direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D21, C35⋊8D6, C70⋊2S3, C30⋊3D7, C15⋊9D14, C210⋊3C2, C42⋊1C10, C105⋊10C22, C6⋊(C5×D7), C14⋊(C5×S3), C7⋊2(S3×C10), C3⋊2(C10×D7), C21⋊2(C2×C10), SmallGroup(420,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C10×D21 |
Generators and relations for C10×D21
G = < a,b,c | a10=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×D21
►On 210 points
Generators in S210
(1 155 88 137 71 124 58 199 32 182)(2 156 89 138 72 125 59 200 33 183)(3 157 90 139 73 126 60 201 34 184)(4 158 91 140 74 106 61 202 35 185)(5 159 92 141 75 107 62 203 36 186)(6 160 93 142 76 108 63 204 37 187)(7 161 94 143 77 109 43 205 38 188)(8 162 95 144 78 110 44 206 39 189)(9 163 96 145 79 111 45 207 40 169)(10 164 97 146 80 112 46 208 41 170)(11 165 98 147 81 113 47 209 42 171)(12 166 99 127 82 114 48 210 22 172)(13 167 100 128 83 115 49 190 23 173)(14 168 101 129 84 116 50 191 24 174)(15 148 102 130 64 117 51 192 25 175)(16 149 103 131 65 118 52 193 26 176)(17 150 104 132 66 119 53 194 27 177)(18 151 105 133 67 120 54 195 28 178)(19 152 85 134 68 121 55 196 29 179)(20 153 86 135 69 122 56 197 30 180)(21 154 87 136 70 123 57 198 31 181)
(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)
(1 123)(2 122)(3 121)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 126)(20 125)(21 124)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 140)(29 139)(30 138)(31 137)(32 136)(33 135)(34 134)(35 133)(36 132)(37 131)(38 130)(39 129)(40 128)(41 127)(42 147)(43 148)(44 168)(45 167)(46 166)(47 165)(48 164)(49 163)(50 162)(51 161)(52 160)(53 159)(54 158)(55 157)(56 156)(57 155)(58 154)(59 153)(60 152)(61 151)(62 150)(63 149)(64 188)(65 187)(66 186)(67 185)(68 184)(69 183)(70 182)(71 181)(72 180)(73 179)(74 178)(75 177)(76 176)(77 175)(78 174)(79 173)(80 172)(81 171)(82 170)(83 169)(84 189)(85 201)(86 200)(87 199)(88 198)(89 197)(90 196)(91 195)(92 194)(93 193)(94 192)(95 191)(96 190)(97 210)(98 209)(99 208)(100 207)(101 206)(102 205)(103 204)(104 203)(105 202)
G:=sub<Sym(210)| (1,155,88,137,71,124,58,199,32,182)(2,156,89,138,72,125,59,200,33,183)(3,157,90,139,73,126,60,201,34,184)(4,158,91,140,74,106,61,202,35,185)(5,159,92,141,75,107,62,203,36,186)(6,160,93,142,76,108,63,204,37,187)(7,161,94,143,77,109,43,205,38,188)(8,162,95,144,78,110,44,206,39,189)(9,163,96,145,79,111,45,207,40,169)(10,164,97,146,80,112,46,208,41,170)(11,165,98,147,81,113,47,209,42,171)(12,166,99,127,82,114,48,210,22,172)(13,167,100,128,83,115,49,190,23,173)(14,168,101,129,84,116,50,191,24,174)(15,148,102,130,64,117,51,192,25,175)(16,149,103,131,65,118,52,193,26,176)(17,150,104,132,66,119,53,194,27,177)(18,151,105,133,67,120,54,195,28,178)(19,152,85,134,68,121,55,196,29,179)(20,153,86,135,69,122,56,197,30,180)(21,154,87,136,70,123,57,198,31,181), (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), (1,123)(2,122)(3,121)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,126)(20,125)(21,124)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,127)(42,147)(43,148)(44,168)(45,167)(46,166)(47,165)(48,164)(49,163)(50,162)(51,161)(52,160)(53,159)(54,158)(55,157)(56,156)(57,155)(58,154)(59,153)(60,152)(61,151)(62,150)(63,149)(64,188)(65,187)(66,186)(67,185)(68,184)(69,183)(70,182)(71,181)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(81,171)(82,170)(83,169)(84,189)(85,201)(86,200)(87,199)(88,198)(89,197)(90,196)(91,195)(92,194)(93,193)(94,192)(95,191)(96,190)(97,210)(98,209)(99,208)(100,207)(101,206)(102,205)(103,204)(104,203)(105,202)>;
G:=Group( (1,155,88,137,71,124,58,199,32,182)(2,156,89,138,72,125,59,200,33,183)(3,157,90,139,73,126,60,201,34,184)(4,158,91,140,74,106,61,202,35,185)(5,159,92,141,75,107,62,203,36,186)(6,160,93,142,76,108,63,204,37,187)(7,161,94,143,77,109,43,205,38,188)(8,162,95,144,78,110,44,206,39,189)(9,163,96,145,79,111,45,207,40,169)(10,164,97,146,80,112,46,208,41,170)(11,165,98,147,81,113,47,209,42,171)(12,166,99,127,82,114,48,210,22,172)(13,167,100,128,83,115,49,190,23,173)(14,168,101,129,84,116,50,191,24,174)(15,148,102,130,64,117,51,192,25,175)(16,149,103,131,65,118,52,193,26,176)(17,150,104,132,66,119,53,194,27,177)(18,151,105,133,67,120,54,195,28,178)(19,152,85,134,68,121,55,196,29,179)(20,153,86,135,69,122,56,197,30,180)(21,154,87,136,70,123,57,198,31,181), (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), (1,123)(2,122)(3,121)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,126)(20,125)(21,124)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,127)(42,147)(43,148)(44,168)(45,167)(46,166)(47,165)(48,164)(49,163)(50,162)(51,161)(52,160)(53,159)(54,158)(55,157)(56,156)(57,155)(58,154)(59,153)(60,152)(61,151)(62,150)(63,149)(64,188)(65,187)(66,186)(67,185)(68,184)(69,183)(70,182)(71,181)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(81,171)(82,170)(83,169)(84,189)(85,201)(86,200)(87,199)(88,198)(89,197)(90,196)(91,195)(92,194)(93,193)(94,192)(95,191)(96,190)(97,210)(98,209)(99,208)(100,207)(101,206)(102,205)(103,204)(104,203)(105,202) );
G=PermutationGroup([[(1,155,88,137,71,124,58,199,32,182),(2,156,89,138,72,125,59,200,33,183),(3,157,90,139,73,126,60,201,34,184),(4,158,91,140,74,106,61,202,35,185),(5,159,92,141,75,107,62,203,36,186),(6,160,93,142,76,108,63,204,37,187),(7,161,94,143,77,109,43,205,38,188),(8,162,95,144,78,110,44,206,39,189),(9,163,96,145,79,111,45,207,40,169),(10,164,97,146,80,112,46,208,41,170),(11,165,98,147,81,113,47,209,42,171),(12,166,99,127,82,114,48,210,22,172),(13,167,100,128,83,115,49,190,23,173),(14,168,101,129,84,116,50,191,24,174),(15,148,102,130,64,117,51,192,25,175),(16,149,103,131,65,118,52,193,26,176),(17,150,104,132,66,119,53,194,27,177),(18,151,105,133,67,120,54,195,28,178),(19,152,85,134,68,121,55,196,29,179),(20,153,86,135,69,122,56,197,30,180),(21,154,87,136,70,123,57,198,31,181)], [(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)], [(1,123),(2,122),(3,121),(4,120),(5,119),(6,118),(7,117),(8,116),(9,115),(10,114),(11,113),(12,112),(13,111),(14,110),(15,109),(16,108),(17,107),(18,106),(19,126),(20,125),(21,124),(22,146),(23,145),(24,144),(25,143),(26,142),(27,141),(28,140),(29,139),(30,138),(31,137),(32,136),(33,135),(34,134),(35,133),(36,132),(37,131),(38,130),(39,129),(40,128),(41,127),(42,147),(43,148),(44,168),(45,167),(46,166),(47,165),(48,164),(49,163),(50,162),(51,161),(52,160),(53,159),(54,158),(55,157),(56,156),(57,155),(58,154),(59,153),(60,152),(61,151),(62,150),(63,149),(64,188),(65,187),(66,186),(67,185),(68,184),(69,183),(70,182),(71,181),(72,180),(73,179),(74,178),(75,177),(76,176),(77,175),(78,174),(79,173),(80,172),(81,171),(82,170),(83,169),(84,189),(85,201),(86,200),(87,199),(88,198),(89,197),(90,196),(91,195),(92,194),(93,193),(94,192),(95,191),(96,190),(97,210),(98,209),(99,208),(100,207),(101,206),(102,205),(103,204),(104,203),(105,202)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 6 | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 14A | 14B | 14C | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 30A | 30B | 30C | 30D | 35A | ··· | 35L | 42A | ··· | 42F | 70A | ··· | 70L | 105A | ··· | 105X | 210A | ··· | 210X |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 6 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 14 | 14 | 14 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 30 | 30 | 30 | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| size | 1 | 1 | 21 | 21 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 21 | ··· | 21 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | D7 | D14 | C5×S3 | D21 | S3×C10 | C5×D7 | D42 | C10×D7 | C5×D21 | C10×D21 |
| kernel | C10×D21 | C5×D21 | C210 | D42 | D21 | C42 | C70 | C35 | C30 | C15 | C14 | C10 | C7 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 3 | 3 | 4 | 6 | 4 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of C10×D21 ►in GL2(𝔽41) generated by
| 31 | 0 |
| 0 | 31 |
| 40 | 18 |
| 4 | 9 |
| 9 | 20 |
| 37 | 32 |
G:=sub<GL(2,GF(41))| [31,0,0,31],[40,4,18,9],[9,37,20,32] >;
C10×D21 in GAP, Magma, Sage, TeX
C_{10}\times D_{21} % in TeX
G:=Group("C10xD21"); // GroupNames label
G:=SmallGroup(420,38);
// by ID
G=gap.SmallGroup(420,38);
# by ID
G:=PCGroup([5,-2,-2,-5,-3,-7,803,9004]);
// Polycyclic
G:=Group<a,b,c|a^10=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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