direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D35, C70⋊5C6, C42⋊2D5, C30⋊2D7, C15⋊8D14, C21⋊8D10, C210⋊2C2, C105⋊9C22, C10⋊(C3×D7), C5⋊2(C6×D7), C7⋊5(C6×D5), C35⋊7(C2×C6), C14⋊3(C3×D5), SmallGroup(420,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C6×D35 |
Generators and relations for C6×D35
G = < a,b,c | a6=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D35
►On 210 points
Generators in S210
(1 163 104 131 42 191)(2 164 105 132 43 192)(3 165 71 133 44 193)(4 166 72 134 45 194)(5 167 73 135 46 195)(6 168 74 136 47 196)(7 169 75 137 48 197)(8 170 76 138 49 198)(9 171 77 139 50 199)(10 172 78 140 51 200)(11 173 79 106 52 201)(12 174 80 107 53 202)(13 175 81 108 54 203)(14 141 82 109 55 204)(15 142 83 110 56 205)(16 143 84 111 57 206)(17 144 85 112 58 207)(18 145 86 113 59 208)(19 146 87 114 60 209)(20 147 88 115 61 210)(21 148 89 116 62 176)(22 149 90 117 63 177)(23 150 91 118 64 178)(24 151 92 119 65 179)(25 152 93 120 66 180)(26 153 94 121 67 181)(27 154 95 122 68 182)(28 155 96 123 69 183)(29 156 97 124 70 184)(30 157 98 125 36 185)(31 158 99 126 37 186)(32 159 100 127 38 187)(33 160 101 128 39 188)(34 161 102 129 40 189)(35 162 103 130 41 190)
(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)
(1 130)(2 129)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 122)(10 121)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 140)(27 139)(28 138)(29 137)(30 136)(31 135)(32 134)(33 133)(34 132)(35 131)(36 168)(37 167)(38 166)(39 165)(40 164)(41 163)(42 162)(43 161)(44 160)(45 159)(46 158)(47 157)(48 156)(49 155)(50 154)(51 153)(52 152)(53 151)(54 150)(55 149)(56 148)(57 147)(58 146)(59 145)(60 144)(61 143)(62 142)(63 141)(64 175)(65 174)(66 173)(67 172)(68 171)(69 170)(70 169)(71 188)(72 187)(73 186)(74 185)(75 184)(76 183)(77 182)(78 181)(79 180)(80 179)(81 178)(82 177)(83 176)(84 210)(85 209)(86 208)(87 207)(88 206)(89 205)(90 204)(91 203)(92 202)(93 201)(94 200)(95 199)(96 198)(97 197)(98 196)(99 195)(100 194)(101 193)(102 192)(103 191)(104 190)(105 189)
G:=sub<Sym(210)| (1,163,104,131,42,191)(2,164,105,132,43,192)(3,165,71,133,44,193)(4,166,72,134,45,194)(5,167,73,135,46,195)(6,168,74,136,47,196)(7,169,75,137,48,197)(8,170,76,138,49,198)(9,171,77,139,50,199)(10,172,78,140,51,200)(11,173,79,106,52,201)(12,174,80,107,53,202)(13,175,81,108,54,203)(14,141,82,109,55,204)(15,142,83,110,56,205)(16,143,84,111,57,206)(17,144,85,112,58,207)(18,145,86,113,59,208)(19,146,87,114,60,209)(20,147,88,115,61,210)(21,148,89,116,62,176)(22,149,90,117,63,177)(23,150,91,118,64,178)(24,151,92,119,65,179)(25,152,93,120,66,180)(26,153,94,121,67,181)(27,154,95,122,68,182)(28,155,96,123,69,183)(29,156,97,124,70,184)(30,157,98,125,36,185)(31,158,99,126,37,186)(32,159,100,127,38,187)(33,160,101,128,39,188)(34,161,102,129,40,189)(35,162,103,130,41,190), (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), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,140)(27,139)(28,138)(29,137)(30,136)(31,135)(32,134)(33,133)(34,132)(35,131)(36,168)(37,167)(38,166)(39,165)(40,164)(41,163)(42,162)(43,161)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(61,143)(62,142)(63,141)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,182)(78,181)(79,180)(80,179)(81,178)(82,177)(83,176)(84,210)(85,209)(86,208)(87,207)(88,206)(89,205)(90,204)(91,203)(92,202)(93,201)(94,200)(95,199)(96,198)(97,197)(98,196)(99,195)(100,194)(101,193)(102,192)(103,191)(104,190)(105,189)>;
G:=Group( (1,163,104,131,42,191)(2,164,105,132,43,192)(3,165,71,133,44,193)(4,166,72,134,45,194)(5,167,73,135,46,195)(6,168,74,136,47,196)(7,169,75,137,48,197)(8,170,76,138,49,198)(9,171,77,139,50,199)(10,172,78,140,51,200)(11,173,79,106,52,201)(12,174,80,107,53,202)(13,175,81,108,54,203)(14,141,82,109,55,204)(15,142,83,110,56,205)(16,143,84,111,57,206)(17,144,85,112,58,207)(18,145,86,113,59,208)(19,146,87,114,60,209)(20,147,88,115,61,210)(21,148,89,116,62,176)(22,149,90,117,63,177)(23,150,91,118,64,178)(24,151,92,119,65,179)(25,152,93,120,66,180)(26,153,94,121,67,181)(27,154,95,122,68,182)(28,155,96,123,69,183)(29,156,97,124,70,184)(30,157,98,125,36,185)(31,158,99,126,37,186)(32,159,100,127,38,187)(33,160,101,128,39,188)(34,161,102,129,40,189)(35,162,103,130,41,190), (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), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,140)(27,139)(28,138)(29,137)(30,136)(31,135)(32,134)(33,133)(34,132)(35,131)(36,168)(37,167)(38,166)(39,165)(40,164)(41,163)(42,162)(43,161)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(61,143)(62,142)(63,141)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,182)(78,181)(79,180)(80,179)(81,178)(82,177)(83,176)(84,210)(85,209)(86,208)(87,207)(88,206)(89,205)(90,204)(91,203)(92,202)(93,201)(94,200)(95,199)(96,198)(97,197)(98,196)(99,195)(100,194)(101,193)(102,192)(103,191)(104,190)(105,189) );
G=PermutationGroup([[(1,163,104,131,42,191),(2,164,105,132,43,192),(3,165,71,133,44,193),(4,166,72,134,45,194),(5,167,73,135,46,195),(6,168,74,136,47,196),(7,169,75,137,48,197),(8,170,76,138,49,198),(9,171,77,139,50,199),(10,172,78,140,51,200),(11,173,79,106,52,201),(12,174,80,107,53,202),(13,175,81,108,54,203),(14,141,82,109,55,204),(15,142,83,110,56,205),(16,143,84,111,57,206),(17,144,85,112,58,207),(18,145,86,113,59,208),(19,146,87,114,60,209),(20,147,88,115,61,210),(21,148,89,116,62,176),(22,149,90,117,63,177),(23,150,91,118,64,178),(24,151,92,119,65,179),(25,152,93,120,66,180),(26,153,94,121,67,181),(27,154,95,122,68,182),(28,155,96,123,69,183),(29,156,97,124,70,184),(30,157,98,125,36,185),(31,158,99,126,37,186),(32,159,100,127,38,187),(33,160,101,128,39,188),(34,161,102,129,40,189),(35,162,103,130,41,190)], [(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)], [(1,130),(2,129),(3,128),(4,127),(5,126),(6,125),(7,124),(8,123),(9,122),(10,121),(11,120),(12,119),(13,118),(14,117),(15,116),(16,115),(17,114),(18,113),(19,112),(20,111),(21,110),(22,109),(23,108),(24,107),(25,106),(26,140),(27,139),(28,138),(29,137),(30,136),(31,135),(32,134),(33,133),(34,132),(35,131),(36,168),(37,167),(38,166),(39,165),(40,164),(41,163),(42,162),(43,161),(44,160),(45,159),(46,158),(47,157),(48,156),(49,155),(50,154),(51,153),(52,152),(53,151),(54,150),(55,149),(56,148),(57,147),(58,146),(59,145),(60,144),(61,143),(62,142),(63,141),(64,175),(65,174),(66,173),(67,172),(68,171),(69,170),(70,169),(71,188),(72,187),(73,186),(74,185),(75,184),(76,183),(77,182),(78,181),(79,180),(80,179),(81,178),(82,177),(83,176),(84,210),(85,209),(86,208),(87,207),(88,206),(89,205),(90,204),(91,203),(92,202),(93,201),(94,200),(95,199),(96,198),(97,197),(98,196),(99,195),(100,194),(101,193),(102,192),(103,191),(104,190),(105,189)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 10A | 10B | 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 | 3 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 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 | 35 | 35 | 1 | 1 | 2 | 2 | 1 | 1 | 35 | 35 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 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 | C3 | C6 | C6 | D5 | D7 | D10 | D14 | C3×D5 | C3×D7 | C6×D5 | D35 | C6×D7 | D70 | C3×D35 | C6×D35 |
| kernel | C6×D35 | C3×D35 | C210 | D70 | D35 | C70 | C42 | C30 | C21 | C15 | C14 | C10 | C7 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 3 | 2 | 3 | 4 | 6 | 4 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of C6×D35 ►in GL3(𝔽211) generated by
| 210 | 0 | 0 |
| 0 | 196 | 0 |
| 0 | 0 | 196 |
| 1 | 0 | 0 |
| 0 | 184 | 9 |
| 0 | 202 | 128 |
| 1 | 0 | 0 |
| 0 | 184 | 9 |
| 0 | 177 | 27 |
G:=sub<GL(3,GF(211))| [210,0,0,0,196,0,0,0,196],[1,0,0,0,184,202,0,9,128],[1,0,0,0,184,177,0,9,27] >;
C6×D35 in GAP, Magma, Sage, TeX
C_6\times D_{35} % in TeX
G:=Group("C6xD35"); // GroupNames label
G:=SmallGroup(420,36);
// by ID
G=gap.SmallGroup(420,36);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,963,9004]);
// Polycyclic
G:=Group<a,b,c|a^6=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export