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G = D26.6D4order 416 = 25·13

2nd non-split extension by D26 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D26.6D4
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D4⋊2D13 — D26.6D4
 Lower central C13 — C26 — C52 — D26.6D4
 Upper central C1 — C2 — C4 — SD16

Generators and relations for D26.6D4
G = < a,b,c,d | a26=b2=d2=1, c4=a13, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a13b, dcd=c3 >

Subgroups: 464 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, Q8, C13, C2×C8, D8, SD16, SD16, Q16, C4○D4, D13, C26, C26, C4○D8, Dic13, Dic13, C52, C52, D26, D26, C2×C26, C132C8, C104, Dic26, C4×D13, C4×D13, D52, D52, C2×Dic13, C13⋊D4, D4×C13, Q8×C13, C8×D13, C104⋊C2, D4⋊D13, C13⋊Q16, C13×SD16, D42D13, D52⋊C2, D26.6D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C4○D8, D26, C22×D13, D4×D13, D26.6D4

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D13 C4○D8 D26 D26 D26 D4×D13 D26.6D4 kernel D26.6D4 C8×D13 C104⋊C2 D4⋊D13 C13⋊Q16 C13×SD16 D4⋊2D13 D52⋊C2 Dic13 D26 SD16 C13 C8 D4 Q8 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 6 4 6 6 6 6 12

Matrix representation of D26.6D4 in GL4(𝔽313) generated by

 300 300 0 0 13 37 0 0 0 0 312 0 0 0 0 312
,
 300 300 0 0 37 13 0 0 0 0 312 0 0 0 280 1
,
 312 0 0 0 0 312 0 0 0 0 125 0 0 0 102 5
,
 312 0 0 0 0 312 0 0 0 0 201 187 0 0 107 112
`G:=sub<GL(4,GF(313))| [300,13,0,0,300,37,0,0,0,0,312,0,0,0,0,312],[300,37,0,0,300,13,0,0,0,0,312,280,0,0,0,1],[312,0,0,0,0,312,0,0,0,0,125,102,0,0,0,5],[312,0,0,0,0,312,0,0,0,0,201,107,0,0,187,112] >;`

D26.6D4 in GAP, Magma, Sage, TeX

`D_{26}._6D_4`
`% in TeX`

`G:=Group("D26.6D4");`
`// GroupNames label`

`G:=SmallGroup(416,137);`
`// by ID`

`G=gap.SmallGroup(416,137);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,362,116,86,297,159,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^26=b^2=d^2=1,c^4=a^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^13*b,d*c*d=c^3>;`
`// generators/relations`

׿
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