direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13×D17, C17⋊C26, C221⋊2C2, SmallGroup(442,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C13×D17 |
Generators and relations for C13×D17
G = < a,b,c | a13=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C13×D17
►On 221 points
Generators in S221
(1 208 189 176 158 147 122 118 87 73 53 50 26)(2 209 190 177 159 148 123 119 88 74 54 51 27)(3 210 191 178 160 149 124 103 89 75 55 35 28)(4 211 192 179 161 150 125 104 90 76 56 36 29)(5 212 193 180 162 151 126 105 91 77 57 37 30)(6 213 194 181 163 152 127 106 92 78 58 38 31)(7 214 195 182 164 153 128 107 93 79 59 39 32)(8 215 196 183 165 137 129 108 94 80 60 40 33)(9 216 197 184 166 138 130 109 95 81 61 41 34)(10 217 198 185 167 139 131 110 96 82 62 42 18)(11 218 199 186 168 140 132 111 97 83 63 43 19)(12 219 200 187 169 141 133 112 98 84 64 44 20)(13 220 201 171 170 142 134 113 99 85 65 45 21)(14 221 202 172 154 143 135 114 100 69 66 46 22)(15 205 203 173 155 144 136 115 101 70 67 47 23)(16 206 204 174 156 145 120 116 102 71 68 48 24)(17 207 188 175 157 146 121 117 86 72 52 49 25)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(48 51)(49 50)(52 53)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(69 76)(70 75)(71 74)(72 73)(77 85)(78 84)(79 83)(80 82)(86 87)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(116 119)(117 118)(120 123)(121 122)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(137 139)(140 153)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(154 161)(155 160)(156 159)(157 158)(162 170)(163 169)(164 168)(165 167)(171 180)(172 179)(173 178)(174 177)(175 176)(181 187)(182 186)(183 185)(188 189)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)(205 210)(206 209)(207 208)(211 221)(212 220)(213 219)(214 218)(215 217)
G:=sub<Sym(221)| (1,208,189,176,158,147,122,118,87,73,53,50,26)(2,209,190,177,159,148,123,119,88,74,54,51,27)(3,210,191,178,160,149,124,103,89,75,55,35,28)(4,211,192,179,161,150,125,104,90,76,56,36,29)(5,212,193,180,162,151,126,105,91,77,57,37,30)(6,213,194,181,163,152,127,106,92,78,58,38,31)(7,214,195,182,164,153,128,107,93,79,59,39,32)(8,215,196,183,165,137,129,108,94,80,60,40,33)(9,216,197,184,166,138,130,109,95,81,61,41,34)(10,217,198,185,167,139,131,110,96,82,62,42,18)(11,218,199,186,168,140,132,111,97,83,63,43,19)(12,219,200,187,169,141,133,112,98,84,64,44,20)(13,220,201,171,170,142,134,113,99,85,65,45,21)(14,221,202,172,154,143,135,114,100,69,66,46,22)(15,205,203,173,155,144,136,115,101,70,67,47,23)(16,206,204,174,156,145,120,116,102,71,68,48,24)(17,207,188,175,157,146,121,117,86,72,52,49,25), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(48,51)(49,50)(52,53)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(69,76)(70,75)(71,74)(72,73)(77,85)(78,84)(79,83)(80,82)(86,87)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,123)(121,122)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(137,139)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(154,161)(155,160)(156,159)(157,158)(162,170)(163,169)(164,168)(165,167)(171,180)(172,179)(173,178)(174,177)(175,176)(181,187)(182,186)(183,185)(188,189)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(205,210)(206,209)(207,208)(211,221)(212,220)(213,219)(214,218)(215,217)>;
G:=Group( (1,208,189,176,158,147,122,118,87,73,53,50,26)(2,209,190,177,159,148,123,119,88,74,54,51,27)(3,210,191,178,160,149,124,103,89,75,55,35,28)(4,211,192,179,161,150,125,104,90,76,56,36,29)(5,212,193,180,162,151,126,105,91,77,57,37,30)(6,213,194,181,163,152,127,106,92,78,58,38,31)(7,214,195,182,164,153,128,107,93,79,59,39,32)(8,215,196,183,165,137,129,108,94,80,60,40,33)(9,216,197,184,166,138,130,109,95,81,61,41,34)(10,217,198,185,167,139,131,110,96,82,62,42,18)(11,218,199,186,168,140,132,111,97,83,63,43,19)(12,219,200,187,169,141,133,112,98,84,64,44,20)(13,220,201,171,170,142,134,113,99,85,65,45,21)(14,221,202,172,154,143,135,114,100,69,66,46,22)(15,205,203,173,155,144,136,115,101,70,67,47,23)(16,206,204,174,156,145,120,116,102,71,68,48,24)(17,207,188,175,157,146,121,117,86,72,52,49,25), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(48,51)(49,50)(52,53)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(69,76)(70,75)(71,74)(72,73)(77,85)(78,84)(79,83)(80,82)(86,87)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,123)(121,122)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(137,139)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(154,161)(155,160)(156,159)(157,158)(162,170)(163,169)(164,168)(165,167)(171,180)(172,179)(173,178)(174,177)(175,176)(181,187)(182,186)(183,185)(188,189)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(205,210)(206,209)(207,208)(211,221)(212,220)(213,219)(214,218)(215,217) );
G=PermutationGroup([[(1,208,189,176,158,147,122,118,87,73,53,50,26),(2,209,190,177,159,148,123,119,88,74,54,51,27),(3,210,191,178,160,149,124,103,89,75,55,35,28),(4,211,192,179,161,150,125,104,90,76,56,36,29),(5,212,193,180,162,151,126,105,91,77,57,37,30),(6,213,194,181,163,152,127,106,92,78,58,38,31),(7,214,195,182,164,153,128,107,93,79,59,39,32),(8,215,196,183,165,137,129,108,94,80,60,40,33),(9,216,197,184,166,138,130,109,95,81,61,41,34),(10,217,198,185,167,139,131,110,96,82,62,42,18),(11,218,199,186,168,140,132,111,97,83,63,43,19),(12,219,200,187,169,141,133,112,98,84,64,44,20),(13,220,201,171,170,142,134,113,99,85,65,45,21),(14,221,202,172,154,143,135,114,100,69,66,46,22),(15,205,203,173,155,144,136,115,101,70,67,47,23),(16,206,204,174,156,145,120,116,102,71,68,48,24),(17,207,188,175,157,146,121,117,86,72,52,49,25)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(48,51),(49,50),(52,53),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(69,76),(70,75),(71,74),(72,73),(77,85),(78,84),(79,83),(80,82),(86,87),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(116,119),(117,118),(120,123),(121,122),(124,136),(125,135),(126,134),(127,133),(128,132),(129,131),(137,139),(140,153),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147),(154,161),(155,160),(156,159),(157,158),(162,170),(163,169),(164,168),(165,167),(171,180),(172,179),(173,178),(174,177),(175,176),(181,187),(182,186),(183,185),(188,189),(190,204),(191,203),(192,202),(193,201),(194,200),(195,199),(196,198),(205,210),(206,209),(207,208),(211,221),(212,220),(213,219),(214,218),(215,217)]])
130 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13L | 17A | ··· | 17H | 26A | ··· | 26L | 221A | ··· | 221CR |
| order | 1 | 2 | 13 | ··· | 13 | 17 | ··· | 17 | 26 | ··· | 26 | 221 | ··· | 221 |
| size | 1 | 17 | 1 | ··· | 1 | 2 | ··· | 2 | 17 | ··· | 17 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C13 | C26 | D17 | C13×D17 |
| kernel | C13×D17 | C221 | D17 | C17 | C13 | C1 |
| # reps | 1 | 1 | 12 | 12 | 8 | 96 |
Matrix representation of C13×D17 ►in GL2(𝔽443) generated by
| 356 | 0 |
| 0 | 356 |
| 0 | 1 |
| 442 | 44 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(443))| [356,0,0,356],[0,442,1,44],[0,1,1,0] >;
C13×D17 in GAP, Magma, Sage, TeX
C_{13}\times D_{17} % in TeX
G:=Group("C13xD17"); // GroupNames label
G:=SmallGroup(442,2);
// by ID
G=gap.SmallGroup(442,2);
# by ID
G:=PCGroup([3,-2,-13,-17,3746]);
// Polycyclic
G:=Group<a,b,c|a^13=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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