direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C41, C5⋊C82, C205⋊3C2, SmallGroup(410,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C41 |
Generators and relations for D5×C41
G = < a,b,c | a41=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C41
►On 205 points
Generators in S205
(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)
(1 184 78 154 98)(2 185 79 155 99)(3 186 80 156 100)(4 187 81 157 101)(5 188 82 158 102)(6 189 42 159 103)(7 190 43 160 104)(8 191 44 161 105)(9 192 45 162 106)(10 193 46 163 107)(11 194 47 164 108)(12 195 48 124 109)(13 196 49 125 110)(14 197 50 126 111)(15 198 51 127 112)(16 199 52 128 113)(17 200 53 129 114)(18 201 54 130 115)(19 202 55 131 116)(20 203 56 132 117)(21 204 57 133 118)(22 205 58 134 119)(23 165 59 135 120)(24 166 60 136 121)(25 167 61 137 122)(26 168 62 138 123)(27 169 63 139 83)(28 170 64 140 84)(29 171 65 141 85)(30 172 66 142 86)(31 173 67 143 87)(32 174 68 144 88)(33 175 69 145 89)(34 176 70 146 90)(35 177 71 147 91)(36 178 72 148 92)(37 179 73 149 93)(38 180 74 150 94)(39 181 75 151 95)(40 182 76 152 96)(41 183 77 153 97)
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(124 195)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(131 202)(132 203)(133 204)(134 205)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)(157 187)(158 188)(159 189)(160 190)(161 191)(162 192)(163 193)(164 194)
G:=sub<Sym(205)| (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), (1,184,78,154,98)(2,185,79,155,99)(3,186,80,156,100)(4,187,81,157,101)(5,188,82,158,102)(6,189,42,159,103)(7,190,43,160,104)(8,191,44,161,105)(9,192,45,162,106)(10,193,46,163,107)(11,194,47,164,108)(12,195,48,124,109)(13,196,49,125,110)(14,197,50,126,111)(15,198,51,127,112)(16,199,52,128,113)(17,200,53,129,114)(18,201,54,130,115)(19,202,55,131,116)(20,203,56,132,117)(21,204,57,133,118)(22,205,58,134,119)(23,165,59,135,120)(24,166,60,136,121)(25,167,61,137,122)(26,168,62,138,123)(27,169,63,139,83)(28,170,64,140,84)(29,171,65,141,85)(30,172,66,142,86)(31,173,67,143,87)(32,174,68,144,88)(33,175,69,145,89)(34,176,70,146,90)(35,177,71,147,91)(36,178,72,148,92)(37,179,73,149,93)(38,180,74,150,94)(39,181,75,151,95)(40,182,76,152,96)(41,183,77,153,97), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190)(161,191)(162,192)(163,193)(164,194)>;
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), (1,184,78,154,98)(2,185,79,155,99)(3,186,80,156,100)(4,187,81,157,101)(5,188,82,158,102)(6,189,42,159,103)(7,190,43,160,104)(8,191,44,161,105)(9,192,45,162,106)(10,193,46,163,107)(11,194,47,164,108)(12,195,48,124,109)(13,196,49,125,110)(14,197,50,126,111)(15,198,51,127,112)(16,199,52,128,113)(17,200,53,129,114)(18,201,54,130,115)(19,202,55,131,116)(20,203,56,132,117)(21,204,57,133,118)(22,205,58,134,119)(23,165,59,135,120)(24,166,60,136,121)(25,167,61,137,122)(26,168,62,138,123)(27,169,63,139,83)(28,170,64,140,84)(29,171,65,141,85)(30,172,66,142,86)(31,173,67,143,87)(32,174,68,144,88)(33,175,69,145,89)(34,176,70,146,90)(35,177,71,147,91)(36,178,72,148,92)(37,179,73,149,93)(38,180,74,150,94)(39,181,75,151,95)(40,182,76,152,96)(41,183,77,153,97), (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,121)(25,122)(26,123)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190)(161,191)(162,192)(163,193)(164,194) );
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)], [(1,184,78,154,98),(2,185,79,155,99),(3,186,80,156,100),(4,187,81,157,101),(5,188,82,158,102),(6,189,42,159,103),(7,190,43,160,104),(8,191,44,161,105),(9,192,45,162,106),(10,193,46,163,107),(11,194,47,164,108),(12,195,48,124,109),(13,196,49,125,110),(14,197,50,126,111),(15,198,51,127,112),(16,199,52,128,113),(17,200,53,129,114),(18,201,54,130,115),(19,202,55,131,116),(20,203,56,132,117),(21,204,57,133,118),(22,205,58,134,119),(23,165,59,135,120),(24,166,60,136,121),(25,167,61,137,122),(26,168,62,138,123),(27,169,63,139,83),(28,170,64,140,84),(29,171,65,141,85),(30,172,66,142,86),(31,173,67,143,87),(32,174,68,144,88),(33,175,69,145,89),(34,176,70,146,90),(35,177,71,147,91),(36,178,72,148,92),(37,179,73,149,93),(38,180,74,150,94),(39,181,75,151,95),(40,182,76,152,96),(41,183,77,153,97)], [(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,113),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,121),(25,122),(26,123),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(124,195),(125,196),(126,197),(127,198),(128,199),(129,200),(130,201),(131,202),(132,203),(133,204),(134,205),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(151,181),(152,182),(153,183),(154,184),(155,185),(156,186),(157,187),(158,188),(159,189),(160,190),(161,191),(162,192),(163,193),(164,194)]])
164 conjugacy classes
| class | 1 | 2 | 5A | 5B | 41A | ··· | 41AN | 82A | ··· | 82AN | 205A | ··· | 205CB |
| order | 1 | 2 | 5 | 5 | 41 | ··· | 41 | 82 | ··· | 82 | 205 | ··· | 205 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
164 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C41 | C82 | D5 | D5×C41 |
| kernel | D5×C41 | C205 | D5 | C5 | C41 | C1 |
| # reps | 1 | 1 | 40 | 40 | 2 | 80 |
Matrix representation of D5×C41 ►in GL2(𝔽821) generated by
| 566 | 0 |
| 0 | 566 |
| 0 | 1 |
| 820 | 212 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(821))| [566,0,0,566],[0,820,1,212],[0,1,1,0] >;
D5×C41 in GAP, Magma, Sage, TeX
D_5\times C_{41} % in TeX
G:=Group("D5xC41"); // GroupNames label
G:=SmallGroup(410,3);
// by ID
G=gap.SmallGroup(410,3);
# by ID
G:=PCGroup([3,-2,-41,-5,2954]);
// Polycyclic
G:=Group<a,b,c|a^41=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export