metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊1Q8, Dic13.15D4, C4⋊C4⋊4D13, C2.6(Q8×D13), C2.14(D4×D13), (C2×C4).13D26, C26.26(C2×D4), C13⋊2(C22⋊Q8), C26.13(C2×Q8), (C2×Dic26)⋊4C2, D26⋊C4.2C2, C26.13(C4○D4), C26.D4⋊12C2, (C2×C26).37C23, (C2×C52).58C22, C2.15(D52⋊5C2), C22.51(C22×D13), (C2×Dic13).12C22, (C22×D13).24C22, (C13×C4⋊C4)⋊7C2, (C2×C4×D13).11C2, SmallGroup(416,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26⋊Q8
G = < a,b,c,d | a26=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=a11b, dbd-1=a24b, dcd-1=c-1 >
Subgroups: 520 in 74 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C13, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, D13, C26, C22⋊Q8, Dic13, Dic13, C52, D26, D26, C2×C26, Dic26, C4×D13, C2×Dic13, C2×C52, C22×D13, C26.D4, D26⋊C4, C13×C4⋊C4, C2×Dic26, C2×C4×D13, D26⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26, C22×D13, D52⋊5C2, D4×D13, Q8×D13, D26⋊Q8
►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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(53 166)(54 165)(55 164)(56 163)(57 162)(58 161)(59 160)(60 159)(61 158)(62 157)(63 182)(64 181)(65 180)(66 179)(67 178)(68 177)(69 176)(70 175)(71 174)(72 173)(73 172)(74 171)(75 170)(76 169)(77 168)(78 167)(79 203)(80 202)(81 201)(82 200)(83 199)(84 198)(85 197)(86 196)(87 195)(88 194)(89 193)(90 192)(91 191)(92 190)(93 189)(94 188)(95 187)(96 186)(97 185)(98 184)(99 183)(100 208)(101 207)(102 206)(103 205)(104 204)(105 147)(106 146)(107 145)(108 144)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 135)(118 134)(119 133)(120 132)(121 131)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)
(1 190 48 93)(2 189 49 92)(3 188 50 91)(4 187 51 90)(5 186 52 89)(6 185 27 88)(7 184 28 87)(8 183 29 86)(9 208 30 85)(10 207 31 84)(11 206 32 83)(12 205 33 82)(13 204 34 81)(14 203 35 80)(15 202 36 79)(16 201 37 104)(17 200 38 103)(18 199 39 102)(19 198 40 101)(20 197 41 100)(21 196 42 99)(22 195 43 98)(23 194 44 97)(24 193 45 96)(25 192 46 95)(26 191 47 94)(53 149 171 113)(54 148 172 112)(55 147 173 111)(56 146 174 110)(57 145 175 109)(58 144 176 108)(59 143 177 107)(60 142 178 106)(61 141 179 105)(62 140 180 130)(63 139 181 129)(64 138 182 128)(65 137 157 127)(66 136 158 126)(67 135 159 125)(68 134 160 124)(69 133 161 123)(70 132 162 122)(71 131 163 121)(72 156 164 120)(73 155 165 119)(74 154 166 118)(75 153 167 117)(76 152 168 116)(77 151 169 115)(78 150 170 114)
(1 115 48 151)(2 114 49 150)(3 113 50 149)(4 112 51 148)(5 111 52 147)(6 110 27 146)(7 109 28 145)(8 108 29 144)(9 107 30 143)(10 106 31 142)(11 105 32 141)(12 130 33 140)(13 129 34 139)(14 128 35 138)(15 127 36 137)(16 126 37 136)(17 125 38 135)(18 124 39 134)(19 123 40 133)(20 122 41 132)(21 121 42 131)(22 120 43 156)(23 119 44 155)(24 118 45 154)(25 117 46 153)(26 116 47 152)(53 188 171 91)(54 187 172 90)(55 186 173 89)(56 185 174 88)(57 184 175 87)(58 183 176 86)(59 208 177 85)(60 207 178 84)(61 206 179 83)(62 205 180 82)(63 204 181 81)(64 203 182 80)(65 202 157 79)(66 201 158 104)(67 200 159 103)(68 199 160 102)(69 198 161 101)(70 197 162 100)(71 196 163 99)(72 195 164 98)(73 194 165 97)(74 193 166 96)(75 192 167 95)(76 191 168 94)(77 190 169 93)(78 189 170 92)
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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(53,166)(54,165)(55,164)(56,163)(57,162)(58,161)(59,160)(60,159)(61,158)(62,157)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,203)(80,202)(81,201)(82,200)(83,199)(84,198)(85,197)(86,196)(87,195)(88,194)(89,193)(90,192)(91,191)(92,190)(93,189)(94,188)(95,187)(96,186)(97,185)(98,184)(99,183)(100,208)(101,207)(102,206)(103,205)(104,204)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148), (1,190,48,93)(2,189,49,92)(3,188,50,91)(4,187,51,90)(5,186,52,89)(6,185,27,88)(7,184,28,87)(8,183,29,86)(9,208,30,85)(10,207,31,84)(11,206,32,83)(12,205,33,82)(13,204,34,81)(14,203,35,80)(15,202,36,79)(16,201,37,104)(17,200,38,103)(18,199,39,102)(19,198,40,101)(20,197,41,100)(21,196,42,99)(22,195,43,98)(23,194,44,97)(24,193,45,96)(25,192,46,95)(26,191,47,94)(53,149,171,113)(54,148,172,112)(55,147,173,111)(56,146,174,110)(57,145,175,109)(58,144,176,108)(59,143,177,107)(60,142,178,106)(61,141,179,105)(62,140,180,130)(63,139,181,129)(64,138,182,128)(65,137,157,127)(66,136,158,126)(67,135,159,125)(68,134,160,124)(69,133,161,123)(70,132,162,122)(71,131,163,121)(72,156,164,120)(73,155,165,119)(74,154,166,118)(75,153,167,117)(76,152,168,116)(77,151,169,115)(78,150,170,114), (1,115,48,151)(2,114,49,150)(3,113,50,149)(4,112,51,148)(5,111,52,147)(6,110,27,146)(7,109,28,145)(8,108,29,144)(9,107,30,143)(10,106,31,142)(11,105,32,141)(12,130,33,140)(13,129,34,139)(14,128,35,138)(15,127,36,137)(16,126,37,136)(17,125,38,135)(18,124,39,134)(19,123,40,133)(20,122,41,132)(21,121,42,131)(22,120,43,156)(23,119,44,155)(24,118,45,154)(25,117,46,153)(26,116,47,152)(53,188,171,91)(54,187,172,90)(55,186,173,89)(56,185,174,88)(57,184,175,87)(58,183,176,86)(59,208,177,85)(60,207,178,84)(61,206,179,83)(62,205,180,82)(63,204,181,81)(64,203,182,80)(65,202,157,79)(66,201,158,104)(67,200,159,103)(68,199,160,102)(69,198,161,101)(70,197,162,100)(71,196,163,99)(72,195,164,98)(73,194,165,97)(74,193,166,96)(75,192,167,95)(76,191,168,94)(77,190,169,93)(78,189,170,92)>;
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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(53,166)(54,165)(55,164)(56,163)(57,162)(58,161)(59,160)(60,159)(61,158)(62,157)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,203)(80,202)(81,201)(82,200)(83,199)(84,198)(85,197)(86,196)(87,195)(88,194)(89,193)(90,192)(91,191)(92,190)(93,189)(94,188)(95,187)(96,186)(97,185)(98,184)(99,183)(100,208)(101,207)(102,206)(103,205)(104,204)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148), (1,190,48,93)(2,189,49,92)(3,188,50,91)(4,187,51,90)(5,186,52,89)(6,185,27,88)(7,184,28,87)(8,183,29,86)(9,208,30,85)(10,207,31,84)(11,206,32,83)(12,205,33,82)(13,204,34,81)(14,203,35,80)(15,202,36,79)(16,201,37,104)(17,200,38,103)(18,199,39,102)(19,198,40,101)(20,197,41,100)(21,196,42,99)(22,195,43,98)(23,194,44,97)(24,193,45,96)(25,192,46,95)(26,191,47,94)(53,149,171,113)(54,148,172,112)(55,147,173,111)(56,146,174,110)(57,145,175,109)(58,144,176,108)(59,143,177,107)(60,142,178,106)(61,141,179,105)(62,140,180,130)(63,139,181,129)(64,138,182,128)(65,137,157,127)(66,136,158,126)(67,135,159,125)(68,134,160,124)(69,133,161,123)(70,132,162,122)(71,131,163,121)(72,156,164,120)(73,155,165,119)(74,154,166,118)(75,153,167,117)(76,152,168,116)(77,151,169,115)(78,150,170,114), (1,115,48,151)(2,114,49,150)(3,113,50,149)(4,112,51,148)(5,111,52,147)(6,110,27,146)(7,109,28,145)(8,108,29,144)(9,107,30,143)(10,106,31,142)(11,105,32,141)(12,130,33,140)(13,129,34,139)(14,128,35,138)(15,127,36,137)(16,126,37,136)(17,125,38,135)(18,124,39,134)(19,123,40,133)(20,122,41,132)(21,121,42,131)(22,120,43,156)(23,119,44,155)(24,118,45,154)(25,117,46,153)(26,116,47,152)(53,188,171,91)(54,187,172,90)(55,186,173,89)(56,185,174,88)(57,184,175,87)(58,183,176,86)(59,208,177,85)(60,207,178,84)(61,206,179,83)(62,205,180,82)(63,204,181,81)(64,203,182,80)(65,202,157,79)(66,201,158,104)(67,200,159,103)(68,199,160,102)(69,198,161,101)(70,197,162,100)(71,196,163,99)(72,195,164,98)(73,194,165,97)(74,193,166,96)(75,192,167,95)(76,191,168,94)(77,190,169,93)(78,189,170,92) );
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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(53,166),(54,165),(55,164),(56,163),(57,162),(58,161),(59,160),(60,159),(61,158),(62,157),(63,182),(64,181),(65,180),(66,179),(67,178),(68,177),(69,176),(70,175),(71,174),(72,173),(73,172),(74,171),(75,170),(76,169),(77,168),(78,167),(79,203),(80,202),(81,201),(82,200),(83,199),(84,198),(85,197),(86,196),(87,195),(88,194),(89,193),(90,192),(91,191),(92,190),(93,189),(94,188),(95,187),(96,186),(97,185),(98,184),(99,183),(100,208),(101,207),(102,206),(103,205),(104,204),(105,147),(106,146),(107,145),(108,144),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,135),(118,134),(119,133),(120,132),(121,131),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148)], [(1,190,48,93),(2,189,49,92),(3,188,50,91),(4,187,51,90),(5,186,52,89),(6,185,27,88),(7,184,28,87),(8,183,29,86),(9,208,30,85),(10,207,31,84),(11,206,32,83),(12,205,33,82),(13,204,34,81),(14,203,35,80),(15,202,36,79),(16,201,37,104),(17,200,38,103),(18,199,39,102),(19,198,40,101),(20,197,41,100),(21,196,42,99),(22,195,43,98),(23,194,44,97),(24,193,45,96),(25,192,46,95),(26,191,47,94),(53,149,171,113),(54,148,172,112),(55,147,173,111),(56,146,174,110),(57,145,175,109),(58,144,176,108),(59,143,177,107),(60,142,178,106),(61,141,179,105),(62,140,180,130),(63,139,181,129),(64,138,182,128),(65,137,157,127),(66,136,158,126),(67,135,159,125),(68,134,160,124),(69,133,161,123),(70,132,162,122),(71,131,163,121),(72,156,164,120),(73,155,165,119),(74,154,166,118),(75,153,167,117),(76,152,168,116),(77,151,169,115),(78,150,170,114)], [(1,115,48,151),(2,114,49,150),(3,113,50,149),(4,112,51,148),(5,111,52,147),(6,110,27,146),(7,109,28,145),(8,108,29,144),(9,107,30,143),(10,106,31,142),(11,105,32,141),(12,130,33,140),(13,129,34,139),(14,128,35,138),(15,127,36,137),(16,126,37,136),(17,125,38,135),(18,124,39,134),(19,123,40,133),(20,122,41,132),(21,121,42,131),(22,120,43,156),(23,119,44,155),(24,118,45,154),(25,117,46,153),(26,116,47,152),(53,188,171,91),(54,187,172,90),(55,186,173,89),(56,185,174,88),(57,184,175,87),(58,183,176,86),(59,208,177,85),(60,207,178,84),(61,206,179,83),(62,205,180,82),(63,204,181,81),(64,203,182,80),(65,202,157,79),(66,201,158,104),(67,200,159,103),(68,199,160,102),(69,198,161,101),(70,197,162,100),(71,196,163,99),(72,195,164,98),(73,194,165,97),(74,193,166,96),(75,192,167,95),(76,191,168,94),(77,190,169,93),(78,189,170,92)]])
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⋊5C2 | D4×D13 | Q8×D13 |
| kernel | D26⋊Q8 | C26.D4 | D26⋊C4 | C13×C4⋊C4 | C2×Dic26 | C2×C4×D13 | Dic13 | D26 | C26 | C4⋊C4 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 18 | 24 | 6 | 6 |
Matrix representation of D26⋊Q8 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 41 | 2 |
| 0 | 0 | 4 | 39 |
| 1 | 0 | 0 | 0 |
| 9 | 52 | 0 | 0 |
| 0 | 0 | 13 | 15 |
| 0 | 0 | 10 | 40 |
| 1 | 41 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 2 | 22 |
| 0 | 0 | 7 | 51 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 1 | 21 |
| 0 | 0 | 10 | 52 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,41,4,0,0,2,39],[1,9,0,0,0,52,0,0,0,0,13,10,0,0,15,40],[1,0,0,0,41,52,0,0,0,0,2,7,0,0,22,51],[52,0,0,0,0,52,0,0,0,0,1,10,0,0,21,52] >;
D26⋊Q8 in GAP, Magma, Sage, TeX
D_{26}\rtimes Q_8 % in TeX
G:=Group("D26:Q8"); // GroupNames label
G:=SmallGroup(416,117);
// by ID
G=gap.SmallGroup(416,117);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,55,506,188,86,13829]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^11*b,d*b*d^-1=a^24*b,d*c*d^-1=c^-1>;
// generators/relations