direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D49, C49⋊C10, C245⋊2C2, C35.2D7, C7.(C5×D7), SmallGroup(490,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C5×D49 |
Generators and relations for C5×D49
G = < a,b,c | a5=b49=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D49
►On 245 points
Generators in S245
(1 210 191 128 65)(2 211 192 129 66)(3 212 193 130 67)(4 213 194 131 68)(5 214 195 132 69)(6 215 196 133 70)(7 216 148 134 71)(8 217 149 135 72)(9 218 150 136 73)(10 219 151 137 74)(11 220 152 138 75)(12 221 153 139 76)(13 222 154 140 77)(14 223 155 141 78)(15 224 156 142 79)(16 225 157 143 80)(17 226 158 144 81)(18 227 159 145 82)(19 228 160 146 83)(20 229 161 147 84)(21 230 162 99 85)(22 231 163 100 86)(23 232 164 101 87)(24 233 165 102 88)(25 234 166 103 89)(26 235 167 104 90)(27 236 168 105 91)(28 237 169 106 92)(29 238 170 107 93)(30 239 171 108 94)(31 240 172 109 95)(32 241 173 110 96)(33 242 174 111 97)(34 243 175 112 98)(35 244 176 113 50)(36 245 177 114 51)(37 197 178 115 52)(38 198 179 116 53)(39 199 180 117 54)(40 200 181 118 55)(41 201 182 119 56)(42 202 183 120 57)(43 203 184 121 58)(44 204 185 122 59)(45 205 186 123 60)(46 206 187 124 61)(47 207 188 125 62)(48 208 189 126 63)(49 209 190 127 64)
(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)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(99 107)(100 106)(101 105)(102 104)(108 147)(109 146)(110 145)(111 144)(112 143)(113 142)(114 141)(115 140)(116 139)(117 138)(118 137)(119 136)(120 135)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(148 184)(149 183)(150 182)(151 181)(152 180)(153 179)(154 178)(155 177)(156 176)(157 175)(158 174)(159 173)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(197 222)(198 221)(199 220)(200 219)(201 218)(202 217)(203 216)(204 215)(205 214)(206 213)(207 212)(208 211)(209 210)(223 245)(224 244)(225 243)(226 242)(227 241)(228 240)(229 239)(230 238)(231 237)(232 236)(233 235)
G:=sub<Sym(245)| (1,210,191,128,65)(2,211,192,129,66)(3,212,193,130,67)(4,213,194,131,68)(5,214,195,132,69)(6,215,196,133,70)(7,216,148,134,71)(8,217,149,135,72)(9,218,150,136,73)(10,219,151,137,74)(11,220,152,138,75)(12,221,153,139,76)(13,222,154,140,77)(14,223,155,141,78)(15,224,156,142,79)(16,225,157,143,80)(17,226,158,144,81)(18,227,159,145,82)(19,228,160,146,83)(20,229,161,147,84)(21,230,162,99,85)(22,231,163,100,86)(23,232,164,101,87)(24,233,165,102,88)(25,234,166,103,89)(26,235,167,104,90)(27,236,168,105,91)(28,237,169,106,92)(29,238,170,107,93)(30,239,171,108,94)(31,240,172,109,95)(32,241,173,110,96)(33,242,174,111,97)(34,243,175,112,98)(35,244,176,113,50)(36,245,177,114,51)(37,197,178,115,52)(38,198,179,116,53)(39,199,180,117,54)(40,200,181,118,55)(41,201,182,119,56)(42,202,183,120,57)(43,203,184,121,58)(44,204,185,122,59)(45,205,186,123,60)(46,206,187,124,61)(47,207,188,125,62)(48,208,189,126,63)(49,209,190,127,64), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(99,107)(100,106)(101,105)(102,104)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(148,184)(149,183)(150,182)(151,181)(152,180)(153,179)(154,178)(155,177)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(197,222)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)(204,215)(205,214)(206,213)(207,212)(208,211)(209,210)(223,245)(224,244)(225,243)(226,242)(227,241)(228,240)(229,239)(230,238)(231,237)(232,236)(233,235)>;
G:=Group( (1,210,191,128,65)(2,211,192,129,66)(3,212,193,130,67)(4,213,194,131,68)(5,214,195,132,69)(6,215,196,133,70)(7,216,148,134,71)(8,217,149,135,72)(9,218,150,136,73)(10,219,151,137,74)(11,220,152,138,75)(12,221,153,139,76)(13,222,154,140,77)(14,223,155,141,78)(15,224,156,142,79)(16,225,157,143,80)(17,226,158,144,81)(18,227,159,145,82)(19,228,160,146,83)(20,229,161,147,84)(21,230,162,99,85)(22,231,163,100,86)(23,232,164,101,87)(24,233,165,102,88)(25,234,166,103,89)(26,235,167,104,90)(27,236,168,105,91)(28,237,169,106,92)(29,238,170,107,93)(30,239,171,108,94)(31,240,172,109,95)(32,241,173,110,96)(33,242,174,111,97)(34,243,175,112,98)(35,244,176,113,50)(36,245,177,114,51)(37,197,178,115,52)(38,198,179,116,53)(39,199,180,117,54)(40,200,181,118,55)(41,201,182,119,56)(42,202,183,120,57)(43,203,184,121,58)(44,204,185,122,59)(45,205,186,123,60)(46,206,187,124,61)(47,207,188,125,62)(48,208,189,126,63)(49,209,190,127,64), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(99,107)(100,106)(101,105)(102,104)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(148,184)(149,183)(150,182)(151,181)(152,180)(153,179)(154,178)(155,177)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(197,222)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)(204,215)(205,214)(206,213)(207,212)(208,211)(209,210)(223,245)(224,244)(225,243)(226,242)(227,241)(228,240)(229,239)(230,238)(231,237)(232,236)(233,235) );
G=PermutationGroup([[(1,210,191,128,65),(2,211,192,129,66),(3,212,193,130,67),(4,213,194,131,68),(5,214,195,132,69),(6,215,196,133,70),(7,216,148,134,71),(8,217,149,135,72),(9,218,150,136,73),(10,219,151,137,74),(11,220,152,138,75),(12,221,153,139,76),(13,222,154,140,77),(14,223,155,141,78),(15,224,156,142,79),(16,225,157,143,80),(17,226,158,144,81),(18,227,159,145,82),(19,228,160,146,83),(20,229,161,147,84),(21,230,162,99,85),(22,231,163,100,86),(23,232,164,101,87),(24,233,165,102,88),(25,234,166,103,89),(26,235,167,104,90),(27,236,168,105,91),(28,237,169,106,92),(29,238,170,107,93),(30,239,171,108,94),(31,240,172,109,95),(32,241,173,110,96),(33,242,174,111,97),(34,243,175,112,98),(35,244,176,113,50),(36,245,177,114,51),(37,197,178,115,52),(38,198,179,116,53),(39,199,180,117,54),(40,200,181,118,55),(41,201,182,119,56),(42,202,183,120,57),(43,203,184,121,58),(44,204,185,122,59),(45,205,186,123,60),(46,206,187,124,61),(47,207,188,125,62),(48,208,189,126,63),(49,209,190,127,64)], [(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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(99,107),(100,106),(101,105),(102,104),(108,147),(109,146),(110,145),(111,144),(112,143),(113,142),(114,141),(115,140),(116,139),(117,138),(118,137),(119,136),(120,135),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(148,184),(149,183),(150,182),(151,181),(152,180),(153,179),(154,178),(155,177),(156,176),(157,175),(158,174),(159,173),(160,172),(161,171),(162,170),(163,169),(164,168),(165,167),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(197,222),(198,221),(199,220),(200,219),(201,218),(202,217),(203,216),(204,215),(205,214),(206,213),(207,212),(208,211),(209,210),(223,245),(224,244),(225,243),(226,242),(227,241),(228,240),(229,239),(230,238),(231,237),(232,236),(233,235)]])
130 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 35A | ··· | 35L | 49A | ··· | 49U | 245A | ··· | 245CF |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 35 | ··· | 35 | 49 | ··· | 49 | 245 | ··· | 245 |
| size | 1 | 49 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 49 | 49 | 49 | 49 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C5 | C10 | D7 | C5×D7 | D49 | C5×D49 |
| kernel | C5×D49 | C245 | D49 | C49 | C35 | C7 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 3 | 12 | 21 | 84 |
Matrix representation of C5×D49 ►in GL2(𝔽491) generated by
| 101 | 0 |
| 0 | 101 |
| 373 | 477 |
| 14 | 114 |
| 127 | 332 |
| 472 | 364 |
G:=sub<GL(2,GF(491))| [101,0,0,101],[373,14,477,114],[127,472,332,364] >;
C5×D49 in GAP, Magma, Sage, TeX
C_5\times D_{49} % in TeX
G:=Group("C5xD49"); // GroupNames label
G:=SmallGroup(490,2);
// by ID
G=gap.SmallGroup(490,2);
# by ID
G:=PCGroup([4,-2,-5,-7,-7,1562,514,6723]);
// Polycyclic
G:=Group<a,b,c|a^5=b^49=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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