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## G = D26⋊Q8order 416 = 25·13

### 1st semidirect product of D26 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — D26⋊Q8
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — D26⋊Q8
 Lower central C13 — C2×C26 — D26⋊Q8
 Upper central C1 — C22 — C4⋊C4

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 [×3], C2 [×2], C4 [×7], C22, C22 [×4], C2×C4 [×3], C2×C4 [×5], Q8 [×2], C23, C13, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, D13 [×2], C26 [×3], C22⋊Q8, Dic13 [×2], Dic13 [×2], C52 [×3], D26 [×2], D26 [×2], C2×C26, Dic26 [×2], C4×D13 [×2], C2×Dic13 [×3], C2×C52 [×3], C22×D13, C26.D4 [×2], D26⋊C4 [×2], C13×C4⋊C4, C2×Dic26, C2×C4×D13, D26⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26 [×3], C22×D13, D525C2, D4×D13, Q8×D13, D26⋊Q8

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

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

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

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

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`

׿
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