direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D43, C43⋊C10, C215⋊2C2, SmallGroup(430,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C5×D43 |
Generators and relations for C5×D43
G = < a,b,c | a5=b43=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D43
►On 215 points
Generators in S215
(1 180 145 112 54)(2 181 146 113 55)(3 182 147 114 56)(4 183 148 115 57)(5 184 149 116 58)(6 185 150 117 59)(7 186 151 118 60)(8 187 152 119 61)(9 188 153 120 62)(10 189 154 121 63)(11 190 155 122 64)(12 191 156 123 65)(13 192 157 124 66)(14 193 158 125 67)(15 194 159 126 68)(16 195 160 127 69)(17 196 161 128 70)(18 197 162 129 71)(19 198 163 87 72)(20 199 164 88 73)(21 200 165 89 74)(22 201 166 90 75)(23 202 167 91 76)(24 203 168 92 77)(25 204 169 93 78)(26 205 170 94 79)(27 206 171 95 80)(28 207 172 96 81)(29 208 130 97 82)(30 209 131 98 83)(31 210 132 99 84)(32 211 133 100 85)(33 212 134 101 86)(34 213 135 102 44)(35 214 136 103 45)(36 215 137 104 46)(37 173 138 105 47)(38 174 139 106 48)(39 175 140 107 49)(40 176 141 108 50)(41 177 142 109 51)(42 178 143 110 52)(43 179 144 111 53)
(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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(87 93)(88 92)(89 91)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(130 159)(131 158)(132 157)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)(173 186)(174 185)(175 184)(176 183)(177 182)(178 181)(179 180)(187 215)(188 214)(189 213)(190 212)(191 211)(192 210)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)
G:=sub<Sym(215)| (1,180,145,112,54)(2,181,146,113,55)(3,182,147,114,56)(4,183,148,115,57)(5,184,149,116,58)(6,185,150,117,59)(7,186,151,118,60)(8,187,152,119,61)(9,188,153,120,62)(10,189,154,121,63)(11,190,155,122,64)(12,191,156,123,65)(13,192,157,124,66)(14,193,158,125,67)(15,194,159,126,68)(16,195,160,127,69)(17,196,161,128,70)(18,197,162,129,71)(19,198,163,87,72)(20,199,164,88,73)(21,200,165,89,74)(22,201,166,90,75)(23,202,167,91,76)(24,203,168,92,77)(25,204,169,93,78)(26,205,170,94,79)(27,206,171,95,80)(28,207,172,96,81)(29,208,130,97,82)(30,209,131,98,83)(31,210,132,99,84)(32,211,133,100,85)(33,212,134,101,86)(34,213,135,102,44)(35,214,136,103,45)(36,215,137,104,46)(37,173,138,105,47)(38,174,139,106,48)(39,175,140,107,49)(40,176,141,108,50)(41,177,142,109,51)(42,178,143,110,52)(43,179,144,111,53), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(87,93)(88,92)(89,91)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)>;
G:=Group( (1,180,145,112,54)(2,181,146,113,55)(3,182,147,114,56)(4,183,148,115,57)(5,184,149,116,58)(6,185,150,117,59)(7,186,151,118,60)(8,187,152,119,61)(9,188,153,120,62)(10,189,154,121,63)(11,190,155,122,64)(12,191,156,123,65)(13,192,157,124,66)(14,193,158,125,67)(15,194,159,126,68)(16,195,160,127,69)(17,196,161,128,70)(18,197,162,129,71)(19,198,163,87,72)(20,199,164,88,73)(21,200,165,89,74)(22,201,166,90,75)(23,202,167,91,76)(24,203,168,92,77)(25,204,169,93,78)(26,205,170,94,79)(27,206,171,95,80)(28,207,172,96,81)(29,208,130,97,82)(30,209,131,98,83)(31,210,132,99,84)(32,211,133,100,85)(33,212,134,101,86)(34,213,135,102,44)(35,214,136,103,45)(36,215,137,104,46)(37,173,138,105,47)(38,174,139,106,48)(39,175,140,107,49)(40,176,141,108,50)(41,177,142,109,51)(42,178,143,110,52)(43,179,144,111,53), (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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(87,93)(88,92)(89,91)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202) );
G=PermutationGroup([[(1,180,145,112,54),(2,181,146,113,55),(3,182,147,114,56),(4,183,148,115,57),(5,184,149,116,58),(6,185,150,117,59),(7,186,151,118,60),(8,187,152,119,61),(9,188,153,120,62),(10,189,154,121,63),(11,190,155,122,64),(12,191,156,123,65),(13,192,157,124,66),(14,193,158,125,67),(15,194,159,126,68),(16,195,160,127,69),(17,196,161,128,70),(18,197,162,129,71),(19,198,163,87,72),(20,199,164,88,73),(21,200,165,89,74),(22,201,166,90,75),(23,202,167,91,76),(24,203,168,92,77),(25,204,169,93,78),(26,205,170,94,79),(27,206,171,95,80),(28,207,172,96,81),(29,208,130,97,82),(30,209,131,98,83),(31,210,132,99,84),(32,211,133,100,85),(33,212,134,101,86),(34,213,135,102,44),(35,214,136,103,45),(36,215,137,104,46),(37,173,138,105,47),(38,174,139,106,48),(39,175,140,107,49),(40,176,141,108,50),(41,177,142,109,51),(42,178,143,110,52),(43,179,144,111,53)], [(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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(87,93),(88,92),(89,91),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123),(101,122),(102,121),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(130,159),(131,158),(132,157),(133,156),(134,155),(135,154),(136,153),(137,152),(138,151),(139,150),(140,149),(141,148),(142,147),(143,146),(144,145),(160,172),(161,171),(162,170),(163,169),(164,168),(165,167),(173,186),(174,185),(175,184),(176,183),(177,182),(178,181),(179,180),(187,215),(188,214),(189,213),(190,212),(191,211),(192,210),(193,209),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202)]])
115 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 43A | ··· | 43U | 215A | ··· | 215CF |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 43 | ··· | 43 | 215 | ··· | 215 |
| size | 1 | 43 | 1 | 1 | 1 | 1 | 43 | 43 | 43 | 43 | 2 | ··· | 2 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D43 | C5×D43 |
| kernel | C5×D43 | C215 | D43 | C43 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 21 | 84 |
Matrix representation of C5×D43 ►in GL2(𝔽431) generated by
| 405 | 0 |
| 0 | 405 |
| 0 | 1 |
| 430 | 147 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(431))| [405,0,0,405],[0,430,1,147],[0,1,1,0] >;
C5×D43 in GAP, Magma, Sage, TeX
C_5\times D_{43} % in TeX
G:=Group("C5xD43"); // GroupNames label
G:=SmallGroup(430,2);
// by ID
G=gap.SmallGroup(430,2);
# by ID
G:=PCGroup([3,-2,-5,-43,3782]);
// Polycyclic
G:=Group<a,b,c|a^5=b^43=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export