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## G = C13×Dic5order 260 = 22·5·13

### Direct product of C13 and Dic5

Aliases: C13×Dic5, C52C52, C659C4, C10.C26, C26.2D5, C130.3C2, C2.(D5×C13), SmallGroup(260,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C13×Dic5
 Chief series C1 — C5 — C10 — C130 — C13×Dic5
 Lower central C5 — C13×Dic5
 Upper central C1 — C26

Generators and relations for C13×Dic5
G = < a,b,c | a13=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C13×Dic5
Regular action on 260 points
Generators in S260
(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)(209 210 211 212 213 214 215 216 217 218 219 220 221)(222 223 224 225 226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243 244 245 246 247)(248 249 250 251 252 253 254 255 256 257 258 259 260)
(1 84 19 165 42 231 108 136 36 74)(2 85 20 166 43 232 109 137 37 75)(3 86 21 167 44 233 110 138 38 76)(4 87 22 168 45 234 111 139 39 77)(5 88 23 169 46 222 112 140 27 78)(6 89 24 157 47 223 113 141 28 66)(7 90 25 158 48 224 114 142 29 67)(8 91 26 159 49 225 115 143 30 68)(9 79 14 160 50 226 116 131 31 69)(10 80 15 161 51 227 117 132 32 70)(11 81 16 162 52 228 105 133 33 71)(12 82 17 163 40 229 106 134 34 72)(13 83 18 164 41 230 107 135 35 73)(53 152 201 95 237 174 220 187 118 258)(54 153 202 96 238 175 221 188 119 259)(55 154 203 97 239 176 209 189 120 260)(56 155 204 98 240 177 210 190 121 248)(57 156 205 99 241 178 211 191 122 249)(58 144 206 100 242 179 212 192 123 250)(59 145 207 101 243 180 213 193 124 251)(60 146 208 102 244 181 214 194 125 252)(61 147 196 103 245 182 215 195 126 253)(62 148 197 104 246 170 216 183 127 254)(63 149 198 92 247 171 217 184 128 255)(64 150 199 93 235 172 218 185 129 256)(65 151 200 94 236 173 219 186 130 257)
(1 245 231 253)(2 246 232 254)(3 247 233 255)(4 235 234 256)(5 236 222 257)(6 237 223 258)(7 238 224 259)(8 239 225 260)(9 240 226 248)(10 241 227 249)(11 242 228 250)(12 243 229 251)(13 244 230 252)(14 204 131 190)(15 205 132 191)(16 206 133 192)(17 207 134 193)(18 208 135 194)(19 196 136 195)(20 197 137 183)(21 198 138 184)(22 199 139 185)(23 200 140 186)(24 201 141 187)(25 202 142 188)(26 203 143 189)(27 219 169 151)(28 220 157 152)(29 221 158 153)(30 209 159 154)(31 210 160 155)(32 211 161 156)(33 212 162 144)(34 213 163 145)(35 214 164 146)(36 215 165 147)(37 216 166 148)(38 217 167 149)(39 218 168 150)(40 59 72 180)(41 60 73 181)(42 61 74 182)(43 62 75 170)(44 63 76 171)(45 64 77 172)(46 65 78 173)(47 53 66 174)(48 54 67 175)(49 55 68 176)(50 56 69 177)(51 57 70 178)(52 58 71 179)(79 98 116 121)(80 99 117 122)(81 100 105 123)(82 101 106 124)(83 102 107 125)(84 103 108 126)(85 104 109 127)(86 92 110 128)(87 93 111 129)(88 94 112 130)(89 95 113 118)(90 96 114 119)(91 97 115 120)

G:=sub<Sym(260)| (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)(209,210,211,212,213,214,215,216,217,218,219,220,221)(222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247)(248,249,250,251,252,253,254,255,256,257,258,259,260), (1,84,19,165,42,231,108,136,36,74)(2,85,20,166,43,232,109,137,37,75)(3,86,21,167,44,233,110,138,38,76)(4,87,22,168,45,234,111,139,39,77)(5,88,23,169,46,222,112,140,27,78)(6,89,24,157,47,223,113,141,28,66)(7,90,25,158,48,224,114,142,29,67)(8,91,26,159,49,225,115,143,30,68)(9,79,14,160,50,226,116,131,31,69)(10,80,15,161,51,227,117,132,32,70)(11,81,16,162,52,228,105,133,33,71)(12,82,17,163,40,229,106,134,34,72)(13,83,18,164,41,230,107,135,35,73)(53,152,201,95,237,174,220,187,118,258)(54,153,202,96,238,175,221,188,119,259)(55,154,203,97,239,176,209,189,120,260)(56,155,204,98,240,177,210,190,121,248)(57,156,205,99,241,178,211,191,122,249)(58,144,206,100,242,179,212,192,123,250)(59,145,207,101,243,180,213,193,124,251)(60,146,208,102,244,181,214,194,125,252)(61,147,196,103,245,182,215,195,126,253)(62,148,197,104,246,170,216,183,127,254)(63,149,198,92,247,171,217,184,128,255)(64,150,199,93,235,172,218,185,129,256)(65,151,200,94,236,173,219,186,130,257), (1,245,231,253)(2,246,232,254)(3,247,233,255)(4,235,234,256)(5,236,222,257)(6,237,223,258)(7,238,224,259)(8,239,225,260)(9,240,226,248)(10,241,227,249)(11,242,228,250)(12,243,229,251)(13,244,230,252)(14,204,131,190)(15,205,132,191)(16,206,133,192)(17,207,134,193)(18,208,135,194)(19,196,136,195)(20,197,137,183)(21,198,138,184)(22,199,139,185)(23,200,140,186)(24,201,141,187)(25,202,142,188)(26,203,143,189)(27,219,169,151)(28,220,157,152)(29,221,158,153)(30,209,159,154)(31,210,160,155)(32,211,161,156)(33,212,162,144)(34,213,163,145)(35,214,164,146)(36,215,165,147)(37,216,166,148)(38,217,167,149)(39,218,168,150)(40,59,72,180)(41,60,73,181)(42,61,74,182)(43,62,75,170)(44,63,76,171)(45,64,77,172)(46,65,78,173)(47,53,66,174)(48,54,67,175)(49,55,68,176)(50,56,69,177)(51,57,70,178)(52,58,71,179)(79,98,116,121)(80,99,117,122)(81,100,105,123)(82,101,106,124)(83,102,107,125)(84,103,108,126)(85,104,109,127)(86,92,110,128)(87,93,111,129)(88,94,112,130)(89,95,113,118)(90,96,114,119)(91,97,115,120)>;

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)(209,210,211,212,213,214,215,216,217,218,219,220,221)(222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247)(248,249,250,251,252,253,254,255,256,257,258,259,260), (1,84,19,165,42,231,108,136,36,74)(2,85,20,166,43,232,109,137,37,75)(3,86,21,167,44,233,110,138,38,76)(4,87,22,168,45,234,111,139,39,77)(5,88,23,169,46,222,112,140,27,78)(6,89,24,157,47,223,113,141,28,66)(7,90,25,158,48,224,114,142,29,67)(8,91,26,159,49,225,115,143,30,68)(9,79,14,160,50,226,116,131,31,69)(10,80,15,161,51,227,117,132,32,70)(11,81,16,162,52,228,105,133,33,71)(12,82,17,163,40,229,106,134,34,72)(13,83,18,164,41,230,107,135,35,73)(53,152,201,95,237,174,220,187,118,258)(54,153,202,96,238,175,221,188,119,259)(55,154,203,97,239,176,209,189,120,260)(56,155,204,98,240,177,210,190,121,248)(57,156,205,99,241,178,211,191,122,249)(58,144,206,100,242,179,212,192,123,250)(59,145,207,101,243,180,213,193,124,251)(60,146,208,102,244,181,214,194,125,252)(61,147,196,103,245,182,215,195,126,253)(62,148,197,104,246,170,216,183,127,254)(63,149,198,92,247,171,217,184,128,255)(64,150,199,93,235,172,218,185,129,256)(65,151,200,94,236,173,219,186,130,257), (1,245,231,253)(2,246,232,254)(3,247,233,255)(4,235,234,256)(5,236,222,257)(6,237,223,258)(7,238,224,259)(8,239,225,260)(9,240,226,248)(10,241,227,249)(11,242,228,250)(12,243,229,251)(13,244,230,252)(14,204,131,190)(15,205,132,191)(16,206,133,192)(17,207,134,193)(18,208,135,194)(19,196,136,195)(20,197,137,183)(21,198,138,184)(22,199,139,185)(23,200,140,186)(24,201,141,187)(25,202,142,188)(26,203,143,189)(27,219,169,151)(28,220,157,152)(29,221,158,153)(30,209,159,154)(31,210,160,155)(32,211,161,156)(33,212,162,144)(34,213,163,145)(35,214,164,146)(36,215,165,147)(37,216,166,148)(38,217,167,149)(39,218,168,150)(40,59,72,180)(41,60,73,181)(42,61,74,182)(43,62,75,170)(44,63,76,171)(45,64,77,172)(46,65,78,173)(47,53,66,174)(48,54,67,175)(49,55,68,176)(50,56,69,177)(51,57,70,178)(52,58,71,179)(79,98,116,121)(80,99,117,122)(81,100,105,123)(82,101,106,124)(83,102,107,125)(84,103,108,126)(85,104,109,127)(86,92,110,128)(87,93,111,129)(88,94,112,130)(89,95,113,118)(90,96,114,119)(91,97,115,120) );

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),(209,210,211,212,213,214,215,216,217,218,219,220,221),(222,223,224,225,226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243,244,245,246,247),(248,249,250,251,252,253,254,255,256,257,258,259,260)], [(1,84,19,165,42,231,108,136,36,74),(2,85,20,166,43,232,109,137,37,75),(3,86,21,167,44,233,110,138,38,76),(4,87,22,168,45,234,111,139,39,77),(5,88,23,169,46,222,112,140,27,78),(6,89,24,157,47,223,113,141,28,66),(7,90,25,158,48,224,114,142,29,67),(8,91,26,159,49,225,115,143,30,68),(9,79,14,160,50,226,116,131,31,69),(10,80,15,161,51,227,117,132,32,70),(11,81,16,162,52,228,105,133,33,71),(12,82,17,163,40,229,106,134,34,72),(13,83,18,164,41,230,107,135,35,73),(53,152,201,95,237,174,220,187,118,258),(54,153,202,96,238,175,221,188,119,259),(55,154,203,97,239,176,209,189,120,260),(56,155,204,98,240,177,210,190,121,248),(57,156,205,99,241,178,211,191,122,249),(58,144,206,100,242,179,212,192,123,250),(59,145,207,101,243,180,213,193,124,251),(60,146,208,102,244,181,214,194,125,252),(61,147,196,103,245,182,215,195,126,253),(62,148,197,104,246,170,216,183,127,254),(63,149,198,92,247,171,217,184,128,255),(64,150,199,93,235,172,218,185,129,256),(65,151,200,94,236,173,219,186,130,257)], [(1,245,231,253),(2,246,232,254),(3,247,233,255),(4,235,234,256),(5,236,222,257),(6,237,223,258),(7,238,224,259),(8,239,225,260),(9,240,226,248),(10,241,227,249),(11,242,228,250),(12,243,229,251),(13,244,230,252),(14,204,131,190),(15,205,132,191),(16,206,133,192),(17,207,134,193),(18,208,135,194),(19,196,136,195),(20,197,137,183),(21,198,138,184),(22,199,139,185),(23,200,140,186),(24,201,141,187),(25,202,142,188),(26,203,143,189),(27,219,169,151),(28,220,157,152),(29,221,158,153),(30,209,159,154),(31,210,160,155),(32,211,161,156),(33,212,162,144),(34,213,163,145),(35,214,164,146),(36,215,165,147),(37,216,166,148),(38,217,167,149),(39,218,168,150),(40,59,72,180),(41,60,73,181),(42,61,74,182),(43,62,75,170),(44,63,76,171),(45,64,77,172),(46,65,78,173),(47,53,66,174),(48,54,67,175),(49,55,68,176),(50,56,69,177),(51,57,70,178),(52,58,71,179),(79,98,116,121),(80,99,117,122),(81,100,105,123),(82,101,106,124),(83,102,107,125),(84,103,108,126),(85,104,109,127),(86,92,110,128),(87,93,111,129),(88,94,112,130),(89,95,113,118),(90,96,114,119),(91,97,115,120)]])

104 conjugacy classes

 class 1 2 4A 4B 5A 5B 10A 10B 13A ··· 13L 26A ··· 26L 52A ··· 52X 65A ··· 65X 130A ··· 130X order 1 2 4 4 5 5 10 10 13 ··· 13 26 ··· 26 52 ··· 52 65 ··· 65 130 ··· 130 size 1 1 5 5 2 2 2 2 1 ··· 1 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C13 C26 C52 D5 Dic5 D5×C13 C13×Dic5 kernel C13×Dic5 C130 C65 Dic5 C10 C5 C26 C13 C2 C1 # reps 1 1 2 12 12 24 2 2 24 24

Matrix representation of C13×Dic5 in GL3(𝔽521) generated by

 1 0 0 0 423 0 0 0 423
,
 520 0 0 0 520 1 0 98 422
,
 286 0 0 0 438 39 0 144 83
G:=sub<GL(3,GF(521))| [1,0,0,0,423,0,0,0,423],[520,0,0,0,520,98,0,1,422],[286,0,0,0,438,144,0,39,83] >;

C13×Dic5 in GAP, Magma, Sage, TeX

C_{13}\times {\rm Dic}_5
% in TeX

G:=Group("C13xDic5");
// GroupNames label

G:=SmallGroup(260,1);
// by ID

G=gap.SmallGroup(260,1);
# by ID

G:=PCGroup([4,-2,-13,-2,-5,104,3331]);
// Polycyclic

G:=Group<a,b,c|a^13=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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