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G = C6×D35order 420 = 22·3·5·7

Direct product of C6 and D35

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

Aliases: C6×D35, C705C6, C422D5, C302D7, C158D14, C218D10, C2102C2, C1059C22, C10⋊(C3×D7), C52(C6×D7), C75(C6×D5), C357(C2×C6), C143(C3×D5), SmallGroup(420,36)

Series: Derived Chief Lower central Upper central

C1C35 — C6×D35
C1C7C35C105C3×D35 — C6×D35
C35 — C6×D35
C1C6

Generators and relations for C6×D35
 G = < a,b,c | a6=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

35C2
35C2
35C22
35C6
35C6
7D5
7D5
5D7
5D7
35C2×C6
7D10
5D14
7C3×D5
7C3×D5
5C3×D7
5C3×D7
7C6×D5
5C6×D7

Smallest permutation representation of C6×D35
On 210 points
Generators in S210
(1 160 100 126 36 190)(2 161 101 127 37 191)(3 162 102 128 38 192)(4 163 103 129 39 193)(5 164 104 130 40 194)(6 165 105 131 41 195)(7 166 71 132 42 196)(8 167 72 133 43 197)(9 168 73 134 44 198)(10 169 74 135 45 199)(11 170 75 136 46 200)(12 171 76 137 47 201)(13 172 77 138 48 202)(14 173 78 139 49 203)(15 174 79 140 50 204)(16 175 80 106 51 205)(17 141 81 107 52 206)(18 142 82 108 53 207)(19 143 83 109 54 208)(20 144 84 110 55 209)(21 145 85 111 56 210)(22 146 86 112 57 176)(23 147 87 113 58 177)(24 148 88 114 59 178)(25 149 89 115 60 179)(26 150 90 116 61 180)(27 151 91 117 62 181)(28 152 92 118 63 182)(29 153 93 119 64 183)(30 154 94 120 65 184)(31 155 95 121 66 185)(32 156 96 122 67 186)(33 157 97 123 68 187)(34 158 98 124 69 188)(35 159 99 125 70 189)
(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)
(1 125)(2 124)(3 123)(4 122)(5 121)(6 120)(7 119)(8 118)(9 117)(10 116)(11 115)(12 114)(13 113)(14 112)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 140)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 128)(34 127)(35 126)(36 159)(37 158)(38 157)(39 156)(40 155)(41 154)(42 153)(43 152)(44 151)(45 150)(46 149)(47 148)(48 147)(49 146)(50 145)(51 144)(52 143)(53 142)(54 141)(55 175)(56 174)(57 173)(58 172)(59 171)(60 170)(61 169)(62 168)(63 167)(64 166)(65 165)(66 164)(67 163)(68 162)(69 161)(70 160)(71 183)(72 182)(73 181)(74 180)(75 179)(76 178)(77 177)(78 176)(79 210)(80 209)(81 208)(82 207)(83 206)(84 205)(85 204)(86 203)(87 202)(88 201)(89 200)(90 199)(91 198)(92 197)(93 196)(94 195)(95 194)(96 193)(97 192)(98 191)(99 190)(100 189)(101 188)(102 187)(103 186)(104 185)(105 184)

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

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

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

114 conjugacy classes

class 1 2A2B2C3A3B5A5B6A6B6C6D6E6F7A7B7C10A10B14A14B14C15A15B15C15D21A···21F30A30B30C30D35A···35L42A···42F70A···70L105A···105X210A···210X
order1222335566666677710101414141515151521···213030303035···3542···4270···70105···105210···210
size113535112211353535352222222222222···222222···22···22···22···22···2

114 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6D5D7D10D14C3×D5C3×D7C6×D5D35C6×D7D70C3×D35C6×D35
kernelC6×D35C3×D35C210D70D35C70C42C30C21C15C14C10C7C6C5C3C2C1
# reps1212422323464126122424

Matrix representation of C6×D35 in GL3(𝔽211) generated by

21000
01960
00196
,
100
01849
0202128
,
100
01849
017727
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

Subgroup lattice of C6×D35 in TeX

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