direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D41, C205⋊2C2, C41⋊3C10, SmallGroup(410,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C5×D41 |
Generators and relations for C5×D41
G = < a,b,c | a5=b41=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D41
►On 205 points
Generators in S205
(1 179 129 83 66)(2 180 130 84 67)(3 181 131 85 68)(4 182 132 86 69)(5 183 133 87 70)(6 184 134 88 71)(7 185 135 89 72)(8 186 136 90 73)(9 187 137 91 74)(10 188 138 92 75)(11 189 139 93 76)(12 190 140 94 77)(13 191 141 95 78)(14 192 142 96 79)(15 193 143 97 80)(16 194 144 98 81)(17 195 145 99 82)(18 196 146 100 42)(19 197 147 101 43)(20 198 148 102 44)(21 199 149 103 45)(22 200 150 104 46)(23 201 151 105 47)(24 202 152 106 48)(25 203 153 107 49)(26 204 154 108 50)(27 205 155 109 51)(28 165 156 110 52)(29 166 157 111 53)(30 167 158 112 54)(31 168 159 113 55)(32 169 160 114 56)(33 170 161 115 57)(34 171 162 116 58)(35 172 163 117 59)(36 173 164 118 60)(37 174 124 119 61)(38 175 125 120 62)(39 176 126 121 63)(40 177 127 122 64)(41 178 128 123 65)
(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)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(42 48)(43 47)(44 46)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(83 123)(84 122)(85 121)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(124 133)(125 132)(126 131)(127 130)(128 129)(134 164)(135 163)(136 162)(137 161)(138 160)(139 159)(140 158)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(165 192)(166 191)(167 190)(168 189)(169 188)(170 187)(171 186)(172 185)(173 184)(174 183)(175 182)(176 181)(177 180)(178 179)(193 205)(194 204)(195 203)(196 202)(197 201)(198 200)
G:=sub<Sym(205)| (1,179,129,83,66)(2,180,130,84,67)(3,181,131,85,68)(4,182,132,86,69)(5,183,133,87,70)(6,184,134,88,71)(7,185,135,89,72)(8,186,136,90,73)(9,187,137,91,74)(10,188,138,92,75)(11,189,139,93,76)(12,190,140,94,77)(13,191,141,95,78)(14,192,142,96,79)(15,193,143,97,80)(16,194,144,98,81)(17,195,145,99,82)(18,196,146,100,42)(19,197,147,101,43)(20,198,148,102,44)(21,199,149,103,45)(22,200,150,104,46)(23,201,151,105,47)(24,202,152,106,48)(25,203,153,107,49)(26,204,154,108,50)(27,205,155,109,51)(28,165,156,110,52)(29,166,157,111,53)(30,167,158,112,54)(31,168,159,113,55)(32,169,160,114,56)(33,170,161,115,57)(34,171,162,116,58)(35,172,163,117,59)(36,173,164,118,60)(37,174,124,119,61)(38,175,125,120,62)(39,176,126,121,63)(40,177,127,122,64)(41,178,128,123,65), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(42,48)(43,47)(44,46)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(83,123)(84,122)(85,121)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(124,133)(125,132)(126,131)(127,130)(128,129)(134,164)(135,163)(136,162)(137,161)(138,160)(139,159)(140,158)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(165,192)(166,191)(167,190)(168,189)(169,188)(170,187)(171,186)(172,185)(173,184)(174,183)(175,182)(176,181)(177,180)(178,179)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200)>;
G:=Group( (1,179,129,83,66)(2,180,130,84,67)(3,181,131,85,68)(4,182,132,86,69)(5,183,133,87,70)(6,184,134,88,71)(7,185,135,89,72)(8,186,136,90,73)(9,187,137,91,74)(10,188,138,92,75)(11,189,139,93,76)(12,190,140,94,77)(13,191,141,95,78)(14,192,142,96,79)(15,193,143,97,80)(16,194,144,98,81)(17,195,145,99,82)(18,196,146,100,42)(19,197,147,101,43)(20,198,148,102,44)(21,199,149,103,45)(22,200,150,104,46)(23,201,151,105,47)(24,202,152,106,48)(25,203,153,107,49)(26,204,154,108,50)(27,205,155,109,51)(28,165,156,110,52)(29,166,157,111,53)(30,167,158,112,54)(31,168,159,113,55)(32,169,160,114,56)(33,170,161,115,57)(34,171,162,116,58)(35,172,163,117,59)(36,173,164,118,60)(37,174,124,119,61)(38,175,125,120,62)(39,176,126,121,63)(40,177,127,122,64)(41,178,128,123,65), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(42,48)(43,47)(44,46)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(83,123)(84,122)(85,121)(86,120)(87,119)(88,118)(89,117)(90,116)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(124,133)(125,132)(126,131)(127,130)(128,129)(134,164)(135,163)(136,162)(137,161)(138,160)(139,159)(140,158)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(165,192)(166,191)(167,190)(168,189)(169,188)(170,187)(171,186)(172,185)(173,184)(174,183)(175,182)(176,181)(177,180)(178,179)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200) );
G=PermutationGroup([[(1,179,129,83,66),(2,180,130,84,67),(3,181,131,85,68),(4,182,132,86,69),(5,183,133,87,70),(6,184,134,88,71),(7,185,135,89,72),(8,186,136,90,73),(9,187,137,91,74),(10,188,138,92,75),(11,189,139,93,76),(12,190,140,94,77),(13,191,141,95,78),(14,192,142,96,79),(15,193,143,97,80),(16,194,144,98,81),(17,195,145,99,82),(18,196,146,100,42),(19,197,147,101,43),(20,198,148,102,44),(21,199,149,103,45),(22,200,150,104,46),(23,201,151,105,47),(24,202,152,106,48),(25,203,153,107,49),(26,204,154,108,50),(27,205,155,109,51),(28,165,156,110,52),(29,166,157,111,53),(30,167,158,112,54),(31,168,159,113,55),(32,169,160,114,56),(33,170,161,115,57),(34,171,162,116,58),(35,172,163,117,59),(36,173,164,118,60),(37,174,124,119,61),(38,175,125,120,62),(39,176,126,121,63),(40,177,127,122,64),(41,178,128,123,65)], [(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)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(42,48),(43,47),(44,46),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,70),(62,69),(63,68),(64,67),(65,66),(83,123),(84,122),(85,121),(86,120),(87,119),(88,118),(89,117),(90,116),(91,115),(92,114),(93,113),(94,112),(95,111),(96,110),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(124,133),(125,132),(126,131),(127,130),(128,129),(134,164),(135,163),(136,162),(137,161),(138,160),(139,159),(140,158),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,150),(165,192),(166,191),(167,190),(168,189),(169,188),(170,187),(171,186),(172,185),(173,184),(174,183),(175,182),(176,181),(177,180),(178,179),(193,205),(194,204),(195,203),(196,202),(197,201),(198,200)]])
110 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 41A | ··· | 41T | 205A | ··· | 205CB |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 41 | ··· | 41 | 205 | ··· | 205 |
| size | 1 | 41 | 1 | 1 | 1 | 1 | 41 | 41 | 41 | 41 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D41 | C5×D41 |
| kernel | C5×D41 | C205 | D41 | C41 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 20 | 80 |
Matrix representation of C5×D41 ►in GL2(𝔽821) generated by
| 161 | 0 |
| 0 | 161 |
| 349 | 1 |
| 820 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(821))| [161,0,0,161],[349,820,1,0],[0,1,1,0] >;
C5×D41 in GAP, Magma, Sage, TeX
C_5\times D_{41} % in TeX
G:=Group("C5xD41"); // GroupNames label
G:=SmallGroup(410,4);
// by ID
G=gap.SmallGroup(410,4);
# by ID
G:=PCGroup([3,-2,-5,-41,3602]);
// Polycyclic
G:=Group<a,b,c|a^5=b^41=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export