45,194)(46,195)(47,196)(48,197)(49,198)(50,199)(51,200)(52,201)(53,118)(54,119)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,152)(80,153)(81,154)(82,155)(83,156)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)>;
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), (1,106,14,119)(2,105,15,118)(3,130,16,117)(4,129,17,116)(5,128,18,115)(6,127,19,114)(7,126,20,113)(8,125,21,112)(9,124,22,111)(10,123,23,110)(11,122,24,109)(12,121,25,108)(13,120,26,107)(27,101,40,88)(28,100,41,87)(29,99,42,86)(30,98,43,85)(31,97,44,84)(32,96,45,83)(33,95,46,82)(34,94,47,81)(35,93,48,80)(36,92,49,79)(37,91,50,104)(38,90,51,103)(39,89,52,102)(53,182,66,169)(54,181,67,168)(55,180,68,167)(56,179,69,166)(57,178,70,165)(58,177,71,164)(59,176,72,163)(60,175,73,162)(61,174,74,161)(62,173,75,160)(63,172,76,159)(64,171,77,158)(65,170,78,157)(131,206,144,193)(132,205,145,192)(133,204,146,191)(134,203,147,190)(135,202,148,189)(136,201,149,188)(137,200,150,187)(138,199,151,186)(139,198,152,185)(140,197,153,184)(141,196,154,183)(142,195,155,208)(143,194,156,207), (1,144,97,181)(2,143,98,180)(3,142,99,179)(4,141,100,178)(5,140,101,177)(6,139,102,176)(7,138,103,175)(8,137,104,174)(9,136,79,173)(10,135,80,172)(11,134,81,171)(12,133,82,170)(13,132,83,169)(14,131,84,168)(15,156,85,167)(16,155,86,166)(17,154,87,165)(18,153,88,164)(19,152,89,163)(20,151,90,162)(21,150,91,161)(22,149,92,160)(23,148,93,159)(24,147,94,158)(25,146,95,157)(26,145,96,182)(27,58,115,184)(28,57,116,183)(29,56,117,208)(30,55,118,207)(31,54,119,206)(32,53,120,205)(33,78,121,204)(34,77,122,203)(35,76,123,202)(36,75,124,201)(37,74,125,200)(38,73,126,199)(39,72,127,198)(40,71,128,197)(41,70,129,196)(42,69,130,195)(43,68,105,194)(44,67,106,193)(45,66,107,192)(46,65,108,191)(47,64,109,190)(48,63,110,189)(49,62,111,188)(50,61,112,187)(51,60,113,186)(52,59,114,185), (1,181)(2,182)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,169)(16,170)(17,171)(18,172)(19,173)(20,174)(21,175)(22,176)(23,177)(24,178)(25,179)(26,180)(27,202)(28,203)(29,204)(30,205)(31,206)(32,207)(33,208)(34,183)(35,184)(36,185)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,193)(45,194)(46,195)(47,196)(48,197)(49,198)(50,199)(51,200)(52,201)(53,118)(54,119)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,152)(80,153)(81,154)(82,155)(83,156)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151) );
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)], [(1,106,14,119),(2,105,15,118),(3,130,16,117),(4,129,17,116),(5,128,18,115),(6,127,19,114),(7,126,20,113),(8,125,21,112),(9,124,22,111),(10,123,23,110),(11,122,24,109),(12,121,25,108),(13,120,26,107),(27,101,40,88),(28,100,41,87),(29,99,42,86),(30,98,43,85),(31,97,44,84),(32,96,45,83),(33,95,46,82),(34,94,47,81),(35,93,48,80),(36,92,49,79),(37,91,50,104),(38,90,51,103),(39,89,52,102),(53,182,66,169),(54,181,67,168),(55,180,68,167),(56,179,69,166),(57,178,70,165),(58,177,71,164),(59,176,72,163),(60,175,73,162),(61,174,74,161),(62,173,75,160),(63,172,76,159),(64,171,77,158),(65,170,78,157),(131,206,144,193),(132,205,145,192),(133,204,146,191),(134,203,147,190),(135,202,148,189),(136,201,149,188),(137,200,150,187),(138,199,151,186),(139,198,152,185),(140,197,153,184),(141,196,154,183),(142,195,155,208),(143,194,156,207)], [(1,144,97,181),(2,143,98,180),(3,142,99,179),(4,141,100,178),(5,140,101,177),(6,139,102,176),(7,138,103,175),(8,137,104,174),(9,136,79,173),(10,135,80,172),(11,134,81,171),(12,133,82,170),(13,132,83,169),(14,131,84,168),(15,156,85,167),(16,155,86,166),(17,154,87,165),(18,153,88,164),(19,152,89,163),(20,151,90,162),(21,150,91,161),(22,149,92,160),(23,148,93,159),(24,147,94,158),(25,146,95,157),(26,145,96,182),(27,58,115,184),(28,57,116,183),(29,56,117,208),(30,55,118,207),(31,54,119,206),(32,53,120,205),(33,78,121,204),(34,77,122,203),(35,76,123,202),(36,75,124,201),(37,74,125,200),(38,73,126,199),(39,72,127,198),(40,71,128,197),(41,70,129,196),(42,69,130,195),(43,68,105,194),(44,67,106,193),(45,66,107,192),(46,65,108,191),(47,64,109,190),(48,63,110,189),(49,62,111,188),(50,61,112,187),(51,60,113,186),(52,59,114,185)], [(1,181),(2,182),(3,157),(4,158),(5,159),(6,160),(7,161),(8,162),(9,163),(10,164),(11,165),(12,166),(13,167),(14,168),(15,169),(16,170),(17,171),(18,172),(19,173),(20,174),(21,175),(22,176),(23,177),(24,178),(25,179),(26,180),(27,202),(28,203),(29,204),(30,205),(31,206),(32,207),(33,208),(34,183),(35,184),(36,185),(37,186),(38,187),(39,188),(40,189),(41,190),(42,191),(43,192),(44,193),(45,194),(46,195),(47,196),(48,197),(49,198),(50,199),(51,200),(52,201),(53,118),(54,119),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,127),(63,128),(64,129),(65,130),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,152),(80,153),(81,154),(82,155),(83,156),(84,131),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AD | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 26 | 26 | 2 | 2 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | D13 | D26 | D26 | C4×D13 | D4×D13 | D4⋊2D13 |
| kernel | Dic13⋊4D4 | C4×Dic13 | C26.D4 | D26⋊C4 | C13×C22⋊C4 | C2×C4×D13 | C22×Dic13 | C2×C13⋊D4 | C13⋊D4 | Dic13 | C26 | C22⋊C4 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 6 | 12 | 6 | 24 | 6 | 6 |
Matrix representation of Dic13⋊4D4 ►in GL4(𝔽53) generated by
| 0 | 52 | 0 | 0 |
| 1 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 0 | 0 | 0 |
| 15 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 26 | 52 | 0 | 0 |
| 0 | 0 | 21 | 9 |
| 0 | 0 | 51 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 21 | 9 |
| 0 | 0 | 10 | 32 |
G:=sub<GL(4,GF(53))| [0,1,0,0,52,27,0,0,0,0,1,0,0,0,0,1],[23,15,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[1,26,0,0,0,52,0,0,0,0,21,51,0,0,9,32],[1,0,0,0,0,1,0,0,0,0,21,10,0,0,9,32] >;
Dic13⋊4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{13}\rtimes_4D_4 % in TeX
G:=Group("Dic13:4D4"); // GroupNames label
G:=SmallGroup(416,102);
// by ID
G=gap.SmallGroup(416,102);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,188,50,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^26=c^4=d^2=1,b^2=a^13,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations