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## G = C2×C8×D13order 416 = 25·13

### Direct product of C2×C8 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C2×C8×D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — C2×C4×D13 — C2×C8×D13
 Lower central C13 — C2×C8×D13
 Upper central C1 — C2×C8

Generators and relations for C2×C8×D13
G = < a,b,c,d | a2=b8=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 400 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C13, C2×C8, C2×C8 [×5], C22×C4, D13 [×4], C26, C26 [×2], C22×C8, Dic13 [×2], C52 [×2], D26 [×6], C2×C26, C132C8 [×2], C104 [×2], C4×D13 [×4], C2×Dic13, C2×C52, C22×D13, C8×D13 [×4], C2×C132C8, C2×C104, C2×C4×D13, C2×C8×D13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, D13, C22×C8, D26 [×3], C4×D13 [×2], C22×D13, C8×D13 [×2], C2×C4×D13, C2×C8×D13

Smallest permutation representation of C2×C8×D13
On 208 points
Generators in S208
(1 117)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 123)(15 124)(16 125)(17 126)(18 127)(19 128)(20 129)(21 130)(22 118)(23 119)(24 120)(25 121)(26 122)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 141)(36 142)(37 143)(38 131)(39 132)(40 147)(41 148)(42 149)(43 150)(44 151)(45 152)(46 153)(47 154)(48 155)(49 156)(50 144)(51 145)(52 146)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 157)(63 158)(64 159)(65 160)(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)(79 188)(80 189)(81 190)(82 191)(83 192)(84 193)(85 194)(86 195)(87 183)(88 184)(89 185)(90 186)(91 187)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)
(1 97 45 66 15 83 31 63)(2 98 46 67 16 84 32 64)(3 99 47 68 17 85 33 65)(4 100 48 69 18 86 34 53)(5 101 49 70 19 87 35 54)(6 102 50 71 20 88 36 55)(7 103 51 72 21 89 37 56)(8 104 52 73 22 90 38 57)(9 92 40 74 23 91 39 58)(10 93 41 75 24 79 27 59)(11 94 42 76 25 80 28 60)(12 95 43 77 26 81 29 61)(13 96 44 78 14 82 30 62)(105 202 153 180 125 193 138 159)(106 203 154 181 126 194 139 160)(107 204 155 182 127 195 140 161)(108 205 156 170 128 183 141 162)(109 206 144 171 129 184 142 163)(110 207 145 172 130 185 143 164)(111 208 146 173 118 186 131 165)(112 196 147 174 119 187 132 166)(113 197 148 175 120 188 133 167)(114 198 149 176 121 189 134 168)(115 199 150 177 122 190 135 169)(116 200 151 178 123 191 136 157)(117 201 152 179 124 192 137 158)
(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)
(1 123)(2 122)(3 121)(4 120)(5 119)(6 118)(7 130)(8 129)(9 128)(10 127)(11 126)(12 125)(13 124)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 155)(28 154)(29 153)(30 152)(31 151)(32 150)(33 149)(34 148)(35 147)(36 146)(37 145)(38 144)(39 156)(40 141)(41 140)(42 139)(43 138)(44 137)(45 136)(46 135)(47 134)(48 133)(49 132)(50 131)(51 143)(52 142)(53 175)(54 174)(55 173)(56 172)(57 171)(58 170)(59 182)(60 181)(61 180)(62 179)(63 178)(64 177)(65 176)(66 157)(67 169)(68 168)(69 167)(70 166)(71 165)(72 164)(73 163)(74 162)(75 161)(76 160)(77 159)(78 158)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 208)(89 207)(90 206)(91 205)(92 183)(93 195)(94 194)(95 193)(96 192)(97 191)(98 190)(99 189)(100 188)(101 187)(102 186)(103 185)(104 184)

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

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

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

128 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 8I ··· 8P 13A ··· 13F 26A ··· 26R 52A ··· 52X 104A ··· 104AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 8 ··· 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 1 1 13 13 13 13 1 1 1 1 13 13 13 13 1 ··· 1 13 ··· 13 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D13 D26 D26 C4×D13 C4×D13 C8×D13 kernel C2×C8×D13 C8×D13 C2×C13⋊2C8 C2×C104 C2×C4×D13 C4×D13 C2×Dic13 C22×D13 D26 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 16 6 12 6 12 12 48

Matrix representation of C2×C8×D13 in GL4(𝔽313) generated by

 1 0 0 0 0 312 0 0 0 0 1 0 0 0 0 1
,
 125 0 0 0 0 1 0 0 0 0 288 0 0 0 0 288
,
 1 0 0 0 0 1 0 0 0 0 66 43 0 0 312 227
,
 1 0 0 0 0 1 0 0 0 0 69 238 0 0 293 244
G:=sub<GL(4,GF(313))| [1,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[125,0,0,0,0,1,0,0,0,0,288,0,0,0,0,288],[1,0,0,0,0,1,0,0,0,0,66,312,0,0,43,227],[1,0,0,0,0,1,0,0,0,0,69,293,0,0,238,244] >;

C2×C8×D13 in GAP, Magma, Sage, TeX

C_2\times C_8\times D_{13}
% in TeX

G:=Group("C2xC8xD13");
// GroupNames label

G:=SmallGroup(416,120);
// by ID

G=gap.SmallGroup(416,120);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,50,69,13829]);
// Polycyclic

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

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