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, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C26, C4⋊D4, Dic13, Dic13, C52, D26, D26, C2×C26, C2×C26, C4×D13, D52, C2×Dic13, C13⋊D4, C2×C52, C22×D13, C22×C26, C26.D4, D26⋊C4, C13×C22⋊C4, C2×C4×D13, C2×D52, C2×C13⋊D4, D26⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4⋊D4, D26, C22×D13, D52⋊5C2, D4×D13, 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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 96)(7 95)(8 94)(9 93)(10 92)(11 91)(12 90)(13 89)(14 88)(15 87)(16 86)(17 85)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 104)(25 103)(26 102)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(105 157)(106 182)(107 181)(108 180)(109 179)(110 178)(111 177)(112 176)(113 175)(114 174)(115 173)(116 172)(117 171)(118 170)(119 169)(120 168)(121 167)(122 166)(123 165)(124 164)(125 163)(126 162)(127 161)(128 160)(129 159)(130 158)(131 187)(132 186)(133 185)(134 184)(135 183)(136 208)(137 207)(138 206)(139 205)(140 204)(141 203)(142 202)(143 201)(144 200)(145 199)(146 198)(147 197)(148 196)(149 195)(150 194)(151 193)(152 192)(153 191)(154 190)(155 189)(156 188)
(1 41 89 58)(2 40 90 57)(3 39 91 56)(4 38 92 55)(5 37 93 54)(6 36 94 53)(7 35 95 78)(8 34 96 77)(9 33 97 76)(10 32 98 75)(11 31 99 74)(12 30 100 73)(13 29 101 72)(14 28 102 71)(15 27 103 70)(16 52 104 69)(17 51 79 68)(18 50 80 67)(19 49 81 66)(20 48 82 65)(21 47 83 64)(22 46 84 63)(23 45 85 62)(24 44 86 61)(25 43 87 60)(26 42 88 59)(105 208 158 137)(106 207 159 136)(107 206 160 135)(108 205 161 134)(109 204 162 133)(110 203 163 132)(111 202 164 131)(112 201 165 156)(113 200 166 155)(114 199 167 154)(115 198 168 153)(116 197 169 152)(117 196 170 151)(118 195 171 150)(119 194 172 149)(120 193 173 148)(121 192 174 147)(122 191 175 146)(123 190 176 145)(124 189 177 144)(125 188 178 143)(126 187 179 142)(127 186 180 141)(128 185 181 140)(129 184 182 139)(130 183 157 138)
(1 195)(2 194)(3 193)(4 192)(5 191)(6 190)(7 189)(8 188)(9 187)(10 186)(11 185)(12 184)(13 183)(14 208)(15 207)(16 206)(17 205)(18 204)(19 203)(20 202)(21 201)(22 200)(23 199)(24 198)(25 197)(26 196)(27 106)(28 105)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 120)(40 119)(41 118)(42 117)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 176)(54 175)(55 174)(56 173)(57 172)(58 171)(59 170)(60 169)(61 168)(62 167)(63 166)(64 165)(65 164)(66 163)(67 162)(68 161)(69 160)(70 159)(71 158)(72 157)(73 182)(74 181)(75 180)(76 179)(77 178)(78 177)(79 134)(80 133)(81 132)(82 131)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)
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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,96)(7,95)(8,94)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,104)(25,103)(26,102)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(105,157)(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,187)(132,186)(133,185)(134,184)(135,183)(136,208)(137,207)(138,206)(139,205)(140,204)(141,203)(142,202)(143,201)(144,200)(145,199)(146,198)(147,197)(148,196)(149,195)(150,194)(151,193)(152,192)(153,191)(154,190)(155,189)(156,188), (1,41,89,58)(2,40,90,57)(3,39,91,56)(4,38,92,55)(5,37,93,54)(6,36,94,53)(7,35,95,78)(8,34,96,77)(9,33,97,76)(10,32,98,75)(11,31,99,74)(12,30,100,73)(13,29,101,72)(14,28,102,71)(15,27,103,70)(16,52,104,69)(17,51,79,68)(18,50,80,67)(19,49,81,66)(20,48,82,65)(21,47,83,64)(22,46,84,63)(23,45,85,62)(24,44,86,61)(25,43,87,60)(26,42,88,59)(105,208,158,137)(106,207,159,136)(107,206,160,135)(108,205,161,134)(109,204,162,133)(110,203,163,132)(111,202,164,131)(112,201,165,156)(113,200,166,155)(114,199,167,154)(115,198,168,153)(116,197,169,152)(117,196,170,151)(118,195,171,150)(119,194,172,149)(120,193,173,148)(121,192,174,147)(122,191,175,146)(123,190,176,145)(124,189,177,144)(125,188,178,143)(126,187,179,142)(127,186,180,141)(128,185,181,140)(129,184,182,139)(130,183,157,138), (1,195)(2,194)(3,193)(4,192)(5,191)(6,190)(7,189)(8,188)(9,187)(10,186)(11,185)(12,184)(13,183)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,201)(22,200)(23,199)(24,198)(25,197)(26,196)(27,106)(28,105)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,169)(61,168)(62,167)(63,166)(64,165)(65,164)(66,163)(67,162)(68,161)(69,160)(70,159)(71,158)(72,157)(73,182)(74,181)(75,180)(76,179)(77,178)(78,177)(79,134)(80,133)(81,132)(82,131)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)>;
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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,96)(7,95)(8,94)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,84)(19,83)(20,82)(21,81)(22,80)(23,79)(24,104)(25,103)(26,102)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(105,157)(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,187)(132,186)(133,185)(134,184)(135,183)(136,208)(137,207)(138,206)(139,205)(140,204)(141,203)(142,202)(143,201)(144,200)(145,199)(146,198)(147,197)(148,196)(149,195)(150,194)(151,193)(152,192)(153,191)(154,190)(155,189)(156,188), (1,41,89,58)(2,40,90,57)(3,39,91,56)(4,38,92,55)(5,37,93,54)(6,36,94,53)(7,35,95,78)(8,34,96,77)(9,33,97,76)(10,32,98,75)(11,31,99,74)(12,30,100,73)(13,29,101,72)(14,28,102,71)(15,27,103,70)(16,52,104,69)(17,51,79,68)(18,50,80,67)(19,49,81,66)(20,48,82,65)(21,47,83,64)(22,46,84,63)(23,45,85,62)(24,44,86,61)(25,43,87,60)(26,42,88,59)(105,208,158,137)(106,207,159,136)(107,206,160,135)(108,205,161,134)(109,204,162,133)(110,203,163,132)(111,202,164,131)(112,201,165,156)(113,200,166,155)(114,199,167,154)(115,198,168,153)(116,197,169,152)(117,196,170,151)(118,195,171,150)(119,194,172,149)(120,193,173,148)(121,192,174,147)(122,191,175,146)(123,190,176,145)(124,189,177,144)(125,188,178,143)(126,187,179,142)(127,186,180,141)(128,185,181,140)(129,184,182,139)(130,183,157,138), (1,195)(2,194)(3,193)(4,192)(5,191)(6,190)(7,189)(8,188)(9,187)(10,186)(11,185)(12,184)(13,183)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,201)(22,200)(23,199)(24,198)(25,197)(26,196)(27,106)(28,105)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,169)(61,168)(62,167)(63,166)(64,165)(65,164)(66,163)(67,162)(68,161)(69,160)(70,159)(71,158)(72,157)(73,182)(74,181)(75,180)(76,179)(77,178)(78,177)(79,134)(80,133)(81,132)(82,131)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135) );
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)], [(1,101),(2,100),(3,99),(4,98),(5,97),(6,96),(7,95),(8,94),(9,93),(10,92),(11,91),(12,90),(13,89),(14,88),(15,87),(16,86),(17,85),(18,84),(19,83),(20,82),(21,81),(22,80),(23,79),(24,104),(25,103),(26,102),(27,58),(28,57),(29,56),(30,55),(31,54),(32,53),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(105,157),(106,182),(107,181),(108,180),(109,179),(110,178),(111,177),(112,176),(113,175),(114,174),(115,173),(116,172),(117,171),(118,170),(119,169),(120,168),(121,167),(122,166),(123,165),(124,164),(125,163),(126,162),(127,161),(128,160),(129,159),(130,158),(131,187),(132,186),(133,185),(134,184),(135,183),(136,208),(137,207),(138,206),(139,205),(140,204),(141,203),(142,202),(143,201),(144,200),(145,199),(146,198),(147,197),(148,196),(149,195),(150,194),(151,193),(152,192),(153,191),(154,190),(155,189),(156,188)], [(1,41,89,58),(2,40,90,57),(3,39,91,56),(4,38,92,55),(5,37,93,54),(6,36,94,53),(7,35,95,78),(8,34,96,77),(9,33,97,76),(10,32,98,75),(11,31,99,74),(12,30,100,73),(13,29,101,72),(14,28,102,71),(15,27,103,70),(16,52,104,69),(17,51,79,68),(18,50,80,67),(19,49,81,66),(20,48,82,65),(21,47,83,64),(22,46,84,63),(23,45,85,62),(24,44,86,61),(25,43,87,60),(26,42,88,59),(105,208,158,137),(106,207,159,136),(107,206,160,135),(108,205,161,134),(109,204,162,133),(110,203,163,132),(111,202,164,131),(112,201,165,156),(113,200,166,155),(114,199,167,154),(115,198,168,153),(116,197,169,152),(117,196,170,151),(118,195,171,150),(119,194,172,149),(120,193,173,148),(121,192,174,147),(122,191,175,146),(123,190,176,145),(124,189,177,144),(125,188,178,143),(126,187,179,142),(127,186,180,141),(128,185,181,140),(129,184,182,139),(130,183,157,138)], [(1,195),(2,194),(3,193),(4,192),(5,191),(6,190),(7,189),(8,188),(9,187),(10,186),(11,185),(12,184),(13,183),(14,208),(15,207),(16,206),(17,205),(18,204),(19,203),(20,202),(21,201),(22,200),(23,199),(24,198),(25,197),(26,196),(27,106),(28,105),(29,130),(30,129),(31,128),(32,127),(33,126),(34,125),(35,124),(36,123),(37,122),(38,121),(39,120),(40,119),(41,118),(42,117),(43,116),(44,115),(45,114),(46,113),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,176),(54,175),(55,174),(56,173),(57,172),(58,171),(59,170),(60,169),(61,168),(62,167),(63,166),(64,165),(65,164),(66,163),(67,162),(68,161),(69,160),(70,159),(71,158),(72,157),(73,182),(74,181),(75,180),(76,179),(77,178),(78,177),(79,134),(80,133),(81,132),(82,131),(83,156),(84,155),(85,154),(86,153),(87,152),(88,151),(89,150),(90,149),(91,148),(92,147),(93,146),(94,145),(95,144),(96,143),(97,142),(98,141),(99,140),(100,139),(101,138),(102,137),(103,136),(104,135)]])
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