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## G = C13×D4⋊C4order 416 = 25·13

### Direct product of C13 and D4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×D4⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C52 — C13×C4⋊C4 — C13×D4⋊C4
 Lower central C1 — C2 — C4 — C13×D4⋊C4
 Upper central C1 — C2×C26 — C2×C52 — C13×D4⋊C4

Generators and relations for C13×D4⋊C4
G = < a,b,c,d | a13=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Smallest permutation representation of C13×D4⋊C4
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)
(1 101 84 48)(2 102 85 49)(3 103 86 50)(4 104 87 51)(5 92 88 52)(6 93 89 40)(7 94 90 41)(8 95 91 42)(9 96 79 43)(10 97 80 44)(11 98 81 45)(12 99 82 46)(13 100 83 47)(14 27 197 179)(15 28 198 180)(16 29 199 181)(17 30 200 182)(18 31 201 170)(19 32 202 171)(20 33 203 172)(21 34 204 173)(22 35 205 174)(23 36 206 175)(24 37 207 176)(25 38 208 177)(26 39 196 178)(53 157 75 183)(54 158 76 184)(55 159 77 185)(56 160 78 186)(57 161 66 187)(58 162 67 188)(59 163 68 189)(60 164 69 190)(61 165 70 191)(62 166 71 192)(63 167 72 193)(64 168 73 194)(65 169 74 195)(105 134 150 130)(106 135 151 118)(107 136 152 119)(108 137 153 120)(109 138 154 121)(110 139 155 122)(111 140 156 123)(112 141 144 124)(113 142 145 125)(114 143 146 126)(115 131 147 127)(116 132 148 128)(117 133 149 129)
(1 78)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 116)(15 117)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 128)(28 129)(29 130)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 191)(41 192)(42 193)(43 194)(44 195)(45 183)(46 184)(47 185)(48 186)(49 187)(50 188)(51 189)(52 190)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 79)(65 80)(92 164)(93 165)(94 166)(95 167)(96 168)(97 169)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(131 178)(132 179)(133 180)(134 181)(135 182)(136 170)(137 171)(138 172)(139 173)(140 174)(141 175)(142 176)(143 177)(144 206)(145 207)(146 208)(147 196)(148 197)(149 198)(150 199)(151 200)(152 201)(153 202)(154 203)(155 204)(156 205)
(1 121 186 172)(2 122 187 173)(3 123 188 174)(4 124 189 175)(5 125 190 176)(6 126 191 177)(7 127 192 178)(8 128 193 179)(9 129 194 180)(10 130 195 181)(11 118 183 182)(12 119 184 170)(13 120 185 171)(14 42 116 72)(15 43 117 73)(16 44 105 74)(17 45 106 75)(18 46 107 76)(19 47 108 77)(20 48 109 78)(21 49 110 66)(22 50 111 67)(23 51 112 68)(24 52 113 69)(25 40 114 70)(26 41 115 71)(27 91 132 167)(28 79 133 168)(29 80 134 169)(30 81 135 157)(31 82 136 158)(32 83 137 159)(33 84 138 160)(34 85 139 161)(35 86 140 162)(36 87 141 163)(37 88 142 164)(38 89 143 165)(39 90 131 166)(53 200 98 151)(54 201 99 152)(55 202 100 153)(56 203 101 154)(57 204 102 155)(58 205 103 156)(59 206 104 144)(60 207 92 145)(61 208 93 146)(62 196 94 147)(63 197 95 148)(64 198 96 149)(65 199 97 150)

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

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

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

182 conjugacy classes

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

182 irreducible representations

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

Matrix representation of C13×D4⋊C4 in GL3(𝔽313) generated by

 1 0 0 0 44 0 0 0 44
,
 1 0 0 0 1 311 0 1 312
,
 1 0 0 0 1 311 0 0 312
,
 288 0 0 0 0 193 0 253 0
G:=sub<GL(3,GF(313))| [1,0,0,0,44,0,0,0,44],[1,0,0,0,1,1,0,311,312],[1,0,0,0,1,0,0,311,312],[288,0,0,0,0,253,0,193,0] >;

C13×D4⋊C4 in GAP, Magma, Sage, TeX

C_{13}\times D_4\rtimes C_4
% in TeX

G:=Group("C13xD4:C4");
// GroupNames label

G:=SmallGroup(416,52);
// by ID

G=gap.SmallGroup(416,52);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,3129,117]);
// Polycyclic

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

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