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## G = D26.13D4order 416 = 25·13

### 2nd non-split extension by D26 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — D26.13D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — D26.13D4
 Lower central C13 — C2×C26 — D26.13D4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D26.13D4
G = < a,b,c,d | a26=b2=c4=1, d2=a13, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a13b, dcd-1=c-1 >

Subgroups: 632 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C13, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C22.D4, Dic13, C52, D26, D26, C2×C26, C4×D13, D52, C2×Dic13, C2×C52, C22×D13, C26.D4, D26⋊C4, C13×C4⋊C4, C2×C4×D13, C2×D52, D26.13D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C22.D4, D26, C22×D13, D525C2, D4×D13, D52⋊C2, D26.13D4

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 26 26 52 2 2 4 4 26 26 52 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 C4○D4 D13 D26 D52⋊5C2 D4×D13 D52⋊C2 kernel D26.13D4 C26.D4 D26⋊C4 C13×C4⋊C4 C2×C4×D13 C2×D52 D26 C26 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 1 3 1 1 1 2 4 6 18 24 6 6

Matrix representation of D26.13D4 in GL4(𝔽53) generated by

 33 47 0 0 28 11 0 0 0 0 52 0 0 0 0 52
,
 33 52 0 0 28 20 0 0 0 0 1 43 0 0 0 52
,
 20 18 0 0 22 33 0 0 0 0 18 1 0 0 46 35
,
 36 43 0 0 29 17 0 0 0 0 10 29 0 0 2 43
`G:=sub<GL(4,GF(53))| [33,28,0,0,47,11,0,0,0,0,52,0,0,0,0,52],[33,28,0,0,52,20,0,0,0,0,1,0,0,0,43,52],[20,22,0,0,18,33,0,0,0,0,18,46,0,0,1,35],[36,29,0,0,43,17,0,0,0,0,10,2,0,0,29,43] >;`

D26.13D4 in GAP, Magma, Sage, TeX

`D_{26}._{13}D_4`
`% in TeX`

`G:=Group("D26.13D4");`
`// GroupNames label`

`G:=SmallGroup(416,115);`
`// by ID`

`G=gap.SmallGroup(416,115);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,55,218,188,86,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^13*b,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
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