direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17×D13, C13⋊C34, C221⋊3C2, SmallGroup(442,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C17×D13 |
Generators and relations for C17×D13
G = < a,b,c | a17=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C17×D13
►On 221 points
Generators in S221
(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)
(1 48 156 109 202 91 123 140 173 33 215 56 81)(2 49 157 110 203 92 124 141 174 34 216 57 82)(3 50 158 111 204 93 125 142 175 18 217 58 83)(4 51 159 112 188 94 126 143 176 19 218 59 84)(5 35 160 113 189 95 127 144 177 20 219 60 85)(6 36 161 114 190 96 128 145 178 21 220 61 69)(7 37 162 115 191 97 129 146 179 22 221 62 70)(8 38 163 116 192 98 130 147 180 23 205 63 71)(9 39 164 117 193 99 131 148 181 24 206 64 72)(10 40 165 118 194 100 132 149 182 25 207 65 73)(11 41 166 119 195 101 133 150 183 26 208 66 74)(12 42 167 103 196 102 134 151 184 27 209 67 75)(13 43 168 104 197 86 135 152 185 28 210 68 76)(14 44 169 105 198 87 136 153 186 29 211 52 77)(15 45 170 106 199 88 120 137 187 30 212 53 78)(16 46 154 107 200 89 121 138 171 31 213 54 79)(17 47 155 108 201 90 122 139 172 32 214 55 80)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(51 59)(86 152)(87 153)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(154 213)(155 214)(156 215)(157 216)(158 217)(159 218)(160 219)(161 220)(162 221)(163 205)(164 206)(165 207)(166 208)(167 209)(168 210)(169 211)(170 212)(171 200)(172 201)(173 202)(174 203)(175 204)(176 188)(177 189)(178 190)(179 191)(180 192)(181 193)(182 194)(183 195)(184 196)(185 197)(186 198)(187 199)
G:=sub<Sym(221)| (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), (1,48,156,109,202,91,123,140,173,33,215,56,81)(2,49,157,110,203,92,124,141,174,34,216,57,82)(3,50,158,111,204,93,125,142,175,18,217,58,83)(4,51,159,112,188,94,126,143,176,19,218,59,84)(5,35,160,113,189,95,127,144,177,20,219,60,85)(6,36,161,114,190,96,128,145,178,21,220,61,69)(7,37,162,115,191,97,129,146,179,22,221,62,70)(8,38,163,116,192,98,130,147,180,23,205,63,71)(9,39,164,117,193,99,131,148,181,24,206,64,72)(10,40,165,118,194,100,132,149,182,25,207,65,73)(11,41,166,119,195,101,133,150,183,26,208,66,74)(12,42,167,103,196,102,134,151,184,27,209,67,75)(13,43,168,104,197,86,135,152,185,28,210,68,76)(14,44,169,105,198,87,136,153,186,29,211,52,77)(15,45,170,106,199,88,120,137,187,30,212,53,78)(16,46,154,107,200,89,121,138,171,31,213,54,79)(17,47,155,108,201,90,122,139,172,32,214,55,80), (1,81)(2,82)(3,83)(4,84)(5,85)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(51,59)(86,152)(87,153)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(154,213)(155,214)(156,215)(157,216)(158,217)(159,218)(160,219)(161,220)(162,221)(163,205)(164,206)(165,207)(166,208)(167,209)(168,210)(169,211)(170,212)(171,200)(172,201)(173,202)(174,203)(175,204)(176,188)(177,189)(178,190)(179,191)(180,192)(181,193)(182,194)(183,195)(184,196)(185,197)(186,198)(187,199)>;
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), (1,48,156,109,202,91,123,140,173,33,215,56,81)(2,49,157,110,203,92,124,141,174,34,216,57,82)(3,50,158,111,204,93,125,142,175,18,217,58,83)(4,51,159,112,188,94,126,143,176,19,218,59,84)(5,35,160,113,189,95,127,144,177,20,219,60,85)(6,36,161,114,190,96,128,145,178,21,220,61,69)(7,37,162,115,191,97,129,146,179,22,221,62,70)(8,38,163,116,192,98,130,147,180,23,205,63,71)(9,39,164,117,193,99,131,148,181,24,206,64,72)(10,40,165,118,194,100,132,149,182,25,207,65,73)(11,41,166,119,195,101,133,150,183,26,208,66,74)(12,42,167,103,196,102,134,151,184,27,209,67,75)(13,43,168,104,197,86,135,152,185,28,210,68,76)(14,44,169,105,198,87,136,153,186,29,211,52,77)(15,45,170,106,199,88,120,137,187,30,212,53,78)(16,46,154,107,200,89,121,138,171,31,213,54,79)(17,47,155,108,201,90,122,139,172,32,214,55,80), (1,81)(2,82)(3,83)(4,84)(5,85)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(51,59)(86,152)(87,153)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(154,213)(155,214)(156,215)(157,216)(158,217)(159,218)(160,219)(161,220)(162,221)(163,205)(164,206)(165,207)(166,208)(167,209)(168,210)(169,211)(170,212)(171,200)(172,201)(173,202)(174,203)(175,204)(176,188)(177,189)(178,190)(179,191)(180,192)(181,193)(182,194)(183,195)(184,196)(185,197)(186,198)(187,199) );
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)], [(1,48,156,109,202,91,123,140,173,33,215,56,81),(2,49,157,110,203,92,124,141,174,34,216,57,82),(3,50,158,111,204,93,125,142,175,18,217,58,83),(4,51,159,112,188,94,126,143,176,19,218,59,84),(5,35,160,113,189,95,127,144,177,20,219,60,85),(6,36,161,114,190,96,128,145,178,21,220,61,69),(7,37,162,115,191,97,129,146,179,22,221,62,70),(8,38,163,116,192,98,130,147,180,23,205,63,71),(9,39,164,117,193,99,131,148,181,24,206,64,72),(10,40,165,118,194,100,132,149,182,25,207,65,73),(11,41,166,119,195,101,133,150,183,26,208,66,74),(12,42,167,103,196,102,134,151,184,27,209,67,75),(13,43,168,104,197,86,135,152,185,28,210,68,76),(14,44,169,105,198,87,136,153,186,29,211,52,77),(15,45,170,106,199,88,120,137,187,30,212,53,78),(16,46,154,107,200,89,121,138,171,31,213,54,79),(17,47,155,108,201,90,122,139,172,32,214,55,80)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,111),(19,112),(20,113),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,109),(34,110),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(43,68),(44,52),(45,53),(46,54),(47,55),(48,56),(49,57),(50,58),(51,59),(86,152),(87,153),(88,137),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(154,213),(155,214),(156,215),(157,216),(158,217),(159,218),(160,219),(161,220),(162,221),(163,205),(164,206),(165,207),(166,208),(167,209),(168,210),(169,211),(170,212),(171,200),(172,201),(173,202),(174,203),(175,204),(176,188),(177,189),(178,190),(179,191),(180,192),(181,193),(182,194),(183,195),(184,196),(185,197),(186,198),(187,199)]])
136 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13F | 17A | ··· | 17P | 34A | ··· | 34P | 221A | ··· | 221CR |
| order | 1 | 2 | 13 | ··· | 13 | 17 | ··· | 17 | 34 | ··· | 34 | 221 | ··· | 221 |
| size | 1 | 13 | 2 | ··· | 2 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C17 | C34 | D13 | C17×D13 |
| kernel | C17×D13 | C221 | D13 | C13 | C17 | C1 |
| # reps | 1 | 1 | 16 | 16 | 6 | 96 |
Matrix representation of C17×D13 ►in GL2(𝔽443) generated by
| 380 | 0 |
| 0 | 380 |
| 442 | 1 |
| 368 | 74 |
| 442 | 0 |
| 368 | 1 |
G:=sub<GL(2,GF(443))| [380,0,0,380],[442,368,1,74],[442,368,0,1] >;
C17×D13 in GAP, Magma, Sage, TeX
C_{17}\times D_{13} % in TeX
G:=Group("C17xD13"); // GroupNames label
G:=SmallGroup(442,1);
// by ID
G=gap.SmallGroup(442,1);
# by ID
G:=PCGroup([3,-2,-17,-13,3674]);
// Polycyclic
G:=Group<a,b,c|a^17=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export