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## G = S3×C70order 420 = 22·3·5·7

### Direct product of C70 and S3

Aliases: S3×C70, C6⋊C70, C2107C2, C423C10, C303C14, C10514C22, C3⋊(C2×C70), C154(C2×C14), C214(C2×C10), SmallGroup(420,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C70
 Chief series C1 — C3 — C21 — C105 — S3×C35 — S3×C70
 Lower central C3 — S3×C70
 Upper central C1 — C70

Generators and relations for S3×C70
G = < a,b,c | a70=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C70
On 210 points
Generators in S210
(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)
(1 155 130)(2 156 131)(3 157 132)(4 158 133)(5 159 134)(6 160 135)(7 161 136)(8 162 137)(9 163 138)(10 164 139)(11 165 140)(12 166 71)(13 167 72)(14 168 73)(15 169 74)(16 170 75)(17 171 76)(18 172 77)(19 173 78)(20 174 79)(21 175 80)(22 176 81)(23 177 82)(24 178 83)(25 179 84)(26 180 85)(27 181 86)(28 182 87)(29 183 88)(30 184 89)(31 185 90)(32 186 91)(33 187 92)(34 188 93)(35 189 94)(36 190 95)(37 191 96)(38 192 97)(39 193 98)(40 194 99)(41 195 100)(42 196 101)(43 197 102)(44 198 103)(45 199 104)(46 200 105)(47 201 106)(48 202 107)(49 203 108)(50 204 109)(51 205 110)(52 206 111)(53 207 112)(54 208 113)(55 209 114)(56 210 115)(57 141 116)(58 142 117)(59 143 118)(60 144 119)(61 145 120)(62 146 121)(63 147 122)(64 148 123)(65 149 124)(66 150 125)(67 151 126)(68 152 127)(69 153 128)(70 154 129)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 161)(102 162)(103 163)(104 164)(105 165)(106 166)(107 167)(108 168)(109 169)(110 170)(111 171)(112 172)(113 173)(114 174)(115 175)(116 176)(117 177)(118 178)(119 179)(120 180)(121 181)(122 182)(123 183)(124 184)(125 185)(126 186)(127 187)(128 188)(129 189)(130 190)(131 191)(132 192)(133 193)(134 194)(135 195)(136 196)(137 197)(138 198)(139 199)(140 200)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 5C 5D 6 7A ··· 7F 10A 10B 10C 10D 10E ··· 10L 14A ··· 14F 14G ··· 14R 15A 15B 15C 15D 21A ··· 21F 30A 30B 30C 30D 35A ··· 35X 42A ··· 42F 70A ··· 70X 70Y ··· 70BT 105A ··· 105X 210A ··· 210X order 1 2 2 2 3 5 5 5 5 6 7 ··· 7 10 10 10 10 10 ··· 10 14 ··· 14 14 ··· 14 15 15 15 15 21 ··· 21 30 30 30 30 35 ··· 35 42 ··· 42 70 ··· 70 70 ··· 70 105 ··· 105 210 ··· 210 size 1 1 3 3 2 1 1 1 1 2 1 ··· 1 1 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3 2 2 2 2 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

210 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C5 C7 C10 C10 C14 C14 C35 C70 C70 S3 D6 C5×S3 S3×C7 S3×C10 S3×C14 S3×C35 S3×C70 kernel S3×C70 S3×C35 C210 S3×C14 S3×C10 S3×C7 C42 C5×S3 C30 D6 S3 C6 C70 C35 C14 C10 C7 C5 C2 C1 # reps 1 2 1 4 6 8 4 12 6 24 48 24 1 1 4 6 4 6 24 24

Matrix representation of S3×C70 in GL3(𝔽211) generated by

 210 0 0 0 13 0 0 0 13
,
 1 0 0 0 210 210 0 1 0
,
 1 0 0 0 1 0 0 210 210
G:=sub<GL(3,GF(211))| [210,0,0,0,13,0,0,0,13],[1,0,0,0,210,1,0,210,0],[1,0,0,0,1,210,0,0,210] >;

S3×C70 in GAP, Magma, Sage, TeX

S_3\times C_{70}
% in TeX

G:=Group("S3xC70");
// GroupNames label

G:=SmallGroup(420,37);
// by ID

G=gap.SmallGroup(420,37);
# by ID

G:=PCGroup([5,-2,-2,-5,-7,-3,7004]);
// Polycyclic

G:=Group<a,b,c|a^70=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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