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## G = C13×C4○D8order 416 = 25·13

### Direct product of C13 and C4○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×C4○D8
 Chief series C1 — C2 — C4 — C52 — D4×C13 — C13×D8 — C13×C4○D8
 Lower central C1 — C2 — C4 — C13×C4○D8
 Upper central C1 — C52 — C2×C52 — C13×C4○D8

Generators and relations for C13×C4○D8
G = < a,b,c,d | a13=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C13, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C26, C26 [×3], C4○D8, C52 [×2], C52 [×2], C2×C26, C2×C26 [×2], C104 [×2], C2×C52, C2×C52 [×2], D4×C13 [×2], D4×C13 [×2], Q8×C13 [×2], C2×C104, C13×D8, C13×SD16 [×2], C13×Q16, C13×C4○D4 [×2], C13×C4○D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, C2×D4, C26 [×7], C4○D8, C2×C26 [×7], D4×C13 [×2], C22×C26, D4×C26, C13×C4○D8

Smallest permutation representation of C13×C4○D8
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 15 163 67)(2 16 164 68)(3 17 165 69)(4 18 166 70)(5 19 167 71)(6 20 168 72)(7 21 169 73)(8 22 157 74)(9 23 158 75)(10 24 159 76)(11 25 160 77)(12 26 161 78)(13 14 162 66)(27 124 94 89)(28 125 95 90)(29 126 96 91)(30 127 97 79)(31 128 98 80)(32 129 99 81)(33 130 100 82)(34 118 101 83)(35 119 102 84)(36 120 103 85)(37 121 104 86)(38 122 92 87)(39 123 93 88)(40 133 186 57)(41 134 187 58)(42 135 188 59)(43 136 189 60)(44 137 190 61)(45 138 191 62)(46 139 192 63)(47 140 193 64)(48 141 194 65)(49 142 195 53)(50 143 183 54)(51 131 184 55)(52 132 185 56)(105 149 196 173)(106 150 197 174)(107 151 198 175)(108 152 199 176)(109 153 200 177)(110 154 201 178)(111 155 202 179)(112 156 203 180)(113 144 204 181)(114 145 205 182)(115 146 206 170)(116 147 207 171)(117 148 208 172)
(1 92 15 87 163 38 67 122)(2 93 16 88 164 39 68 123)(3 94 17 89 165 27 69 124)(4 95 18 90 166 28 70 125)(5 96 19 91 167 29 71 126)(6 97 20 79 168 30 72 127)(7 98 21 80 169 31 73 128)(8 99 22 81 157 32 74 129)(9 100 23 82 158 33 75 130)(10 101 24 83 159 34 76 118)(11 102 25 84 160 35 77 119)(12 103 26 85 161 36 78 120)(13 104 14 86 162 37 66 121)(40 156 57 112 186 180 133 203)(41 144 58 113 187 181 134 204)(42 145 59 114 188 182 135 205)(43 146 60 115 189 170 136 206)(44 147 61 116 190 171 137 207)(45 148 62 117 191 172 138 208)(46 149 63 105 192 173 139 196)(47 150 64 106 193 174 140 197)(48 151 65 107 194 175 141 198)(49 152 53 108 195 176 142 199)(50 153 54 109 183 177 143 200)(51 154 55 110 184 178 131 201)(52 155 56 111 185 179 132 202)
(1 143)(2 131)(3 132)(4 133)(5 134)(6 135)(7 136)(8 137)(9 138)(10 139)(11 140)(12 141)(13 142)(14 195)(15 183)(16 184)(17 185)(18 186)(19 187)(20 188)(21 189)(22 190)(23 191)(24 192)(25 193)(26 194)(27 155)(28 156)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 66)(50 67)(51 68)(52 69)(53 162)(54 163)(55 164)(56 165)(57 166)(58 167)(59 168)(60 169)(61 157)(62 158)(63 159)(64 160)(65 161)(79 114)(80 115)(81 116)(82 117)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 113)(92 177)(93 178)(94 179)(95 180)(96 181)(97 182)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(118 196)(119 197)(120 198)(121 199)(122 200)(123 201)(124 202)(125 203)(126 204)(127 205)(128 206)(129 207)(130 208)

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

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

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

182 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 13A ··· 13L 26A ··· 26L 26M ··· 26X 26Y ··· 26AV 52A ··· 52X 52Y ··· 52AJ 52AK ··· 52BH 104A ··· 104AV order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 4 4 1 1 2 4 4 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

182 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C13 C26 C26 C26 C26 C26 D4 D4 C4○D8 D4×C13 D4×C13 C13×C4○D8 kernel C13×C4○D8 C2×C104 C13×D8 C13×SD16 C13×Q16 C13×C4○D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C52 C2×C26 C13 C4 C22 C1 # reps 1 1 1 2 1 2 12 12 12 24 12 24 1 1 4 12 12 48

Matrix representation of C13×C4○D8 in GL3(𝔽313) generated by

 48 0 0 0 1 0 0 0 1
,
 312 0 0 0 288 0 0 0 288
,
 1 0 0 0 188 0 0 0 5
,
 312 0 0 0 0 5 0 188 0
G:=sub<GL(3,GF(313))| [48,0,0,0,1,0,0,0,1],[312,0,0,0,288,0,0,0,288],[1,0,0,0,188,0,0,0,5],[312,0,0,0,0,188,0,5,0] >;

C13×C4○D8 in GAP, Magma, Sage, TeX

C_{13}\times C_4\circ D_8
% in TeX

G:=Group("C13xC4oD8");
// GroupNames label

G:=SmallGroup(416,196);
// by ID

G=gap.SmallGroup(416,196);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,950,9364,4690,88]);
// Polycyclic

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

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