direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×Dic13, C65⋊8C4, C13⋊2C20, C26.C10, C130.2C2, C10.2D13, C2.(C5×D13), SmallGroup(260,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C5×Dic13 |
Generators and relations for C5×Dic13
G = < a,b,c | a5=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic13
►Regular action on 260 points
Generators in S260
(1 120 99 73 51)(2 121 100 74 52)(3 122 101 75 27)(4 123 102 76 28)(5 124 103 77 29)(6 125 104 78 30)(7 126 79 53 31)(8 127 80 54 32)(9 128 81 55 33)(10 129 82 56 34)(11 130 83 57 35)(12 105 84 58 36)(13 106 85 59 37)(14 107 86 60 38)(15 108 87 61 39)(16 109 88 62 40)(17 110 89 63 41)(18 111 90 64 42)(19 112 91 65 43)(20 113 92 66 44)(21 114 93 67 45)(22 115 94 68 46)(23 116 95 69 47)(24 117 96 70 48)(25 118 97 71 49)(26 119 98 72 50)(131 248 209 183 157)(132 249 210 184 158)(133 250 211 185 159)(134 251 212 186 160)(135 252 213 187 161)(136 253 214 188 162)(137 254 215 189 163)(138 255 216 190 164)(139 256 217 191 165)(140 257 218 192 166)(141 258 219 193 167)(142 259 220 194 168)(143 260 221 195 169)(144 235 222 196 170)(145 236 223 197 171)(146 237 224 198 172)(147 238 225 199 173)(148 239 226 200 174)(149 240 227 201 175)(150 241 228 202 176)(151 242 229 203 177)(152 243 230 204 178)(153 244 231 205 179)(154 245 232 206 180)(155 246 233 207 181)(156 247 234 208 182)
(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)
(1 144 14 131)(2 143 15 156)(3 142 16 155)(4 141 17 154)(5 140 18 153)(6 139 19 152)(7 138 20 151)(8 137 21 150)(9 136 22 149)(10 135 23 148)(11 134 24 147)(12 133 25 146)(13 132 26 145)(27 168 40 181)(28 167 41 180)(29 166 42 179)(30 165 43 178)(31 164 44 177)(32 163 45 176)(33 162 46 175)(34 161 47 174)(35 160 48 173)(36 159 49 172)(37 158 50 171)(38 157 51 170)(39 182 52 169)(53 190 66 203)(54 189 67 202)(55 188 68 201)(56 187 69 200)(57 186 70 199)(58 185 71 198)(59 184 72 197)(60 183 73 196)(61 208 74 195)(62 207 75 194)(63 206 76 193)(64 205 77 192)(65 204 78 191)(79 216 92 229)(80 215 93 228)(81 214 94 227)(82 213 95 226)(83 212 96 225)(84 211 97 224)(85 210 98 223)(86 209 99 222)(87 234 100 221)(88 233 101 220)(89 232 102 219)(90 231 103 218)(91 230 104 217)(105 250 118 237)(106 249 119 236)(107 248 120 235)(108 247 121 260)(109 246 122 259)(110 245 123 258)(111 244 124 257)(112 243 125 256)(113 242 126 255)(114 241 127 254)(115 240 128 253)(116 239 129 252)(117 238 130 251)
G:=sub<Sym(260)| (1,120,99,73,51)(2,121,100,74,52)(3,122,101,75,27)(4,123,102,76,28)(5,124,103,77,29)(6,125,104,78,30)(7,126,79,53,31)(8,127,80,54,32)(9,128,81,55,33)(10,129,82,56,34)(11,130,83,57,35)(12,105,84,58,36)(13,106,85,59,37)(14,107,86,60,38)(15,108,87,61,39)(16,109,88,62,40)(17,110,89,63,41)(18,111,90,64,42)(19,112,91,65,43)(20,113,92,66,44)(21,114,93,67,45)(22,115,94,68,46)(23,116,95,69,47)(24,117,96,70,48)(25,118,97,71,49)(26,119,98,72,50)(131,248,209,183,157)(132,249,210,184,158)(133,250,211,185,159)(134,251,212,186,160)(135,252,213,187,161)(136,253,214,188,162)(137,254,215,189,163)(138,255,216,190,164)(139,256,217,191,165)(140,257,218,192,166)(141,258,219,193,167)(142,259,220,194,168)(143,260,221,195,169)(144,235,222,196,170)(145,236,223,197,171)(146,237,224,198,172)(147,238,225,199,173)(148,239,226,200,174)(149,240,227,201,175)(150,241,228,202,176)(151,242,229,203,177)(152,243,230,204,178)(153,244,231,205,179)(154,245,232,206,180)(155,246,233,207,181)(156,247,234,208,182), (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), (1,144,14,131)(2,143,15,156)(3,142,16,155)(4,141,17,154)(5,140,18,153)(6,139,19,152)(7,138,20,151)(8,137,21,150)(9,136,22,149)(10,135,23,148)(11,134,24,147)(12,133,25,146)(13,132,26,145)(27,168,40,181)(28,167,41,180)(29,166,42,179)(30,165,43,178)(31,164,44,177)(32,163,45,176)(33,162,46,175)(34,161,47,174)(35,160,48,173)(36,159,49,172)(37,158,50,171)(38,157,51,170)(39,182,52,169)(53,190,66,203)(54,189,67,202)(55,188,68,201)(56,187,69,200)(57,186,70,199)(58,185,71,198)(59,184,72,197)(60,183,73,196)(61,208,74,195)(62,207,75,194)(63,206,76,193)(64,205,77,192)(65,204,78,191)(79,216,92,229)(80,215,93,228)(81,214,94,227)(82,213,95,226)(83,212,96,225)(84,211,97,224)(85,210,98,223)(86,209,99,222)(87,234,100,221)(88,233,101,220)(89,232,102,219)(90,231,103,218)(91,230,104,217)(105,250,118,237)(106,249,119,236)(107,248,120,235)(108,247,121,260)(109,246,122,259)(110,245,123,258)(111,244,124,257)(112,243,125,256)(113,242,126,255)(114,241,127,254)(115,240,128,253)(116,239,129,252)(117,238,130,251)>;
G:=Group( (1,120,99,73,51)(2,121,100,74,52)(3,122,101,75,27)(4,123,102,76,28)(5,124,103,77,29)(6,125,104,78,30)(7,126,79,53,31)(8,127,80,54,32)(9,128,81,55,33)(10,129,82,56,34)(11,130,83,57,35)(12,105,84,58,36)(13,106,85,59,37)(14,107,86,60,38)(15,108,87,61,39)(16,109,88,62,40)(17,110,89,63,41)(18,111,90,64,42)(19,112,91,65,43)(20,113,92,66,44)(21,114,93,67,45)(22,115,94,68,46)(23,116,95,69,47)(24,117,96,70,48)(25,118,97,71,49)(26,119,98,72,50)(131,248,209,183,157)(132,249,210,184,158)(133,250,211,185,159)(134,251,212,186,160)(135,252,213,187,161)(136,253,214,188,162)(137,254,215,189,163)(138,255,216,190,164)(139,256,217,191,165)(140,257,218,192,166)(141,258,219,193,167)(142,259,220,194,168)(143,260,221,195,169)(144,235,222,196,170)(145,236,223,197,171)(146,237,224,198,172)(147,238,225,199,173)(148,239,226,200,174)(149,240,227,201,175)(150,241,228,202,176)(151,242,229,203,177)(152,243,230,204,178)(153,244,231,205,179)(154,245,232,206,180)(155,246,233,207,181)(156,247,234,208,182), (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), (1,144,14,131)(2,143,15,156)(3,142,16,155)(4,141,17,154)(5,140,18,153)(6,139,19,152)(7,138,20,151)(8,137,21,150)(9,136,22,149)(10,135,23,148)(11,134,24,147)(12,133,25,146)(13,132,26,145)(27,168,40,181)(28,167,41,180)(29,166,42,179)(30,165,43,178)(31,164,44,177)(32,163,45,176)(33,162,46,175)(34,161,47,174)(35,160,48,173)(36,159,49,172)(37,158,50,171)(38,157,51,170)(39,182,52,169)(53,190,66,203)(54,189,67,202)(55,188,68,201)(56,187,69,200)(57,186,70,199)(58,185,71,198)(59,184,72,197)(60,183,73,196)(61,208,74,195)(62,207,75,194)(63,206,76,193)(64,205,77,192)(65,204,78,191)(79,216,92,229)(80,215,93,228)(81,214,94,227)(82,213,95,226)(83,212,96,225)(84,211,97,224)(85,210,98,223)(86,209,99,222)(87,234,100,221)(88,233,101,220)(89,232,102,219)(90,231,103,218)(91,230,104,217)(105,250,118,237)(106,249,119,236)(107,248,120,235)(108,247,121,260)(109,246,122,259)(110,245,123,258)(111,244,124,257)(112,243,125,256)(113,242,126,255)(114,241,127,254)(115,240,128,253)(116,239,129,252)(117,238,130,251) );
G=PermutationGroup([[(1,120,99,73,51),(2,121,100,74,52),(3,122,101,75,27),(4,123,102,76,28),(5,124,103,77,29),(6,125,104,78,30),(7,126,79,53,31),(8,127,80,54,32),(9,128,81,55,33),(10,129,82,56,34),(11,130,83,57,35),(12,105,84,58,36),(13,106,85,59,37),(14,107,86,60,38),(15,108,87,61,39),(16,109,88,62,40),(17,110,89,63,41),(18,111,90,64,42),(19,112,91,65,43),(20,113,92,66,44),(21,114,93,67,45),(22,115,94,68,46),(23,116,95,69,47),(24,117,96,70,48),(25,118,97,71,49),(26,119,98,72,50),(131,248,209,183,157),(132,249,210,184,158),(133,250,211,185,159),(134,251,212,186,160),(135,252,213,187,161),(136,253,214,188,162),(137,254,215,189,163),(138,255,216,190,164),(139,256,217,191,165),(140,257,218,192,166),(141,258,219,193,167),(142,259,220,194,168),(143,260,221,195,169),(144,235,222,196,170),(145,236,223,197,171),(146,237,224,198,172),(147,238,225,199,173),(148,239,226,200,174),(149,240,227,201,175),(150,241,228,202,176),(151,242,229,203,177),(152,243,230,204,178),(153,244,231,205,179),(154,245,232,206,180),(155,246,233,207,181),(156,247,234,208,182)], [(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)], [(1,144,14,131),(2,143,15,156),(3,142,16,155),(4,141,17,154),(5,140,18,153),(6,139,19,152),(7,138,20,151),(8,137,21,150),(9,136,22,149),(10,135,23,148),(11,134,24,147),(12,133,25,146),(13,132,26,145),(27,168,40,181),(28,167,41,180),(29,166,42,179),(30,165,43,178),(31,164,44,177),(32,163,45,176),(33,162,46,175),(34,161,47,174),(35,160,48,173),(36,159,49,172),(37,158,50,171),(38,157,51,170),(39,182,52,169),(53,190,66,203),(54,189,67,202),(55,188,68,201),(56,187,69,200),(57,186,70,199),(58,185,71,198),(59,184,72,197),(60,183,73,196),(61,208,74,195),(62,207,75,194),(63,206,76,193),(64,205,77,192),(65,204,78,191),(79,216,92,229),(80,215,93,228),(81,214,94,227),(82,213,95,226),(83,212,96,225),(84,211,97,224),(85,210,98,223),(86,209,99,222),(87,234,100,221),(88,233,101,220),(89,232,102,219),(90,231,103,218),(91,230,104,217),(105,250,118,237),(106,249,119,236),(107,248,120,235),(108,247,121,260),(109,246,122,259),(110,245,123,258),(111,244,124,257),(112,243,125,256),(113,242,126,255),(114,241,127,254),(115,240,128,253),(116,239,129,252),(117,238,130,251)]])
80 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 13A | ··· | 13F | 20A | ··· | 20H | 26A | ··· | 26F | 65A | ··· | 65X | 130A | ··· | 130X |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 13 | ··· | 13 | 20 | ··· | 20 | 26 | ··· | 26 | 65 | ··· | 65 | 130 | ··· | 130 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D13 | Dic13 | C5×D13 | C5×Dic13 |
| kernel | C5×Dic13 | C130 | C65 | Dic13 | C26 | C13 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 6 | 6 | 24 | 24 |
Matrix representation of C5×Dic13 ►in GL2(𝔽521) generated by
| 396 | 0 |
| 0 | 396 |
| 0 | 520 |
| 1 | 474 |
| 39 | 289 |
| 38 | 482 |
G:=sub<GL(2,GF(521))| [396,0,0,396],[0,1,520,474],[39,38,289,482] >;
C5×Dic13 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{13} % in TeX
G:=Group("C5xDic13"); // GroupNames label
G:=SmallGroup(260,2);
// by ID
G=gap.SmallGroup(260,2);
# by ID
G:=PCGroup([4,-2,-5,-2,-13,40,3843]);
// Polycyclic
G:=Group<a,b,c|a^5=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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