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G = C10×D21order 420 = 22·3·5·7

Direct product of C10 and D21

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C10×D21, C358D6, C702S3, C303D7, C159D14, C2103C2, C421C10, C10510C22, C6⋊(C5×D7), C14⋊(C5×S3), C72(S3×C10), C32(C10×D7), C212(C2×C10), SmallGroup(420,38)

Series: Derived Chief Lower central Upper central

C1C21 — C10×D21
C1C7C21C105C5×D21 — C10×D21
C21 — C10×D21
C1C10

Generators and relations for C10×D21
 G = < a,b,c | a10=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >

21C2
21C2
21C22
7S3
7S3
21C10
21C10
3D7
3D7
7D6
21C2×C10
3D14
7C5×S3
7C5×S3
3C5×D7
3C5×D7
7S3×C10
3C10×D7

Smallest permutation representation of C10×D21
On 210 points
Generators in S210
(1 166 87 130 71 112 51 190 26 181)(2 167 88 131 72 113 52 191 27 182)(3 168 89 132 73 114 53 192 28 183)(4 148 90 133 74 115 54 193 29 184)(5 149 91 134 75 116 55 194 30 185)(6 150 92 135 76 117 56 195 31 186)(7 151 93 136 77 118 57 196 32 187)(8 152 94 137 78 119 58 197 33 188)(9 153 95 138 79 120 59 198 34 189)(10 154 96 139 80 121 60 199 35 169)(11 155 97 140 81 122 61 200 36 170)(12 156 98 141 82 123 62 201 37 171)(13 157 99 142 83 124 63 202 38 172)(14 158 100 143 84 125 43 203 39 173)(15 159 101 144 64 126 44 204 40 174)(16 160 102 145 65 106 45 205 41 175)(17 161 103 146 66 107 46 206 42 176)(18 162 104 147 67 108 47 207 22 177)(19 163 105 127 68 109 48 208 23 178)(20 164 85 128 69 110 49 209 24 179)(21 165 86 129 70 111 50 210 25 180)
(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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 126)(8 125)(9 124)(10 123)(11 122)(12 121)(13 120)(14 119)(15 118)(16 117)(17 116)(18 115)(19 114)(20 113)(21 112)(22 133)(23 132)(24 131)(25 130)(26 129)(27 128)(28 127)(29 147)(30 146)(31 145)(32 144)(33 143)(34 142)(35 141)(36 140)(37 139)(38 138)(39 137)(40 136)(41 135)(42 134)(43 152)(44 151)(45 150)(46 149)(47 148)(48 168)(49 167)(50 166)(51 165)(52 164)(53 163)(54 162)(55 161)(56 160)(57 159)(58 158)(59 157)(60 156)(61 155)(62 154)(63 153)(64 187)(65 186)(66 185)(67 184)(68 183)(69 182)(70 181)(71 180)(72 179)(73 178)(74 177)(75 176)(76 175)(77 174)(78 173)(79 172)(80 171)(81 170)(82 169)(83 189)(84 188)(85 191)(86 190)(87 210)(88 209)(89 208)(90 207)(91 206)(92 205)(93 204)(94 203)(95 202)(96 201)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 192)

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 5A5B5C5D 6 7A7B7C10A10B10C10D10E···10L14A14B14C15A15B15C15D21A···21F30A30B30C30D35A···35L42A···42F70A···70L105A···105X210A···210X
order12223555567771010101010···101414141515151521···213030303035···3542···4270···70105···105210···210
size112121211112222111121···2122222222···222222···22···22···22···22···2

120 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C5C10C10S3D6D7D14C5×S3D21S3×C10C5×D7D42C10×D7C5×D21C10×D21
kernelC10×D21C5×D21C210D42D21C42C70C35C30C15C14C10C7C6C5C3C2C1
# reps1214841133464126122424

Matrix representation of C10×D21 in GL2(𝔽41) generated by

310
031
,
4018
49
,
920
3732
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

Export

Subgroup lattice of C10×D21 in TeX

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