metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊2Q8, C4.13D52, C52.11D4, C4⋊C4⋊5D13, C52⋊3C4⋊6C2, C26.8(C2×D4), C2.7(Q8×D13), C2.10(C2×D52), (C2×C4).14D26, C13⋊3(C22⋊Q8), C26.14(C2×Q8), (C2×Dic26)⋊7C2, (C2×C52).6C22, D26⋊C4.3C2, C26.27(C4○D4), (C2×C26).38C23, C2.13(D4⋊2D13), C22.52(C22×D13), (C2×Dic13).13C22, (C22×D13).25C22, (C13×C4⋊C4)⋊8C2, (C2×C4×D13).3C2, SmallGroup(416,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26⋊2Q8
G = < a,b,c,d | a26=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a11b, bd=db, dcd-1=c-1 >
Subgroups: 520 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C13, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, D13, C26, C22⋊Q8, Dic13, C52, C52, D26, D26, C2×C26, Dic26, C4×D13, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C22×D13, C52⋊3C4, D26⋊C4, C13×C4⋊C4, C2×Dic26, C2×C4×D13, D26⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26, D52, C22×D13, C2×D52, D4⋊2D13, Q8×D13, D26⋊2Q8
►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 96)(2 95)(3 94)(4 93)(5 92)(6 91)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 84)(14 83)(15 82)(16 81)(17 80)(18 79)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 98)(26 97)(27 184)(28 183)(29 208)(30 207)(31 206)(32 205)(33 204)(34 203)(35 202)(36 201)(37 200)(38 199)(39 198)(40 197)(41 196)(42 195)(43 194)(44 193)(45 192)(46 191)(47 190)(48 189)(49 188)(50 187)(51 186)(52 185)(53 171)(54 170)(55 169)(56 168)(57 167)(58 166)(59 165)(60 164)(61 163)(62 162)(63 161)(64 160)(65 159)(66 158)(67 157)(68 182)(69 181)(70 180)(71 179)(72 178)(73 177)(74 176)(75 175)(76 174)(77 173)(78 172)(105 138)(106 137)(107 136)(108 135)(109 134)(110 133)(111 132)(112 131)(113 156)(114 155)(115 154)(116 153)(117 152)(118 151)(119 150)(120 149)(121 148)(122 147)(123 146)(124 145)(125 144)(126 143)(127 142)(128 141)(129 140)(130 139)
(1 135 97 122)(2 134 98 121)(3 133 99 120)(4 132 100 119)(5 131 101 118)(6 156 102 117)(7 155 103 116)(8 154 104 115)(9 153 79 114)(10 152 80 113)(11 151 81 112)(12 150 82 111)(13 149 83 110)(14 148 84 109)(15 147 85 108)(16 146 86 107)(17 145 87 106)(18 144 88 105)(19 143 89 130)(20 142 90 129)(21 141 91 128)(22 140 92 127)(23 139 93 126)(24 138 94 125)(25 137 95 124)(26 136 96 123)(27 70 203 176)(28 69 204 175)(29 68 205 174)(30 67 206 173)(31 66 207 172)(32 65 208 171)(33 64 183 170)(34 63 184 169)(35 62 185 168)(36 61 186 167)(37 60 187 166)(38 59 188 165)(39 58 189 164)(40 57 190 163)(41 56 191 162)(42 55 192 161)(43 54 193 160)(44 53 194 159)(45 78 195 158)(46 77 196 157)(47 76 197 182)(48 75 198 181)(49 74 199 180)(50 73 200 179)(51 72 201 178)(52 71 202 177)
(1 44 97 194)(2 45 98 195)(3 46 99 196)(4 47 100 197)(5 48 101 198)(6 49 102 199)(7 50 103 200)(8 51 104 201)(9 52 79 202)(10 27 80 203)(11 28 81 204)(12 29 82 205)(13 30 83 206)(14 31 84 207)(15 32 85 208)(16 33 86 183)(17 34 87 184)(18 35 88 185)(19 36 89 186)(20 37 90 187)(21 38 91 188)(22 39 92 189)(23 40 93 190)(24 41 94 191)(25 42 95 192)(26 43 96 193)(53 135 159 122)(54 136 160 123)(55 137 161 124)(56 138 162 125)(57 139 163 126)(58 140 164 127)(59 141 165 128)(60 142 166 129)(61 143 167 130)(62 144 168 105)(63 145 169 106)(64 146 170 107)(65 147 171 108)(66 148 172 109)(67 149 173 110)(68 150 174 111)(69 151 175 112)(70 152 176 113)(71 153 177 114)(72 154 178 115)(73 155 179 116)(74 156 180 117)(75 131 181 118)(76 132 182 119)(77 133 157 120)(78 134 158 121)
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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,184)(28,183)(29,208)(30,207)(31,206)(32,205)(33,204)(34,203)(35,202)(36,201)(37,200)(38,199)(39,198)(40,197)(41,196)(42,195)(43,194)(44,193)(45,192)(46,191)(47,190)(48,189)(49,188)(50,187)(51,186)(52,185)(53,171)(54,170)(55,169)(56,168)(57,167)(58,166)(59,165)(60,164)(61,163)(62,162)(63,161)(64,160)(65,159)(66,158)(67,157)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(105,138)(106,137)(107,136)(108,135)(109,134)(110,133)(111,132)(112,131)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)(119,150)(120,149)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139), (1,135,97,122)(2,134,98,121)(3,133,99,120)(4,132,100,119)(5,131,101,118)(6,156,102,117)(7,155,103,116)(8,154,104,115)(9,153,79,114)(10,152,80,113)(11,151,81,112)(12,150,82,111)(13,149,83,110)(14,148,84,109)(15,147,85,108)(16,146,86,107)(17,145,87,106)(18,144,88,105)(19,143,89,130)(20,142,90,129)(21,141,91,128)(22,140,92,127)(23,139,93,126)(24,138,94,125)(25,137,95,124)(26,136,96,123)(27,70,203,176)(28,69,204,175)(29,68,205,174)(30,67,206,173)(31,66,207,172)(32,65,208,171)(33,64,183,170)(34,63,184,169)(35,62,185,168)(36,61,186,167)(37,60,187,166)(38,59,188,165)(39,58,189,164)(40,57,190,163)(41,56,191,162)(42,55,192,161)(43,54,193,160)(44,53,194,159)(45,78,195,158)(46,77,196,157)(47,76,197,182)(48,75,198,181)(49,74,199,180)(50,73,200,179)(51,72,201,178)(52,71,202,177), (1,44,97,194)(2,45,98,195)(3,46,99,196)(4,47,100,197)(5,48,101,198)(6,49,102,199)(7,50,103,200)(8,51,104,201)(9,52,79,202)(10,27,80,203)(11,28,81,204)(12,29,82,205)(13,30,83,206)(14,31,84,207)(15,32,85,208)(16,33,86,183)(17,34,87,184)(18,35,88,185)(19,36,89,186)(20,37,90,187)(21,38,91,188)(22,39,92,189)(23,40,93,190)(24,41,94,191)(25,42,95,192)(26,43,96,193)(53,135,159,122)(54,136,160,123)(55,137,161,124)(56,138,162,125)(57,139,163,126)(58,140,164,127)(59,141,165,128)(60,142,166,129)(61,143,167,130)(62,144,168,105)(63,145,169,106)(64,146,170,107)(65,147,171,108)(66,148,172,109)(67,149,173,110)(68,150,174,111)(69,151,175,112)(70,152,176,113)(71,153,177,114)(72,154,178,115)(73,155,179,116)(74,156,180,117)(75,131,181,118)(76,132,182,119)(77,133,157,120)(78,134,158,121)>;
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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,184)(28,183)(29,208)(30,207)(31,206)(32,205)(33,204)(34,203)(35,202)(36,201)(37,200)(38,199)(39,198)(40,197)(41,196)(42,195)(43,194)(44,193)(45,192)(46,191)(47,190)(48,189)(49,188)(50,187)(51,186)(52,185)(53,171)(54,170)(55,169)(56,168)(57,167)(58,166)(59,165)(60,164)(61,163)(62,162)(63,161)(64,160)(65,159)(66,158)(67,157)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(105,138)(106,137)(107,136)(108,135)(109,134)(110,133)(111,132)(112,131)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)(119,150)(120,149)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139), (1,135,97,122)(2,134,98,121)(3,133,99,120)(4,132,100,119)(5,131,101,118)(6,156,102,117)(7,155,103,116)(8,154,104,115)(9,153,79,114)(10,152,80,113)(11,151,81,112)(12,150,82,111)(13,149,83,110)(14,148,84,109)(15,147,85,108)(16,146,86,107)(17,145,87,106)(18,144,88,105)(19,143,89,130)(20,142,90,129)(21,141,91,128)(22,140,92,127)(23,139,93,126)(24,138,94,125)(25,137,95,124)(26,136,96,123)(27,70,203,176)(28,69,204,175)(29,68,205,174)(30,67,206,173)(31,66,207,172)(32,65,208,171)(33,64,183,170)(34,63,184,169)(35,62,185,168)(36,61,186,167)(37,60,187,166)(38,59,188,165)(39,58,189,164)(40,57,190,163)(41,56,191,162)(42,55,192,161)(43,54,193,160)(44,53,194,159)(45,78,195,158)(46,77,196,157)(47,76,197,182)(48,75,198,181)(49,74,199,180)(50,73,200,179)(51,72,201,178)(52,71,202,177), (1,44,97,194)(2,45,98,195)(3,46,99,196)(4,47,100,197)(5,48,101,198)(6,49,102,199)(7,50,103,200)(8,51,104,201)(9,52,79,202)(10,27,80,203)(11,28,81,204)(12,29,82,205)(13,30,83,206)(14,31,84,207)(15,32,85,208)(16,33,86,183)(17,34,87,184)(18,35,88,185)(19,36,89,186)(20,37,90,187)(21,38,91,188)(22,39,92,189)(23,40,93,190)(24,41,94,191)(25,42,95,192)(26,43,96,193)(53,135,159,122)(54,136,160,123)(55,137,161,124)(56,138,162,125)(57,139,163,126)(58,140,164,127)(59,141,165,128)(60,142,166,129)(61,143,167,130)(62,144,168,105)(63,145,169,106)(64,146,170,107)(65,147,171,108)(66,148,172,109)(67,149,173,110)(68,150,174,111)(69,151,175,112)(70,152,176,113)(71,153,177,114)(72,154,178,115)(73,155,179,116)(74,156,180,117)(75,131,181,118)(76,132,182,119)(77,133,157,120)(78,134,158,121) );
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)], [(1,96),(2,95),(3,94),(4,93),(5,92),(6,91),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,84),(14,83),(15,82),(16,81),(17,80),(18,79),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(25,98),(26,97),(27,184),(28,183),(29,208),(30,207),(31,206),(32,205),(33,204),(34,203),(35,202),(36,201),(37,200),(38,199),(39,198),(40,197),(41,196),(42,195),(43,194),(44,193),(45,192),(46,191),(47,190),(48,189),(49,188),(50,187),(51,186),(52,185),(53,171),(54,170),(55,169),(56,168),(57,167),(58,166),(59,165),(60,164),(61,163),(62,162),(63,161),(64,160),(65,159),(66,158),(67,157),(68,182),(69,181),(70,180),(71,179),(72,178),(73,177),(74,176),(75,175),(76,174),(77,173),(78,172),(105,138),(106,137),(107,136),(108,135),(109,134),(110,133),(111,132),(112,131),(113,156),(114,155),(115,154),(116,153),(117,152),(118,151),(119,150),(120,149),(121,148),(122,147),(123,146),(124,145),(125,144),(126,143),(127,142),(128,141),(129,140),(130,139)], [(1,135,97,122),(2,134,98,121),(3,133,99,120),(4,132,100,119),(5,131,101,118),(6,156,102,117),(7,155,103,116),(8,154,104,115),(9,153,79,114),(10,152,80,113),(11,151,81,112),(12,150,82,111),(13,149,83,110),(14,148,84,109),(15,147,85,108),(16,146,86,107),(17,145,87,106),(18,144,88,105),(19,143,89,130),(20,142,90,129),(21,141,91,128),(22,140,92,127),(23,139,93,126),(24,138,94,125),(25,137,95,124),(26,136,96,123),(27,70,203,176),(28,69,204,175),(29,68,205,174),(30,67,206,173),(31,66,207,172),(32,65,208,171),(33,64,183,170),(34,63,184,169),(35,62,185,168),(36,61,186,167),(37,60,187,166),(38,59,188,165),(39,58,189,164),(40,57,190,163),(41,56,191,162),(42,55,192,161),(43,54,193,160),(44,53,194,159),(45,78,195,158),(46,77,196,157),(47,76,197,182),(48,75,198,181),(49,74,199,180),(50,73,200,179),(51,72,201,178),(52,71,202,177)], [(1,44,97,194),(2,45,98,195),(3,46,99,196),(4,47,100,197),(5,48,101,198),(6,49,102,199),(7,50,103,200),(8,51,104,201),(9,52,79,202),(10,27,80,203),(11,28,81,204),(12,29,82,205),(13,30,83,206),(14,31,84,207),(15,32,85,208),(16,33,86,183),(17,34,87,184),(18,35,88,185),(19,36,89,186),(20,37,90,187),(21,38,91,188),(22,39,92,189),(23,40,93,190),(24,41,94,191),(25,42,95,192),(26,43,96,193),(53,135,159,122),(54,136,160,123),(55,137,161,124),(56,138,162,125),(57,139,163,126),(58,140,164,127),(59,141,165,128),(60,142,166,129),(61,143,167,130),(62,144,168,105),(63,145,169,106),(64,146,170,107),(65,147,171,108),(66,148,172,109),(67,149,173,110),(68,150,174,111),(69,151,175,112),(70,152,176,113),(71,153,177,114),(72,154,178,115),(73,155,179,116),(74,156,180,117),(75,131,181,118),(76,132,182,119),(77,133,157,120),(78,134,158,121)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 2 | 2 | 4 | 4 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | D13 | D26 | D52 | D4⋊2D13 | Q8×D13 |
| kernel | D26⋊2Q8 | C52⋊3C4 | D26⋊C4 | C13×C4⋊C4 | C2×Dic26 | C2×C4×D13 | C52 | D26 | C26 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 18 | 24 | 6 | 6 |
Matrix representation of D26⋊2Q8 ►in GL4(𝔽53) generated by
| 50 | 47 | 0 | 0 |
| 6 | 47 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 14 | 19 | 0 | 0 |
| 26 | 39 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 52 |
| 13 | 22 | 0 | 0 |
| 2 | 40 | 0 | 0 |
| 0 | 0 | 52 | 39 |
| 0 | 0 | 38 | 1 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 26 | 23 |
G:=sub<GL(4,GF(53))| [50,6,0,0,47,47,0,0,0,0,52,0,0,0,0,52],[14,26,0,0,19,39,0,0,0,0,1,15,0,0,0,52],[13,2,0,0,22,40,0,0,0,0,52,38,0,0,39,1],[52,0,0,0,0,52,0,0,0,0,30,26,0,0,0,23] >;
D26⋊2Q8 in GAP, Magma, Sage, TeX
D_{26}\rtimes_2Q_8 % in TeX
G:=Group("D26:2Q8"); // GroupNames label
G:=SmallGroup(416,118);
// by ID
G=gap.SmallGroup(416,118);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,122,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^11*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations