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## G = C14×D15order 420 = 22·3·5·7

### Direct product of C14 and D15

Aliases: C14×D15, C359D6, C703S3, C423D5, C219D10, C301C14, C2104C2, C10511C22, C6⋊(C7×D5), C10⋊(S3×C7), C52(S3×C14), C32(D5×C14), C152(C2×C14), SmallGroup(420,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C14×D15
 Chief series C1 — C5 — C15 — C105 — C7×D15 — C14×D15
 Lower central C15 — C14×D15
 Upper central C1 — C14

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
(1 151 92 144 88 121 65 118 46 208 43 193 24 166)(2 152 93 145 89 122 66 119 47 209 44 194 25 167)(3 153 94 146 90 123 67 120 48 210 45 195 26 168)(4 154 95 147 76 124 68 106 49 196 31 181 27 169)(5 155 96 148 77 125 69 107 50 197 32 182 28 170)(6 156 97 149 78 126 70 108 51 198 33 183 29 171)(7 157 98 150 79 127 71 109 52 199 34 184 30 172)(8 158 99 136 80 128 72 110 53 200 35 185 16 173)(9 159 100 137 81 129 73 111 54 201 36 186 17 174)(10 160 101 138 82 130 74 112 55 202 37 187 18 175)(11 161 102 139 83 131 75 113 56 203 38 188 19 176)(12 162 103 140 84 132 61 114 57 204 39 189 20 177)(13 163 104 141 85 133 62 115 58 205 40 190 21 178)(14 164 105 142 86 134 63 116 59 206 41 191 22 179)(15 165 91 143 87 135 64 117 60 207 42 192 23 180)
(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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 110)(9 109)(10 108)(11 107)(12 106)(13 120)(14 119)(15 118)(16 128)(17 127)(18 126)(19 125)(20 124)(21 123)(22 122)(23 121)(24 135)(25 134)(26 133)(27 132)(28 131)(29 130)(30 129)(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)(46 165)(47 164)(48 163)(49 162)(50 161)(51 160)(52 159)(53 158)(54 157)(55 156)(56 155)(57 154)(58 153)(59 152)(60 151)(61 169)(62 168)(63 167)(64 166)(65 180)(66 179)(67 178)(68 177)(69 176)(70 175)(71 174)(72 173)(73 172)(74 171)(75 170)(76 189)(77 188)(78 187)(79 186)(80 185)(81 184)(82 183)(83 182)(84 181)(85 195)(86 194)(87 193)(88 192)(89 191)(90 190)(91 208)(92 207)(93 206)(94 205)(95 204)(96 203)(97 202)(98 201)(99 200)(100 199)(101 198)(102 197)(103 196)(104 210)(105 209)

G:=sub<Sym(210)| (1,151,92,144,88,121,65,118,46,208,43,193,24,166)(2,152,93,145,89,122,66,119,47,209,44,194,25,167)(3,153,94,146,90,123,67,120,48,210,45,195,26,168)(4,154,95,147,76,124,68,106,49,196,31,181,27,169)(5,155,96,148,77,125,69,107,50,197,32,182,28,170)(6,156,97,149,78,126,70,108,51,198,33,183,29,171)(7,157,98,150,79,127,71,109,52,199,34,184,30,172)(8,158,99,136,80,128,72,110,53,200,35,185,16,173)(9,159,100,137,81,129,73,111,54,201,36,186,17,174)(10,160,101,138,82,130,74,112,55,202,37,187,18,175)(11,161,102,139,83,131,75,113,56,203,38,188,19,176)(12,162,103,140,84,132,61,114,57,204,39,189,20,177)(13,163,104,141,85,133,62,115,58,205,40,190,21,178)(14,164,105,142,86,134,63,116,59,206,41,191,22,179)(15,165,91,143,87,135,64,117,60,207,42,192,23,180), (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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,108)(11,107)(12,106)(13,120)(14,119)(15,118)(16,128)(17,127)(18,126)(19,125)(20,124)(21,123)(22,122)(23,121)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(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)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,154)(58,153)(59,152)(60,151)(61,169)(62,168)(63,167)(64,166)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,208)(92,207)(93,206)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,196)(104,210)(105,209)>;

G:=Group( (1,151,92,144,88,121,65,118,46,208,43,193,24,166)(2,152,93,145,89,122,66,119,47,209,44,194,25,167)(3,153,94,146,90,123,67,120,48,210,45,195,26,168)(4,154,95,147,76,124,68,106,49,196,31,181,27,169)(5,155,96,148,77,125,69,107,50,197,32,182,28,170)(6,156,97,149,78,126,70,108,51,198,33,183,29,171)(7,157,98,150,79,127,71,109,52,199,34,184,30,172)(8,158,99,136,80,128,72,110,53,200,35,185,16,173)(9,159,100,137,81,129,73,111,54,201,36,186,17,174)(10,160,101,138,82,130,74,112,55,202,37,187,18,175)(11,161,102,139,83,131,75,113,56,203,38,188,19,176)(12,162,103,140,84,132,61,114,57,204,39,189,20,177)(13,163,104,141,85,133,62,115,58,205,40,190,21,178)(14,164,105,142,86,134,63,116,59,206,41,191,22,179)(15,165,91,143,87,135,64,117,60,207,42,192,23,180), (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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,108)(11,107)(12,106)(13,120)(14,119)(15,118)(16,128)(17,127)(18,126)(19,125)(20,124)(21,123)(22,122)(23,121)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(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)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,154)(58,153)(59,152)(60,151)(61,169)(62,168)(63,167)(64,166)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,208)(92,207)(93,206)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,196)(104,210)(105,209) );

G=PermutationGroup([(1,151,92,144,88,121,65,118,46,208,43,193,24,166),(2,152,93,145,89,122,66,119,47,209,44,194,25,167),(3,153,94,146,90,123,67,120,48,210,45,195,26,168),(4,154,95,147,76,124,68,106,49,196,31,181,27,169),(5,155,96,148,77,125,69,107,50,197,32,182,28,170),(6,156,97,149,78,126,70,108,51,198,33,183,29,171),(7,157,98,150,79,127,71,109,52,199,34,184,30,172),(8,158,99,136,80,128,72,110,53,200,35,185,16,173),(9,159,100,137,81,129,73,111,54,201,36,186,17,174),(10,160,101,138,82,130,74,112,55,202,37,187,18,175),(11,161,102,139,83,131,75,113,56,203,38,188,19,176),(12,162,103,140,84,132,61,114,57,204,39,189,20,177),(13,163,104,141,85,133,62,115,58,205,40,190,21,178),(14,164,105,142,86,134,63,116,59,206,41,191,22,179),(15,165,91,143,87,135,64,117,60,207,42,192,23,180)], [(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)], [(1,117),(2,116),(3,115),(4,114),(5,113),(6,112),(7,111),(8,110),(9,109),(10,108),(11,107),(12,106),(13,120),(14,119),(15,118),(16,128),(17,127),(18,126),(19,125),(20,124),(21,123),(22,122),(23,121),(24,135),(25,134),(26,133),(27,132),(28,131),(29,130),(30,129),(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),(46,165),(47,164),(48,163),(49,162),(50,161),(51,160),(52,159),(53,158),(54,157),(55,156),(56,155),(57,154),(58,153),(59,152),(60,151),(61,169),(62,168),(63,167),(64,166),(65,180),(66,179),(67,178),(68,177),(69,176),(70,175),(71,174),(72,173),(73,172),(74,171),(75,170),(76,189),(77,188),(78,187),(79,186),(80,185),(81,184),(82,183),(83,182),(84,181),(85,195),(86,194),(87,193),(88,192),(89,191),(90,190),(91,208),(92,207),(93,206),(94,205),(95,204),(96,203),(97,202),(98,201),(99,200),(100,199),(101,198),(102,197),(103,196),(104,210),(105,209)])

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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