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## G = D4.D26order 416 = 25·13

### 4th non-split extension by D4 of D26 acting via D26/D13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D4.D26
 Chief series C1 — C13 — C26 — C52 — C4×D13 — Q8×D13 — D4.D26
 Lower central C13 — C26 — C52 — D4.D26
 Upper central C1 — C2 — C4 — SD16

Generators and relations for D4.D26
G = < a,b,c,d | a4=b2=1, c26=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c25 >

Subgroups: 408 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, Q8, C13, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, D13, C26, C26, C8.C22, Dic13, Dic13, C52, C52, D26, C2×C26, C132C8, C104, Dic26, Dic26, C4×D13, C4×D13, C2×Dic13, C13⋊D4, D4×C13, Q8×C13, C8⋊D13, Dic52, D4.D13, C13⋊Q16, C13×SD16, D42D13, Q8×D13, D4.D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8.C22, D26, C22×D13, D4×D13, D4.D26

Smallest permutation representation of D4.D26
On 208 points
Generators in S208
```(1 194 27 168)(2 169 28 195)(3 196 29 170)(4 171 30 197)(5 198 31 172)(6 173 32 199)(7 200 33 174)(8 175 34 201)(9 202 35 176)(10 177 36 203)(11 204 37 178)(12 179 38 205)(13 206 39 180)(14 181 40 207)(15 208 41 182)(16 183 42 157)(17 158 43 184)(18 185 44 159)(19 160 45 186)(20 187 46 161)(21 162 47 188)(22 189 48 163)(23 164 49 190)(24 191 50 165)(25 166 51 192)(26 193 52 167)(53 151 79 125)(54 126 80 152)(55 153 81 127)(56 128 82 154)(57 155 83 129)(58 130 84 156)(59 105 85 131)(60 132 86 106)(61 107 87 133)(62 134 88 108)(63 109 89 135)(64 136 90 110)(65 111 91 137)(66 138 92 112)(67 113 93 139)(68 140 94 114)(69 115 95 141)(70 142 96 116)(71 117 97 143)(72 144 98 118)(73 119 99 145)(74 146 100 120)(75 121 101 147)(76 148 102 122)(77 123 103 149)(78 150 104 124)
(1 154)(2 83)(3 156)(4 85)(5 106)(6 87)(7 108)(8 89)(9 110)(10 91)(11 112)(12 93)(13 114)(14 95)(15 116)(16 97)(17 118)(18 99)(19 120)(20 101)(21 122)(22 103)(23 124)(24 53)(25 126)(26 55)(27 128)(28 57)(29 130)(30 59)(31 132)(32 61)(33 134)(34 63)(35 136)(36 65)(37 138)(38 67)(39 140)(40 69)(41 142)(42 71)(43 144)(44 73)(45 146)(46 75)(47 148)(48 77)(49 150)(50 79)(51 152)(52 81)(54 166)(56 168)(58 170)(60 172)(62 174)(64 176)(66 178)(68 180)(70 182)(72 184)(74 186)(76 188)(78 190)(80 192)(82 194)(84 196)(86 198)(88 200)(90 202)(92 204)(94 206)(96 208)(98 158)(100 160)(102 162)(104 164)(105 171)(107 173)(109 175)(111 177)(113 179)(115 181)(117 183)(119 185)(121 187)(123 189)(125 191)(127 193)(129 195)(131 197)(133 199)(135 201)(137 203)(139 205)(141 207)(143 157)(145 159)(147 161)(149 163)(151 165)(153 167)(155 169)
(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 26 27 52)(2 51 28 25)(3 24 29 50)(4 49 30 23)(5 22 31 48)(6 47 32 21)(7 20 33 46)(8 45 34 19)(9 18 35 44)(10 43 36 17)(11 16 37 42)(12 41 38 15)(13 14 39 40)(53 84 79 58)(54 57 80 83)(55 82 81 56)(59 78 85 104)(60 103 86 77)(61 76 87 102)(62 101 88 75)(63 74 89 100)(64 99 90 73)(65 72 91 98)(66 97 92 71)(67 70 93 96)(68 95 94 69)(105 124 131 150)(106 149 132 123)(107 122 133 148)(108 147 134 121)(109 120 135 146)(110 145 136 119)(111 118 137 144)(112 143 138 117)(113 116 139 142)(114 141 140 115)(125 156 151 130)(126 129 152 155)(127 154 153 128)(157 178 183 204)(158 203 184 177)(159 176 185 202)(160 201 186 175)(161 174 187 200)(162 199 188 173)(163 172 189 198)(164 197 190 171)(165 170 191 196)(166 195 192 169)(167 168 193 194)(179 208 205 182)(180 181 206 207)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 13A ··· 13F 26A ··· 26F 26G ··· 26L 52A ··· 52F 52G ··· 52L 104A ··· 104L order 1 2 2 2 4 4 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 4 26 2 4 26 52 52 4 52 2 ··· 2 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 D26 C8.C22 D4×D13 D4.D26 kernel D4.D26 C8⋊D13 Dic52 D4.D13 C13⋊Q16 C13×SD16 D4⋊2D13 Q8×D13 Dic13 D26 SD16 C8 D4 Q8 C13 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 6 6 6 6 1 6 12

Matrix representation of D4.D26 in GL4(𝔽313) generated by

 279 233 51 201 221 55 111 4 147 154 135 80 238 16 126 157
,
 290 68 230 119 146 34 257 143 42 159 274 245 165 173 236 28
,
 251 252 129 123 142 172 77 189 311 214 12 61 50 194 55 191
,
 86 82 123 129 92 111 189 77 250 246 61 12 216 191 191 55
`G:=sub<GL(4,GF(313))| [279,221,147,238,233,55,154,16,51,111,135,126,201,4,80,157],[290,146,42,165,68,34,159,173,230,257,274,236,119,143,245,28],[251,142,311,50,252,172,214,194,129,77,12,55,123,189,61,191],[86,92,250,216,82,111,246,191,123,189,61,191,129,77,12,55] >;`

D4.D26 in GAP, Magma, Sage, TeX

`D_4.D_{26}`
`% in TeX`

`G:=Group("D4.D26");`
`// GroupNames label`

`G:=SmallGroup(416,136);`
`// by ID`

`G=gap.SmallGroup(416,136);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,362,116,297,159,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^26=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^25>;`
`// generators/relations`

׿
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