metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊1D4, Dic13⋊2D4, C23.5D26, (C2×D52)⋊3C2, C2.9(D4×D13), (C2×C4).7D26, C13⋊1(C4⋊D4), C22⋊C4⋊4D13, C26.20(C2×D4), C26.9(C4○D4), C26.D4⋊5C2, D26⋊C4⋊11C2, (C2×C26).25C23, (C2×C52).53C22, C2.11(D52⋊5C2), (C22×C26).14C22, (C2×Dic13).7C22, C22.43(C22×D13), (C22×D13).21C22, (C2×C4×D13)⋊11C2, (C2×C13⋊D4)⋊2C2, (C13×C22⋊C4)⋊6C2, SmallGroup(416,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26⋊D4
G = < a,b,c,d | a26=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a24b, dbd=a11b, dcd=c-1 >
Subgroups: 760 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×5], C22, C22 [×10], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C13, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], D13 [×3], C26 [×3], C26, C4⋊D4, Dic13 [×2], Dic13, C52 [×2], D26 [×2], D26 [×5], C2×C26, C2×C26 [×3], C4×D13 [×2], D52 [×2], C2×Dic13 [×2], C13⋊D4 [×4], C2×C52 [×2], C22×D13 [×2], C22×C26, C26.D4, D26⋊C4, C13×C22⋊C4, C2×C4×D13, C2×D52, C2×C13⋊D4 [×2], D26⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, D13, C4⋊D4, D26 [×3], C22×D13, D52⋊5C2, D4×D13 [×2], D26⋊D4
►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)
(1 139)(2 138)(3 137)(4 136)(5 135)(6 134)(7 133)(8 132)(9 131)(10 156)(11 155)(12 154)(13 153)(14 152)(15 151)(16 150)(17 149)(18 148)(19 147)(20 146)(21 145)(22 144)(23 143)(24 142)(25 141)(26 140)(27 115)(28 114)(29 113)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 130)(39 129)(40 128)(41 127)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 195)(54 194)(55 193)(56 192)(57 191)(58 190)(59 189)(60 188)(61 187)(62 186)(63 185)(64 184)(65 183)(66 208)(67 207)(68 206)(69 205)(70 204)(71 203)(72 202)(73 201)(74 200)(75 199)(76 198)(77 197)(78 196)(79 170)(80 169)(81 168)(82 167)(83 166)(84 165)(85 164)(86 163)(87 162)(88 161)(89 160)(90 159)(91 158)(92 157)(93 182)(94 181)(95 180)(96 179)(97 178)(98 177)(99 176)(100 175)(101 174)(102 173)(103 172)(104 171)
(1 87 153 176)(2 86 154 175)(3 85 155 174)(4 84 156 173)(5 83 131 172)(6 82 132 171)(7 81 133 170)(8 80 134 169)(9 79 135 168)(10 104 136 167)(11 103 137 166)(12 102 138 165)(13 101 139 164)(14 100 140 163)(15 99 141 162)(16 98 142 161)(17 97 143 160)(18 96 144 159)(19 95 145 158)(20 94 146 157)(21 93 147 182)(22 92 148 181)(23 91 149 180)(24 90 150 179)(25 89 151 178)(26 88 152 177)(27 56 112 197)(28 55 113 196)(29 54 114 195)(30 53 115 194)(31 78 116 193)(32 77 117 192)(33 76 118 191)(34 75 119 190)(35 74 120 189)(36 73 121 188)(37 72 122 187)(38 71 123 186)(39 70 124 185)(40 69 125 184)(41 68 126 183)(42 67 127 208)(43 66 128 207)(44 65 129 206)(45 64 130 205)(46 63 105 204)(47 62 106 203)(48 61 107 202)(49 60 108 201)(50 59 109 200)(51 58 110 199)(52 57 111 198)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 120)(9 119)(10 118)(11 117)(12 116)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 106)(23 105)(24 130)(25 129)(26 128)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 136)(34 135)(35 134)(36 133)(37 132)(38 131)(39 156)(40 155)(41 154)(42 153)(43 152)(44 151)(45 150)(46 149)(47 148)(48 147)(49 146)(50 145)(51 144)(52 143)(53 101)(54 100)(55 99)(56 98)(57 97)(58 96)(59 95)(60 94)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 104)(77 103)(78 102)(157 201)(158 200)(159 199)(160 198)(161 197)(162 196)(163 195)(164 194)(165 193)(166 192)(167 191)(168 190)(169 189)(170 188)(171 187)(172 186)(173 185)(174 184)(175 183)(176 208)(177 207)(178 206)(179 205)(180 204)(181 203)(182 202)
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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,156)(11,155)(12,154)(13,153)(14,152)(15,151)(16,150)(17,149)(18,148)(19,147)(20,146)(21,145)(22,144)(23,143)(24,142)(25,141)(26,140)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,130)(39,129)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,195)(54,194)(55,193)(56,192)(57,191)(58,190)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,208)(67,207)(68,206)(69,205)(70,204)(71,203)(72,202)(73,201)(74,200)(75,199)(76,198)(77,197)(78,196)(79,170)(80,169)(81,168)(82,167)(83,166)(84,165)(85,164)(86,163)(87,162)(88,161)(89,160)(90,159)(91,158)(92,157)(93,182)(94,181)(95,180)(96,179)(97,178)(98,177)(99,176)(100,175)(101,174)(102,173)(103,172)(104,171), (1,87,153,176)(2,86,154,175)(3,85,155,174)(4,84,156,173)(5,83,131,172)(6,82,132,171)(7,81,133,170)(8,80,134,169)(9,79,135,168)(10,104,136,167)(11,103,137,166)(12,102,138,165)(13,101,139,164)(14,100,140,163)(15,99,141,162)(16,98,142,161)(17,97,143,160)(18,96,144,159)(19,95,145,158)(20,94,146,157)(21,93,147,182)(22,92,148,181)(23,91,149,180)(24,90,150,179)(25,89,151,178)(26,88,152,177)(27,56,112,197)(28,55,113,196)(29,54,114,195)(30,53,115,194)(31,78,116,193)(32,77,117,192)(33,76,118,191)(34,75,119,190)(35,74,120,189)(36,73,121,188)(37,72,122,187)(38,71,123,186)(39,70,124,185)(40,69,125,184)(41,68,126,183)(42,67,127,208)(43,66,128,207)(44,65,129,206)(45,64,130,205)(46,63,105,204)(47,62,106,203)(48,61,107,202)(49,60,108,201)(50,59,109,200)(51,58,110,199)(52,57,111,198), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,130)(25,129)(26,128)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,156)(40,155)(41,154)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,101)(54,100)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,104)(77,103)(78,102)(157,201)(158,200)(159,199)(160,198)(161,197)(162,196)(163,195)(164,194)(165,193)(166,192)(167,191)(168,190)(169,189)(170,188)(171,187)(172,186)(173,185)(174,184)(175,183)(176,208)(177,207)(178,206)(179,205)(180,204)(181,203)(182,202)>;
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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,156)(11,155)(12,154)(13,153)(14,152)(15,151)(16,150)(17,149)(18,148)(19,147)(20,146)(21,145)(22,144)(23,143)(24,142)(25,141)(26,140)(27,115)(28,114)(29,113)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,130)(39,129)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,195)(54,194)(55,193)(56,192)(57,191)(58,190)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,208)(67,207)(68,206)(69,205)(70,204)(71,203)(72,202)(73,201)(74,200)(75,199)(76,198)(77,197)(78,196)(79,170)(80,169)(81,168)(82,167)(83,166)(84,165)(85,164)(86,163)(87,162)(88,161)(89,160)(90,159)(91,158)(92,157)(93,182)(94,181)(95,180)(96,179)(97,178)(98,177)(99,176)(100,175)(101,174)(102,173)(103,172)(104,171), (1,87,153,176)(2,86,154,175)(3,85,155,174)(4,84,156,173)(5,83,131,172)(6,82,132,171)(7,81,133,170)(8,80,134,169)(9,79,135,168)(10,104,136,167)(11,103,137,166)(12,102,138,165)(13,101,139,164)(14,100,140,163)(15,99,141,162)(16,98,142,161)(17,97,143,160)(18,96,144,159)(19,95,145,158)(20,94,146,157)(21,93,147,182)(22,92,148,181)(23,91,149,180)(24,90,150,179)(25,89,151,178)(26,88,152,177)(27,56,112,197)(28,55,113,196)(29,54,114,195)(30,53,115,194)(31,78,116,193)(32,77,117,192)(33,76,118,191)(34,75,119,190)(35,74,120,189)(36,73,121,188)(37,72,122,187)(38,71,123,186)(39,70,124,185)(40,69,125,184)(41,68,126,183)(42,67,127,208)(43,66,128,207)(44,65,129,206)(45,64,130,205)(46,63,105,204)(47,62,106,203)(48,61,107,202)(49,60,108,201)(50,59,109,200)(51,58,110,199)(52,57,111,198), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,130)(25,129)(26,128)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,156)(40,155)(41,154)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,101)(54,100)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,104)(77,103)(78,102)(157,201)(158,200)(159,199)(160,198)(161,197)(162,196)(163,195)(164,194)(165,193)(166,192)(167,191)(168,190)(169,189)(170,188)(171,187)(172,186)(173,185)(174,184)(175,183)(176,208)(177,207)(178,206)(179,205)(180,204)(181,203)(182,202) );
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)], [(1,139),(2,138),(3,137),(4,136),(5,135),(6,134),(7,133),(8,132),(9,131),(10,156),(11,155),(12,154),(13,153),(14,152),(15,151),(16,150),(17,149),(18,148),(19,147),(20,146),(21,145),(22,144),(23,143),(24,142),(25,141),(26,140),(27,115),(28,114),(29,113),(30,112),(31,111),(32,110),(33,109),(34,108),(35,107),(36,106),(37,105),(38,130),(39,129),(40,128),(41,127),(42,126),(43,125),(44,124),(45,123),(46,122),(47,121),(48,120),(49,119),(50,118),(51,117),(52,116),(53,195),(54,194),(55,193),(56,192),(57,191),(58,190),(59,189),(60,188),(61,187),(62,186),(63,185),(64,184),(65,183),(66,208),(67,207),(68,206),(69,205),(70,204),(71,203),(72,202),(73,201),(74,200),(75,199),(76,198),(77,197),(78,196),(79,170),(80,169),(81,168),(82,167),(83,166),(84,165),(85,164),(86,163),(87,162),(88,161),(89,160),(90,159),(91,158),(92,157),(93,182),(94,181),(95,180),(96,179),(97,178),(98,177),(99,176),(100,175),(101,174),(102,173),(103,172),(104,171)], [(1,87,153,176),(2,86,154,175),(3,85,155,174),(4,84,156,173),(5,83,131,172),(6,82,132,171),(7,81,133,170),(8,80,134,169),(9,79,135,168),(10,104,136,167),(11,103,137,166),(12,102,138,165),(13,101,139,164),(14,100,140,163),(15,99,141,162),(16,98,142,161),(17,97,143,160),(18,96,144,159),(19,95,145,158),(20,94,146,157),(21,93,147,182),(22,92,148,181),(23,91,149,180),(24,90,150,179),(25,89,151,178),(26,88,152,177),(27,56,112,197),(28,55,113,196),(29,54,114,195),(30,53,115,194),(31,78,116,193),(32,77,117,192),(33,76,118,191),(34,75,119,190),(35,74,120,189),(36,73,121,188),(37,72,122,187),(38,71,123,186),(39,70,124,185),(40,69,125,184),(41,68,126,183),(42,67,127,208),(43,66,128,207),(44,65,129,206),(45,64,130,205),(46,63,105,204),(47,62,106,203),(48,61,107,202),(49,60,108,201),(50,59,109,200),(51,58,110,199),(52,57,111,198)], [(1,127),(2,126),(3,125),(4,124),(5,123),(6,122),(7,121),(8,120),(9,119),(10,118),(11,117),(12,116),(13,115),(14,114),(15,113),(16,112),(17,111),(18,110),(19,109),(20,108),(21,107),(22,106),(23,105),(24,130),(25,129),(26,128),(27,142),(28,141),(29,140),(30,139),(31,138),(32,137),(33,136),(34,135),(35,134),(36,133),(37,132),(38,131),(39,156),(40,155),(41,154),(42,153),(43,152),(44,151),(45,150),(46,149),(47,148),(48,147),(49,146),(50,145),(51,144),(52,143),(53,101),(54,100),(55,99),(56,98),(57,97),(58,96),(59,95),(60,94),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82),(73,81),(74,80),(75,79),(76,104),(77,103),(78,102),(157,201),(158,200),(159,199),(160,198),(161,197),(162,196),(163,195),(164,194),(165,193),(166,192),(167,191),(168,190),(169,189),(170,188),(171,187),(172,186),(173,185),(174,184),(175,183),(176,208),(177,207),(178,206),(179,205),(180,204),(181,203),(182,202)])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AD | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 26 | 26 | 52 | 2 | 2 | 4 | 26 | 26 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D13 | D26 | D26 | D52⋊5C2 | D4×D13 |
| kernel | D26⋊D4 | C26.D4 | D26⋊C4 | C13×C22⋊C4 | C2×C4×D13 | C2×D52 | C2×C13⋊D4 | Dic13 | D26 | C26 | C22⋊C4 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 12 | 6 | 24 | 12 |
Matrix representation of D26⋊D4 ►in GL4(𝔽53) generated by
| 17 | 33 | 0 | 0 |
| 16 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 41 | 44 | 0 | 0 |
| 10 | 12 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 17 | 29 | 0 | 0 |
| 43 | 36 | 0 | 0 |
| 0 | 0 | 7 | 38 |
| 0 | 0 | 21 | 46 |
| 41 | 14 | 0 | 0 |
| 39 | 12 | 0 | 0 |
| 0 | 0 | 46 | 25 |
| 0 | 0 | 32 | 7 |
G:=sub<GL(4,GF(53))| [17,16,0,0,33,28,0,0,0,0,1,0,0,0,0,1],[41,10,0,0,44,12,0,0,0,0,52,0,0,0,0,52],[17,43,0,0,29,36,0,0,0,0,7,21,0,0,38,46],[41,39,0,0,14,12,0,0,0,0,46,32,0,0,25,7] >;
D26⋊D4 in GAP, Magma, Sage, TeX
D_{26}\rtimes D_4 % in TeX
G:=Group("D26:D4"); // GroupNames label
G:=SmallGroup(416,105);
// by ID
G=gap.SmallGroup(416,105);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,55,506,188,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^24*b,d*b*d=a^11*b,d*c*d=c^-1>;
// generators/relations