direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C43, C5⋊C86, C215⋊3C2, SmallGroup(430,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C43 |
Generators and relations for D5×C43
G = < a,b,c | a43=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C43
►On 215 points
Generators in S215
(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)
(1 175 47 113 158)(2 176 48 114 159)(3 177 49 115 160)(4 178 50 116 161)(5 179 51 117 162)(6 180 52 118 163)(7 181 53 119 164)(8 182 54 120 165)(9 183 55 121 166)(10 184 56 122 167)(11 185 57 123 168)(12 186 58 124 169)(13 187 59 125 170)(14 188 60 126 171)(15 189 61 127 172)(16 190 62 128 130)(17 191 63 129 131)(18 192 64 87 132)(19 193 65 88 133)(20 194 66 89 134)(21 195 67 90 135)(22 196 68 91 136)(23 197 69 92 137)(24 198 70 93 138)(25 199 71 94 139)(26 200 72 95 140)(27 201 73 96 141)(28 202 74 97 142)(29 203 75 98 143)(30 204 76 99 144)(31 205 77 100 145)(32 206 78 101 146)(33 207 79 102 147)(34 208 80 103 148)(35 209 81 104 149)(36 210 82 105 150)(37 211 83 106 151)(38 212 84 107 152)(39 213 85 108 153)(40 214 86 109 154)(41 215 44 110 155)(42 173 45 111 156)(43 174 46 112 157)
(1 158)(2 159)(3 160)(4 161)(5 162)(6 163)(7 164)(8 165)(9 166)(10 167)(11 168)(12 169)(13 170)(14 171)(15 172)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 157)(87 192)(88 193)(89 194)(90 195)(91 196)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 211)(107 212)(108 213)(109 214)(110 215)(111 173)(112 174)(113 175)(114 176)(115 177)(116 178)(117 179)(118 180)(119 181)(120 182)(121 183)(122 184)(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)
G:=sub<Sym(215)| (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), (1,175,47,113,158)(2,176,48,114,159)(3,177,49,115,160)(4,178,50,116,161)(5,179,51,117,162)(6,180,52,118,163)(7,181,53,119,164)(8,182,54,120,165)(9,183,55,121,166)(10,184,56,122,167)(11,185,57,123,168)(12,186,58,124,169)(13,187,59,125,170)(14,188,60,126,171)(15,189,61,127,172)(16,190,62,128,130)(17,191,63,129,131)(18,192,64,87,132)(19,193,65,88,133)(20,194,66,89,134)(21,195,67,90,135)(22,196,68,91,136)(23,197,69,92,137)(24,198,70,93,138)(25,199,71,94,139)(26,200,72,95,140)(27,201,73,96,141)(28,202,74,97,142)(29,203,75,98,143)(30,204,76,99,144)(31,205,77,100,145)(32,206,78,101,146)(33,207,79,102,147)(34,208,80,103,148)(35,209,81,104,149)(36,210,82,105,150)(37,211,83,106,151)(38,212,84,107,152)(39,213,85,108,153)(40,214,86,109,154)(41,215,44,110,155)(42,173,45,111,156)(43,174,46,112,157), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,171)(15,172)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(87,192)(88,193)(89,194)(90,195)(91,196)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)>;
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), (1,175,47,113,158)(2,176,48,114,159)(3,177,49,115,160)(4,178,50,116,161)(5,179,51,117,162)(6,180,52,118,163)(7,181,53,119,164)(8,182,54,120,165)(9,183,55,121,166)(10,184,56,122,167)(11,185,57,123,168)(12,186,58,124,169)(13,187,59,125,170)(14,188,60,126,171)(15,189,61,127,172)(16,190,62,128,130)(17,191,63,129,131)(18,192,64,87,132)(19,193,65,88,133)(20,194,66,89,134)(21,195,67,90,135)(22,196,68,91,136)(23,197,69,92,137)(24,198,70,93,138)(25,199,71,94,139)(26,200,72,95,140)(27,201,73,96,141)(28,202,74,97,142)(29,203,75,98,143)(30,204,76,99,144)(31,205,77,100,145)(32,206,78,101,146)(33,207,79,102,147)(34,208,80,103,148)(35,209,81,104,149)(36,210,82,105,150)(37,211,83,106,151)(38,212,84,107,152)(39,213,85,108,153)(40,214,86,109,154)(41,215,44,110,155)(42,173,45,111,156)(43,174,46,112,157), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,171)(15,172)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(87,192)(88,193)(89,194)(90,195)(91,196)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,181)(120,182)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191) );
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)], [(1,175,47,113,158),(2,176,48,114,159),(3,177,49,115,160),(4,178,50,116,161),(5,179,51,117,162),(6,180,52,118,163),(7,181,53,119,164),(8,182,54,120,165),(9,183,55,121,166),(10,184,56,122,167),(11,185,57,123,168),(12,186,58,124,169),(13,187,59,125,170),(14,188,60,126,171),(15,189,61,127,172),(16,190,62,128,130),(17,191,63,129,131),(18,192,64,87,132),(19,193,65,88,133),(20,194,66,89,134),(21,195,67,90,135),(22,196,68,91,136),(23,197,69,92,137),(24,198,70,93,138),(25,199,71,94,139),(26,200,72,95,140),(27,201,73,96,141),(28,202,74,97,142),(29,203,75,98,143),(30,204,76,99,144),(31,205,77,100,145),(32,206,78,101,146),(33,207,79,102,147),(34,208,80,103,148),(35,209,81,104,149),(36,210,82,105,150),(37,211,83,106,151),(38,212,84,107,152),(39,213,85,108,153),(40,214,86,109,154),(41,215,44,110,155),(42,173,45,111,156),(43,174,46,112,157)], [(1,158),(2,159),(3,160),(4,161),(5,162),(6,163),(7,164),(8,165),(9,166),(10,167),(11,168),(12,169),(13,170),(14,171),(15,172),(16,130),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,141),(28,142),(29,143),(30,144),(31,145),(32,146),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,155),(42,156),(43,157),(87,192),(88,193),(89,194),(90,195),(91,196),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,204),(100,205),(101,206),(102,207),(103,208),(104,209),(105,210),(106,211),(107,212),(108,213),(109,214),(110,215),(111,173),(112,174),(113,175),(114,176),(115,177),(116,178),(117,179),(118,180),(119,181),(120,182),(121,183),(122,184),(123,185),(124,186),(125,187),(126,188),(127,189),(128,190),(129,191)]])
172 conjugacy classes
| class | 1 | 2 | 5A | 5B | 43A | ··· | 43AP | 86A | ··· | 86AP | 215A | ··· | 215CF |
| order | 1 | 2 | 5 | 5 | 43 | ··· | 43 | 86 | ··· | 86 | 215 | ··· | 215 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
172 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C43 | C86 | D5 | D5×C43 |
| kernel | D5×C43 | C215 | D5 | C5 | C43 | C1 |
| # reps | 1 | 1 | 42 | 42 | 2 | 84 |
Matrix representation of D5×C43 ►in GL2(𝔽431) generated by
| 220 | 0 |
| 0 | 220 |
| 90 | 1 |
| 430 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(431))| [220,0,0,220],[90,430,1,0],[0,1,1,0] >;
D5×C43 in GAP, Magma, Sage, TeX
D_5\times C_{43} % in TeX
G:=Group("D5xC43"); // GroupNames label
G:=SmallGroup(430,1);
// by ID
G=gap.SmallGroup(430,1);
# by ID
G:=PCGroup([3,-2,-43,-5,3098]);
// Polycyclic
G:=Group<a,b,c|a^43=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export