direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C49, C5⋊C98, C245⋊3C2, C35.C14, C7.(C7×D5), (C7×D5).C7, SmallGroup(490,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C49 |
Generators and relations for D5×C49
G = < a,b,c | a49=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C49
►On 245 points
Generators in S245
(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)
(1 192 132 69 212)(2 193 133 70 213)(3 194 134 71 214)(4 195 135 72 215)(5 196 136 73 216)(6 148 137 74 217)(7 149 138 75 218)(8 150 139 76 219)(9 151 140 77 220)(10 152 141 78 221)(11 153 142 79 222)(12 154 143 80 223)(13 155 144 81 224)(14 156 145 82 225)(15 157 146 83 226)(16 158 147 84 227)(17 159 99 85 228)(18 160 100 86 229)(19 161 101 87 230)(20 162 102 88 231)(21 163 103 89 232)(22 164 104 90 233)(23 165 105 91 234)(24 166 106 92 235)(25 167 107 93 236)(26 168 108 94 237)(27 169 109 95 238)(28 170 110 96 239)(29 171 111 97 240)(30 172 112 98 241)(31 173 113 50 242)(32 174 114 51 243)(33 175 115 52 244)(34 176 116 53 245)(35 177 117 54 197)(36 178 118 55 198)(37 179 119 56 199)(38 180 120 57 200)(39 181 121 58 201)(40 182 122 59 202)(41 183 123 60 203)(42 184 124 61 204)(43 185 125 62 205)(44 186 126 63 206)(45 187 127 64 207)(46 188 128 65 208)(47 189 129 66 209)(48 190 130 67 210)(49 191 131 68 211)
(1 212)(2 213)(3 214)(4 215)(5 216)(6 217)(7 218)(8 219)(9 220)(10 221)(11 222)(12 223)(13 224)(14 225)(15 226)(16 227)(17 228)(18 229)(19 230)(20 231)(21 232)(22 233)(23 234)(24 235)(25 236)(26 237)(27 238)(28 239)(29 240)(30 241)(31 242)(32 243)(33 244)(34 245)(35 197)(36 198)(37 199)(38 200)(39 201)(40 202)(41 203)(42 204)(43 205)(44 206)(45 207)(46 208)(47 209)(48 210)(49 211)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 180)(58 181)(59 182)(60 183)(61 184)(62 185)(63 186)(64 187)(65 188)(66 189)(67 190)(68 191)(69 192)(70 193)(71 194)(72 195)(73 196)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 161)(88 162)(89 163)(90 164)(91 165)(92 166)(93 167)(94 168)(95 169)(96 170)(97 171)(98 172)
G:=sub<Sym(245)| (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), (1,192,132,69,212)(2,193,133,70,213)(3,194,134,71,214)(4,195,135,72,215)(5,196,136,73,216)(6,148,137,74,217)(7,149,138,75,218)(8,150,139,76,219)(9,151,140,77,220)(10,152,141,78,221)(11,153,142,79,222)(12,154,143,80,223)(13,155,144,81,224)(14,156,145,82,225)(15,157,146,83,226)(16,158,147,84,227)(17,159,99,85,228)(18,160,100,86,229)(19,161,101,87,230)(20,162,102,88,231)(21,163,103,89,232)(22,164,104,90,233)(23,165,105,91,234)(24,166,106,92,235)(25,167,107,93,236)(26,168,108,94,237)(27,169,109,95,238)(28,170,110,96,239)(29,171,111,97,240)(30,172,112,98,241)(31,173,113,50,242)(32,174,114,51,243)(33,175,115,52,244)(34,176,116,53,245)(35,177,117,54,197)(36,178,118,55,198)(37,179,119,56,199)(38,180,120,57,200)(39,181,121,58,201)(40,182,122,59,202)(41,183,123,60,203)(42,184,124,61,204)(43,185,125,62,205)(44,186,126,63,206)(45,187,127,64,207)(46,188,128,65,208)(47,189,129,66,209)(48,190,130,67,210)(49,191,131,68,211), (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,225)(15,226)(16,227)(17,228)(18,229)(19,230)(20,231)(21,232)(22,233)(23,234)(24,235)(25,236)(26,237)(27,238)(28,239)(29,240)(30,241)(31,242)(32,243)(33,244)(34,245)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,181)(59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,161)(88,162)(89,163)(90,164)(91,165)(92,166)(93,167)(94,168)(95,169)(96,170)(97,171)(98,172)>;
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), (1,192,132,69,212)(2,193,133,70,213)(3,194,134,71,214)(4,195,135,72,215)(5,196,136,73,216)(6,148,137,74,217)(7,149,138,75,218)(8,150,139,76,219)(9,151,140,77,220)(10,152,141,78,221)(11,153,142,79,222)(12,154,143,80,223)(13,155,144,81,224)(14,156,145,82,225)(15,157,146,83,226)(16,158,147,84,227)(17,159,99,85,228)(18,160,100,86,229)(19,161,101,87,230)(20,162,102,88,231)(21,163,103,89,232)(22,164,104,90,233)(23,165,105,91,234)(24,166,106,92,235)(25,167,107,93,236)(26,168,108,94,237)(27,169,109,95,238)(28,170,110,96,239)(29,171,111,97,240)(30,172,112,98,241)(31,173,113,50,242)(32,174,114,51,243)(33,175,115,52,244)(34,176,116,53,245)(35,177,117,54,197)(36,178,118,55,198)(37,179,119,56,199)(38,180,120,57,200)(39,181,121,58,201)(40,182,122,59,202)(41,183,123,60,203)(42,184,124,61,204)(43,185,125,62,205)(44,186,126,63,206)(45,187,127,64,207)(46,188,128,65,208)(47,189,129,66,209)(48,190,130,67,210)(49,191,131,68,211), (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,225)(15,226)(16,227)(17,228)(18,229)(19,230)(20,231)(21,232)(22,233)(23,234)(24,235)(25,236)(26,237)(27,238)(28,239)(29,240)(30,241)(31,242)(32,243)(33,244)(34,245)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,181)(59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,161)(88,162)(89,163)(90,164)(91,165)(92,166)(93,167)(94,168)(95,169)(96,170)(97,171)(98,172) );
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)], [(1,192,132,69,212),(2,193,133,70,213),(3,194,134,71,214),(4,195,135,72,215),(5,196,136,73,216),(6,148,137,74,217),(7,149,138,75,218),(8,150,139,76,219),(9,151,140,77,220),(10,152,141,78,221),(11,153,142,79,222),(12,154,143,80,223),(13,155,144,81,224),(14,156,145,82,225),(15,157,146,83,226),(16,158,147,84,227),(17,159,99,85,228),(18,160,100,86,229),(19,161,101,87,230),(20,162,102,88,231),(21,163,103,89,232),(22,164,104,90,233),(23,165,105,91,234),(24,166,106,92,235),(25,167,107,93,236),(26,168,108,94,237),(27,169,109,95,238),(28,170,110,96,239),(29,171,111,97,240),(30,172,112,98,241),(31,173,113,50,242),(32,174,114,51,243),(33,175,115,52,244),(34,176,116,53,245),(35,177,117,54,197),(36,178,118,55,198),(37,179,119,56,199),(38,180,120,57,200),(39,181,121,58,201),(40,182,122,59,202),(41,183,123,60,203),(42,184,124,61,204),(43,185,125,62,205),(44,186,126,63,206),(45,187,127,64,207),(46,188,128,65,208),(47,189,129,66,209),(48,190,130,67,210),(49,191,131,68,211)], [(1,212),(2,213),(3,214),(4,215),(5,216),(6,217),(7,218),(8,219),(9,220),(10,221),(11,222),(12,223),(13,224),(14,225),(15,226),(16,227),(17,228),(18,229),(19,230),(20,231),(21,232),(22,233),(23,234),(24,235),(25,236),(26,237),(27,238),(28,239),(29,240),(30,241),(31,242),(32,243),(33,244),(34,245),(35,197),(36,198),(37,199),(38,200),(39,201),(40,202),(41,203),(42,204),(43,205),(44,206),(45,207),(46,208),(47,209),(48,210),(49,211),(50,173),(51,174),(52,175),(53,176),(54,177),(55,178),(56,179),(57,180),(58,181),(59,182),(60,183),(61,184),(62,185),(63,186),(64,187),(65,188),(66,189),(67,190),(68,191),(69,192),(70,193),(71,194),(72,195),(73,196),(74,148),(75,149),(76,150),(77,151),(78,152),(79,153),(80,154),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,161),(88,162),(89,163),(90,164),(91,165),(92,166),(93,167),(94,168),(95,169),(96,170),(97,171),(98,172)]])
196 conjugacy classes
| class | 1 | 2 | 5A | 5B | 7A | ··· | 7F | 14A | ··· | 14F | 35A | ··· | 35L | 49A | ··· | 49AP | 98A | ··· | 98AP | 245A | ··· | 245CF |
| order | 1 | 2 | 5 | 5 | 7 | ··· | 7 | 14 | ··· | 14 | 35 | ··· | 35 | 49 | ··· | 49 | 98 | ··· | 98 | 245 | ··· | 245 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C7 | C14 | C49 | C98 | D5 | C7×D5 | D5×C49 |
| kernel | D5×C49 | C245 | C7×D5 | C35 | D5 | C5 | C49 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 42 | 42 | 2 | 12 | 84 |
Matrix representation of D5×C49 ►in GL2(𝔽491) generated by
| 51 | 0 |
| 0 | 51 |
| 417 | 1 |
| 490 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(491))| [51,0,0,51],[417,490,1,0],[0,1,1,0] >;
D5×C49 in GAP, Magma, Sage, TeX
D_5\times C_{49} % in TeX
G:=Group("D5xC49"); // GroupNames label
G:=SmallGroup(490,1);
// by ID
G=gap.SmallGroup(490,1);
# by ID
G:=PCGroup([4,-2,-7,-7,-5,61,6275]);
// Polycyclic
G:=Group<a,b,c|a^49=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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