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G = S3×C71order 426 = 2·3·71

Direct product of C71 and S3

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

Aliases: S3×C71, C3⋊C142, C2133C2, SmallGroup(426,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C71
C1C3C213 — S3×C71
C3 — S3×C71
C1C71

Generators and relations for S3×C71
 G = < a,b,c | a71=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C142

Smallest permutation representation of S3×C71
On 213 points
Generators in S213
(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)(143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213)
(1 194 131)(2 195 132)(3 196 133)(4 197 134)(5 198 135)(6 199 136)(7 200 137)(8 201 138)(9 202 139)(10 203 140)(11 204 141)(12 205 142)(13 206 72)(14 207 73)(15 208 74)(16 209 75)(17 210 76)(18 211 77)(19 212 78)(20 213 79)(21 143 80)(22 144 81)(23 145 82)(24 146 83)(25 147 84)(26 148 85)(27 149 86)(28 150 87)(29 151 88)(30 152 89)(31 153 90)(32 154 91)(33 155 92)(34 156 93)(35 157 94)(36 158 95)(37 159 96)(38 160 97)(39 161 98)(40 162 99)(41 163 100)(42 164 101)(43 165 102)(44 166 103)(45 167 104)(46 168 105)(47 169 106)(48 170 107)(49 171 108)(50 172 109)(51 173 110)(52 174 111)(53 175 112)(54 176 113)(55 177 114)(56 178 115)(57 179 116)(58 180 117)(59 181 118)(60 182 119)(61 183 120)(62 184 121)(63 185 122)(64 186 123)(65 187 124)(66 188 125)(67 189 126)(68 190 127)(69 191 128)(70 192 129)(71 193 130)
(72 206)(73 207)(74 208)(75 209)(76 210)(77 211)(78 212)(79 213)(80 143)(81 144)(82 145)(83 146)(84 147)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 169)(107 170)(108 171)(109 172)(110 173)(111 174)(112 175)(113 176)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 183)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 190)(128 191)(129 192)(130 193)(131 194)(132 195)(133 196)(134 197)(135 198)(136 199)(137 200)(138 201)(139 202)(140 203)(141 204)(142 205)

G:=sub<Sym(213)| (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)(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213), (1,194,131)(2,195,132)(3,196,133)(4,197,134)(5,198,135)(6,199,136)(7,200,137)(8,201,138)(9,202,139)(10,203,140)(11,204,141)(12,205,142)(13,206,72)(14,207,73)(15,208,74)(16,209,75)(17,210,76)(18,211,77)(19,212,78)(20,213,79)(21,143,80)(22,144,81)(23,145,82)(24,146,83)(25,147,84)(26,148,85)(27,149,86)(28,150,87)(29,151,88)(30,152,89)(31,153,90)(32,154,91)(33,155,92)(34,156,93)(35,157,94)(36,158,95)(37,159,96)(38,160,97)(39,161,98)(40,162,99)(41,163,100)(42,164,101)(43,165,102)(44,166,103)(45,167,104)(46,168,105)(47,169,106)(48,170,107)(49,171,108)(50,172,109)(51,173,110)(52,174,111)(53,175,112)(54,176,113)(55,177,114)(56,178,115)(57,179,116)(58,180,117)(59,181,118)(60,182,119)(61,183,120)(62,184,121)(63,185,122)(64,186,123)(65,187,124)(66,188,125)(67,189,126)(68,190,127)(69,191,128)(70,192,129)(71,193,130), (72,206)(73,207)(74,208)(75,209)(76,210)(77,211)(78,212)(79,213)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)(141,204)(142,205)>;

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)(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213), (1,194,131)(2,195,132)(3,196,133)(4,197,134)(5,198,135)(6,199,136)(7,200,137)(8,201,138)(9,202,139)(10,203,140)(11,204,141)(12,205,142)(13,206,72)(14,207,73)(15,208,74)(16,209,75)(17,210,76)(18,211,77)(19,212,78)(20,213,79)(21,143,80)(22,144,81)(23,145,82)(24,146,83)(25,147,84)(26,148,85)(27,149,86)(28,150,87)(29,151,88)(30,152,89)(31,153,90)(32,154,91)(33,155,92)(34,156,93)(35,157,94)(36,158,95)(37,159,96)(38,160,97)(39,161,98)(40,162,99)(41,163,100)(42,164,101)(43,165,102)(44,166,103)(45,167,104)(46,168,105)(47,169,106)(48,170,107)(49,171,108)(50,172,109)(51,173,110)(52,174,111)(53,175,112)(54,176,113)(55,177,114)(56,178,115)(57,179,116)(58,180,117)(59,181,118)(60,182,119)(61,183,120)(62,184,121)(63,185,122)(64,186,123)(65,187,124)(66,188,125)(67,189,126)(68,190,127)(69,191,128)(70,192,129)(71,193,130), (72,206)(73,207)(74,208)(75,209)(76,210)(77,211)(78,212)(79,213)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)(141,204)(142,205) );

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),(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213)], [(1,194,131),(2,195,132),(3,196,133),(4,197,134),(5,198,135),(6,199,136),(7,200,137),(8,201,138),(9,202,139),(10,203,140),(11,204,141),(12,205,142),(13,206,72),(14,207,73),(15,208,74),(16,209,75),(17,210,76),(18,211,77),(19,212,78),(20,213,79),(21,143,80),(22,144,81),(23,145,82),(24,146,83),(25,147,84),(26,148,85),(27,149,86),(28,150,87),(29,151,88),(30,152,89),(31,153,90),(32,154,91),(33,155,92),(34,156,93),(35,157,94),(36,158,95),(37,159,96),(38,160,97),(39,161,98),(40,162,99),(41,163,100),(42,164,101),(43,165,102),(44,166,103),(45,167,104),(46,168,105),(47,169,106),(48,170,107),(49,171,108),(50,172,109),(51,173,110),(52,174,111),(53,175,112),(54,176,113),(55,177,114),(56,178,115),(57,179,116),(58,180,117),(59,181,118),(60,182,119),(61,183,120),(62,184,121),(63,185,122),(64,186,123),(65,187,124),(66,188,125),(67,189,126),(68,190,127),(69,191,128),(70,192,129),(71,193,130)], [(72,206),(73,207),(74,208),(75,209),(76,210),(77,211),(78,212),(79,213),(80,143),(81,144),(82,145),(83,146),(84,147),(85,148),(86,149),(87,150),(88,151),(89,152),(90,153),(91,154),(92,155),(93,156),(94,157),(95,158),(96,159),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,169),(107,170),(108,171),(109,172),(110,173),(111,174),(112,175),(113,176),(114,177),(115,178),(116,179),(117,180),(118,181),(119,182),(120,183),(121,184),(122,185),(123,186),(124,187),(125,188),(126,189),(127,190),(128,191),(129,192),(130,193),(131,194),(132,195),(133,196),(134,197),(135,198),(136,199),(137,200),(138,201),(139,202),(140,203),(141,204),(142,205)])

213 conjugacy classes

class 1  2  3 71A···71BR142A···142BR213A···213BR
order12371···71142···142213···213
size1321···13···32···2

213 irreducible representations

dim111122
type+++
imageC1C2C71C142S3S3×C71
kernelS3×C71C213S3C3C71C1
# reps117070170

Matrix representation of S3×C71 in GL2(𝔽853) generated by

5410
0541
,
0852
1852
,
01
10
G:=sub<GL(2,GF(853))| [541,0,0,541],[0,1,852,852],[0,1,1,0] >;

S3×C71 in GAP, Magma, Sage, TeX

S_3\times C_{71}
% in TeX

G:=Group("S3xC71");
// GroupNames label

G:=SmallGroup(426,1);
// by ID

G=gap.SmallGroup(426,1);
# by ID

G:=PCGroup([3,-2,-71,-3,2558]);
// Polycyclic

G:=Group<a,b,c|a^71=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C71 in TeX

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