direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C70, C6⋊C70, C210⋊7C2, C42⋊3C10, C30⋊3C14, C105⋊14C22, C3⋊(C2×C70), C15⋊4(C2×C14), C21⋊4(C2×C10), SmallGroup(420,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×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 78 192)(2 79 193)(3 80 194)(4 81 195)(5 82 196)(6 83 197)(7 84 198)(8 85 199)(9 86 200)(10 87 201)(11 88 202)(12 89 203)(13 90 204)(14 91 205)(15 92 206)(16 93 207)(17 94 208)(18 95 209)(19 96 210)(20 97 141)(21 98 142)(22 99 143)(23 100 144)(24 101 145)(25 102 146)(26 103 147)(27 104 148)(28 105 149)(29 106 150)(30 107 151)(31 108 152)(32 109 153)(33 110 154)(34 111 155)(35 112 156)(36 113 157)(37 114 158)(38 115 159)(39 116 160)(40 117 161)(41 118 162)(42 119 163)(43 120 164)(44 121 165)(45 122 166)(46 123 167)(47 124 168)(48 125 169)(49 126 170)(50 127 171)(51 128 172)(52 129 173)(53 130 174)(54 131 175)(55 132 176)(56 133 177)(57 134 178)(58 135 179)(59 136 180)(60 137 181)(61 138 182)(62 139 183)(63 140 184)(64 71 185)(65 72 186)(66 73 187)(67 74 188)(68 75 189)(69 76 190)(70 77 191)
(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)(71 150)(72 151)(73 152)(74 153)(75 154)(76 155)(77 156)(78 157)(79 158)(80 159)(81 160)(82 161)(83 162)(84 163)(85 164)(86 165)(87 166)(88 167)(89 168)(90 169)(91 170)(92 171)(93 172)(94 173)(95 174)(96 175)(97 176)(98 177)(99 178)(100 179)(101 180)(102 181)(103 182)(104 183)(105 184)(106 185)(107 186)(108 187)(109 188)(110 189)(111 190)(112 191)(113 192)(114 193)(115 194)(116 195)(117 196)(118 197)(119 198)(120 199)(121 200)(122 201)(123 202)(124 203)(125 204)(126 205)(127 206)(128 207)(129 208)(130 209)(131 210)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
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), (1,78,192)(2,79,193)(3,80,194)(4,81,195)(5,82,196)(6,83,197)(7,84,198)(8,85,199)(9,86,200)(10,87,201)(11,88,202)(12,89,203)(13,90,204)(14,91,205)(15,92,206)(16,93,207)(17,94,208)(18,95,209)(19,96,210)(20,97,141)(21,98,142)(22,99,143)(23,100,144)(24,101,145)(25,102,146)(26,103,147)(27,104,148)(28,105,149)(29,106,150)(30,107,151)(31,108,152)(32,109,153)(33,110,154)(34,111,155)(35,112,156)(36,113,157)(37,114,158)(38,115,159)(39,116,160)(40,117,161)(41,118,162)(42,119,163)(43,120,164)(44,121,165)(45,122,166)(46,123,167)(47,124,168)(48,125,169)(49,126,170)(50,127,171)(51,128,172)(52,129,173)(53,130,174)(54,131,175)(55,132,176)(56,133,177)(57,134,178)(58,135,179)(59,136,180)(60,137,181)(61,138,182)(62,139,183)(63,140,184)(64,71,185)(65,72,186)(66,73,187)(67,74,188)(68,75,189)(69,76,190)(70,77,191), (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)(71,150)(72,151)(73,152)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158)(80,159)(81,160)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,168)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,181)(103,182)(104,183)(105,184)(106,185)(107,186)(108,187)(109,188)(110,189)(111,190)(112,191)(113,192)(114,193)(115,194)(116,195)(117,196)(118,197)(119,198)(120,199)(121,200)(122,201)(123,202)(124,203)(125,204)(126,205)(127,206)(128,207)(129,208)(130,209)(131,210)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149)>;
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), (1,78,192)(2,79,193)(3,80,194)(4,81,195)(5,82,196)(6,83,197)(7,84,198)(8,85,199)(9,86,200)(10,87,201)(11,88,202)(12,89,203)(13,90,204)(14,91,205)(15,92,206)(16,93,207)(17,94,208)(18,95,209)(19,96,210)(20,97,141)(21,98,142)(22,99,143)(23,100,144)(24,101,145)(25,102,146)(26,103,147)(27,104,148)(28,105,149)(29,106,150)(30,107,151)(31,108,152)(32,109,153)(33,110,154)(34,111,155)(35,112,156)(36,113,157)(37,114,158)(38,115,159)(39,116,160)(40,117,161)(41,118,162)(42,119,163)(43,120,164)(44,121,165)(45,122,166)(46,123,167)(47,124,168)(48,125,169)(49,126,170)(50,127,171)(51,128,172)(52,129,173)(53,130,174)(54,131,175)(55,132,176)(56,133,177)(57,134,178)(58,135,179)(59,136,180)(60,137,181)(61,138,182)(62,139,183)(63,140,184)(64,71,185)(65,72,186)(66,73,187)(67,74,188)(68,75,189)(69,76,190)(70,77,191), (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)(71,150)(72,151)(73,152)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158)(80,159)(81,160)(82,161)(83,162)(84,163)(85,164)(86,165)(87,166)(88,167)(89,168)(90,169)(91,170)(92,171)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(100,179)(101,180)(102,181)(103,182)(104,183)(105,184)(106,185)(107,186)(108,187)(109,188)(110,189)(111,190)(112,191)(113,192)(114,193)(115,194)(116,195)(117,196)(118,197)(119,198)(120,199)(121,200)(122,201)(123,202)(124,203)(125,204)(126,205)(127,206)(128,207)(129,208)(130,209)(131,210)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149) );
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)], [(1,78,192),(2,79,193),(3,80,194),(4,81,195),(5,82,196),(6,83,197),(7,84,198),(8,85,199),(9,86,200),(10,87,201),(11,88,202),(12,89,203),(13,90,204),(14,91,205),(15,92,206),(16,93,207),(17,94,208),(18,95,209),(19,96,210),(20,97,141),(21,98,142),(22,99,143),(23,100,144),(24,101,145),(25,102,146),(26,103,147),(27,104,148),(28,105,149),(29,106,150),(30,107,151),(31,108,152),(32,109,153),(33,110,154),(34,111,155),(35,112,156),(36,113,157),(37,114,158),(38,115,159),(39,116,160),(40,117,161),(41,118,162),(42,119,163),(43,120,164),(44,121,165),(45,122,166),(46,123,167),(47,124,168),(48,125,169),(49,126,170),(50,127,171),(51,128,172),(52,129,173),(53,130,174),(54,131,175),(55,132,176),(56,133,177),(57,134,178),(58,135,179),(59,136,180),(60,137,181),(61,138,182),(62,139,183),(63,140,184),(64,71,185),(65,72,186),(66,73,187),(67,74,188),(68,75,189),(69,76,190),(70,77,191)], [(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),(71,150),(72,151),(73,152),(74,153),(75,154),(76,155),(77,156),(78,157),(79,158),(80,159),(81,160),(82,161),(83,162),(84,163),(85,164),(86,165),(87,166),(88,167),(89,168),(90,169),(91,170),(92,171),(93,172),(94,173),(95,174),(96,175),(97,176),(98,177),(99,178),(100,179),(101,180),(102,181),(103,182),(104,183),(105,184),(106,185),(107,186),(108,187),(109,188),(110,189),(111,190),(112,191),(113,192),(114,193),(115,194),(116,195),(117,196),(118,197),(119,198),(120,199),(121,200),(122,201),(123,202),(124,203),(125,204),(126,205),(127,206),(128,207),(129,208),(130,209),(131,210),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)]])
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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