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## G = D8⋊3D13order 416 = 25·13

### The semidirect product of D8 and D13 acting through Inn(D8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D8⋊3D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D4⋊2D13 — D8⋊3D13
 Lower central C13 — C26 — C52 — D8⋊3D13
 Upper central C1 — C2 — C4 — D8

Generators and relations for D83D13
G = < a,b,c,d | a8=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 416 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C13, C2×C8, D8, SD16, Q16, C4○D4, D13, C26, C26, C4○D8, Dic13, Dic13, C52, D26, C2×C26, C132C8, C104, Dic26, C4×D13, C2×Dic13, C13⋊D4, D4×C13, C8×D13, Dic52, D4.D13, C13×D8, D42D13, D83D13
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C4○D8, D26, C22×D13, D4×D13, D83D13

Smallest permutation representation of D83D13
On 208 points
Generators in S208
```(1 187 37 181 21 208 48 164)(2 188 38 182 22 196 49 165)(3 189 39 170 23 197 50 166)(4 190 27 171 24 198 51 167)(5 191 28 172 25 199 52 168)(6 192 29 173 26 200 40 169)(7 193 30 174 14 201 41 157)(8 194 31 175 15 202 42 158)(9 195 32 176 16 203 43 159)(10 183 33 177 17 204 44 160)(11 184 34 178 18 205 45 161)(12 185 35 179 19 206 46 162)(13 186 36 180 20 207 47 163)(53 154 102 128 76 143 79 109)(54 155 103 129 77 131 80 110)(55 156 104 130 78 132 81 111)(56 144 92 118 66 133 82 112)(57 145 93 119 67 134 83 113)(58 146 94 120 68 135 84 114)(59 147 95 121 69 136 85 115)(60 148 96 122 70 137 86 116)(61 149 97 123 71 138 87 117)(62 150 98 124 72 139 88 105)(63 151 99 125 73 140 89 106)(64 152 100 126 74 141 90 107)(65 153 101 127 75 142 91 108)
(1 114)(2 115)(3 116)(4 117)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 113)(14 126)(15 127)(16 128)(17 129)(18 130)(19 118)(20 119)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 151)(41 152)(42 153)(43 154)(44 155)(45 156)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 159)(54 160)(55 161)(56 162)(57 163)(58 164)(59 165)(60 166)(61 167)(62 168)(63 169)(64 157)(65 158)(66 179)(67 180)(68 181)(69 182)(70 170)(71 171)(72 172)(73 173)(74 174)(75 175)(76 176)(77 177)(78 178)(79 195)(80 183)(81 184)(82 185)(83 186)(84 187)(85 188)(86 189)(87 190)(88 191)(89 192)(90 193)(91 194)(92 206)(93 207)(94 208)(95 196)(96 197)(97 198)(98 199)(99 200)(100 201)(101 202)(102 203)(103 204)(104 205)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 33)(28 32)(29 31)(34 39)(35 38)(36 37)(40 42)(43 52)(44 51)(45 50)(46 49)(47 48)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 130)(117 129)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 156)(138 155)(139 154)(140 153)(141 152)(142 151)(143 150)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)(170 178)(171 177)(172 176)(173 175)(179 182)(180 181)(183 190)(184 189)(185 188)(186 187)(191 195)(192 194)(196 206)(197 205)(198 204)(199 203)(200 202)(207 208)```

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

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

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

56 conjugacy classes

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

56 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 D4 D13 C4○D8 D26 D26 D4×D13 D8⋊3D13 kernel D8⋊3D13 C8×D13 Dic52 D4.D13 C13×D8 D4⋊2D13 Dic13 D26 D8 C13 C8 D4 C2 C1 # reps 1 1 1 2 1 2 1 1 6 4 6 12 6 12

Matrix representation of D83D13 in GL4(𝔽313) generated by

 5 0 0 0 25 188 0 0 0 0 312 0 0 0 0 312
,
 49 246 0 0 204 264 0 0 0 0 312 0 0 0 0 312
,
 1 0 0 0 0 1 0 0 0 0 288 1 0 0 48 286
,
 1 0 0 0 193 312 0 0 0 0 162 261 0 0 294 151
`G:=sub<GL(4,GF(313))| [5,25,0,0,0,188,0,0,0,0,312,0,0,0,0,312],[49,204,0,0,246,264,0,0,0,0,312,0,0,0,0,312],[1,0,0,0,0,1,0,0,0,0,288,48,0,0,1,286],[1,193,0,0,0,312,0,0,0,0,162,294,0,0,261,151] >;`

D83D13 in GAP, Magma, Sage, TeX

`D_8\rtimes_3D_{13}`
`% in TeX`

`G:=Group("D8:3D13");`
`// GroupNames label`

`G:=SmallGroup(416,133);`
`// by ID`

`G=gap.SmallGroup(416,133);`
`# by ID`

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

`G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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