direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C14×D15, C35⋊9D6, C70⋊3S3, C42⋊3D5, C21⋊9D10, C30⋊1C14, C210⋊4C2, C105⋊11C22, C6⋊(C7×D5), C10⋊(S3×C7), C5⋊2(S3×C14), C3⋊2(D5×C14), C15⋊2(C2×C14), SmallGroup(420,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C14×D15 |
Generators and relations for C14×D15
G = < a,b,c | a14=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C14×D15
►On 210 points
Generators in S210
(1 155 95 137 80 125 69 107 54 208 35 185 24 174)(2 156 96 138 81 126 70 108 55 209 36 186 25 175)(3 157 97 139 82 127 71 109 56 210 37 187 26 176)(4 158 98 140 83 128 72 110 57 196 38 188 27 177)(5 159 99 141 84 129 73 111 58 197 39 189 28 178)(6 160 100 142 85 130 74 112 59 198 40 190 29 179)(7 161 101 143 86 131 75 113 60 199 41 191 30 180)(8 162 102 144 87 132 61 114 46 200 42 192 16 166)(9 163 103 145 88 133 62 115 47 201 43 193 17 167)(10 164 104 146 89 134 63 116 48 202 44 194 18 168)(11 165 105 147 90 135 64 117 49 203 45 195 19 169)(12 151 91 148 76 121 65 118 50 204 31 181 20 170)(13 152 92 149 77 122 66 119 51 205 32 182 21 171)(14 153 93 150 78 123 67 120 52 206 33 183 22 172)(15 154 94 136 79 124 68 106 53 207 34 184 23 173)
(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)
(1 106)(2 120)(3 119)(4 118)(5 117)(6 116)(7 115)(8 114)(9 113)(10 112)(11 111)(12 110)(13 109)(14 108)(15 107)(16 132)(17 131)(18 130)(19 129)(20 128)(21 127)(22 126)(23 125)(24 124)(25 123)(26 122)(27 121)(28 135)(29 134)(30 133)(31 140)(32 139)(33 138)(34 137)(35 136)(36 150)(37 149)(38 148)(39 147)(40 146)(41 145)(42 144)(43 143)(44 142)(45 141)(46 162)(47 161)(48 160)(49 159)(50 158)(51 157)(52 156)(53 155)(54 154)(55 153)(56 152)(57 151)(58 165)(59 164)(60 163)(61 166)(62 180)(63 179)(64 178)(65 177)(66 176)(67 175)(68 174)(69 173)(70 172)(71 171)(72 170)(73 169)(74 168)(75 167)(76 188)(77 187)(78 186)(79 185)(80 184)(81 183)(82 182)(83 181)(84 195)(85 194)(86 193)(87 192)(88 191)(89 190)(90 189)(91 196)(92 210)(93 209)(94 208)(95 207)(96 206)(97 205)(98 204)(99 203)(100 202)(101 201)(102 200)(103 199)(104 198)(105 197)
G:=sub<Sym(210)| (1,155,95,137,80,125,69,107,54,208,35,185,24,174)(2,156,96,138,81,126,70,108,55,209,36,186,25,175)(3,157,97,139,82,127,71,109,56,210,37,187,26,176)(4,158,98,140,83,128,72,110,57,196,38,188,27,177)(5,159,99,141,84,129,73,111,58,197,39,189,28,178)(6,160,100,142,85,130,74,112,59,198,40,190,29,179)(7,161,101,143,86,131,75,113,60,199,41,191,30,180)(8,162,102,144,87,132,61,114,46,200,42,192,16,166)(9,163,103,145,88,133,62,115,47,201,43,193,17,167)(10,164,104,146,89,134,63,116,48,202,44,194,18,168)(11,165,105,147,90,135,64,117,49,203,45,195,19,169)(12,151,91,148,76,121,65,118,50,204,31,181,20,170)(13,152,92,149,77,122,66,119,51,205,32,182,21,171)(14,153,93,150,78,123,67,120,52,206,33,183,22,172)(15,154,94,136,79,124,68,106,53,207,34,184,23,173), (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), (1,106)(2,120)(3,119)(4,118)(5,117)(6,116)(7,115)(8,114)(9,113)(10,112)(11,111)(12,110)(13,109)(14,108)(15,107)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,135)(29,134)(30,133)(31,140)(32,139)(33,138)(34,137)(35,136)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,143)(44,142)(45,141)(46,162)(47,161)(48,160)(49,159)(50,158)(51,157)(52,156)(53,155)(54,154)(55,153)(56,152)(57,151)(58,165)(59,164)(60,163)(61,166)(62,180)(63,179)(64,178)(65,177)(66,176)(67,175)(68,174)(69,173)(70,172)(71,171)(72,170)(73,169)(74,168)(75,167)(76,188)(77,187)(78,186)(79,185)(80,184)(81,183)(82,182)(83,181)(84,195)(85,194)(86,193)(87,192)(88,191)(89,190)(90,189)(91,196)(92,210)(93,209)(94,208)(95,207)(96,206)(97,205)(98,204)(99,203)(100,202)(101,201)(102,200)(103,199)(104,198)(105,197)>;
G:=Group( (1,155,95,137,80,125,69,107,54,208,35,185,24,174)(2,156,96,138,81,126,70,108,55,209,36,186,25,175)(3,157,97,139,82,127,71,109,56,210,37,187,26,176)(4,158,98,140,83,128,72,110,57,196,38,188,27,177)(5,159,99,141,84,129,73,111,58,197,39,189,28,178)(6,160,100,142,85,130,74,112,59,198,40,190,29,179)(7,161,101,143,86,131,75,113,60,199,41,191,30,180)(8,162,102,144,87,132,61,114,46,200,42,192,16,166)(9,163,103,145,88,133,62,115,47,201,43,193,17,167)(10,164,104,146,89,134,63,116,48,202,44,194,18,168)(11,165,105,147,90,135,64,117,49,203,45,195,19,169)(12,151,91,148,76,121,65,118,50,204,31,181,20,170)(13,152,92,149,77,122,66,119,51,205,32,182,21,171)(14,153,93,150,78,123,67,120,52,206,33,183,22,172)(15,154,94,136,79,124,68,106,53,207,34,184,23,173), (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), (1,106)(2,120)(3,119)(4,118)(5,117)(6,116)(7,115)(8,114)(9,113)(10,112)(11,111)(12,110)(13,109)(14,108)(15,107)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,135)(29,134)(30,133)(31,140)(32,139)(33,138)(34,137)(35,136)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,143)(44,142)(45,141)(46,162)(47,161)(48,160)(49,159)(50,158)(51,157)(52,156)(53,155)(54,154)(55,153)(56,152)(57,151)(58,165)(59,164)(60,163)(61,166)(62,180)(63,179)(64,178)(65,177)(66,176)(67,175)(68,174)(69,173)(70,172)(71,171)(72,170)(73,169)(74,168)(75,167)(76,188)(77,187)(78,186)(79,185)(80,184)(81,183)(82,182)(83,181)(84,195)(85,194)(86,193)(87,192)(88,191)(89,190)(90,189)(91,196)(92,210)(93,209)(94,208)(95,207)(96,206)(97,205)(98,204)(99,203)(100,202)(101,201)(102,200)(103,199)(104,198)(105,197) );
G=PermutationGroup([[(1,155,95,137,80,125,69,107,54,208,35,185,24,174),(2,156,96,138,81,126,70,108,55,209,36,186,25,175),(3,157,97,139,82,127,71,109,56,210,37,187,26,176),(4,158,98,140,83,128,72,110,57,196,38,188,27,177),(5,159,99,141,84,129,73,111,58,197,39,189,28,178),(6,160,100,142,85,130,74,112,59,198,40,190,29,179),(7,161,101,143,86,131,75,113,60,199,41,191,30,180),(8,162,102,144,87,132,61,114,46,200,42,192,16,166),(9,163,103,145,88,133,62,115,47,201,43,193,17,167),(10,164,104,146,89,134,63,116,48,202,44,194,18,168),(11,165,105,147,90,135,64,117,49,203,45,195,19,169),(12,151,91,148,76,121,65,118,50,204,31,181,20,170),(13,152,92,149,77,122,66,119,51,205,32,182,21,171),(14,153,93,150,78,123,67,120,52,206,33,183,22,172),(15,154,94,136,79,124,68,106,53,207,34,184,23,173)], [(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)], [(1,106),(2,120),(3,119),(4,118),(5,117),(6,116),(7,115),(8,114),(9,113),(10,112),(11,111),(12,110),(13,109),(14,108),(15,107),(16,132),(17,131),(18,130),(19,129),(20,128),(21,127),(22,126),(23,125),(24,124),(25,123),(26,122),(27,121),(28,135),(29,134),(30,133),(31,140),(32,139),(33,138),(34,137),(35,136),(36,150),(37,149),(38,148),(39,147),(40,146),(41,145),(42,144),(43,143),(44,142),(45,141),(46,162),(47,161),(48,160),(49,159),(50,158),(51,157),(52,156),(53,155),(54,154),(55,153),(56,152),(57,151),(58,165),(59,164),(60,163),(61,166),(62,180),(63,179),(64,178),(65,177),(66,176),(67,175),(68,174),(69,173),(70,172),(71,171),(72,170),(73,169),(74,168),(75,167),(76,188),(77,187),(78,186),(79,185),(80,184),(81,183),(82,182),(83,181),(84,195),(85,194),(86,193),(87,192),(88,191),(89,190),(90,189),(91,196),(92,210),(93,209),(94,208),(95,207),(96,206),(97,205),(98,204),(99,203),(100,202),(101,201),(102,200),(103,199),(104,198),(105,197)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 6 | 7A | ··· | 7F | 10A | 10B | 14A | ··· | 14F | 14G | ··· | 14R | 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 | 6 | 7 | ··· | 7 | 10 | 10 | 14 | ··· | 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 | 15 | 15 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 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 | C7 | C14 | C14 | S3 | D5 | D6 | D10 | D15 | S3×C7 | D30 | C7×D5 | S3×C14 | D5×C14 | C7×D15 | C14×D15 |
| kernel | C14×D15 | C7×D15 | C210 | D30 | D15 | C30 | C70 | C42 | C35 | C21 | C14 | C10 | C7 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 1 | 2 | 1 | 2 | 4 | 6 | 4 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of C14×D15 ►in GL2(𝔽29) generated by
| 9 | 0 |
| 0 | 9 |
| 19 | 22 |
| 16 | 14 |
| 14 | 14 |
| 13 | 15 |
G:=sub<GL(2,GF(29))| [9,0,0,9],[19,16,22,14],[14,13,14,15] >;
C14×D15 in GAP, Magma, Sage, TeX
C_{14}\times D_{15} % in TeX
G:=Group("C14xD15"); // GroupNames label
G:=SmallGroup(420,39);
// by ID
G=gap.SmallGroup(420,39);
# by ID
G:=PCGroup([5,-2,-2,-7,-3,-5,1123,8404]);
// Polycyclic
G:=Group<a,b,c|a^14=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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