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G = S3×C83order 498 = 2·3·83

Direct product of C83 and S3

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

Aliases: S3×C83, C3⋊C166, C2493C2, SmallGroup(498,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C83
C1C3C249 — S3×C83
C3 — S3×C83
C1C83

Generators and relations for S3×C83
 G = < a,b,c | a83=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C166

Smallest permutation representation of S3×C83
On 249 points
Generators in S249
(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)(167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249)
(1 109 238)(2 110 239)(3 111 240)(4 112 241)(5 113 242)(6 114 243)(7 115 244)(8 116 245)(9 117 246)(10 118 247)(11 119 248)(12 120 249)(13 121 167)(14 122 168)(15 123 169)(16 124 170)(17 125 171)(18 126 172)(19 127 173)(20 128 174)(21 129 175)(22 130 176)(23 131 177)(24 132 178)(25 133 179)(26 134 180)(27 135 181)(28 136 182)(29 137 183)(30 138 184)(31 139 185)(32 140 186)(33 141 187)(34 142 188)(35 143 189)(36 144 190)(37 145 191)(38 146 192)(39 147 193)(40 148 194)(41 149 195)(42 150 196)(43 151 197)(44 152 198)(45 153 199)(46 154 200)(47 155 201)(48 156 202)(49 157 203)(50 158 204)(51 159 205)(52 160 206)(53 161 207)(54 162 208)(55 163 209)(56 164 210)(57 165 211)(58 166 212)(59 84 213)(60 85 214)(61 86 215)(62 87 216)(63 88 217)(64 89 218)(65 90 219)(66 91 220)(67 92 221)(68 93 222)(69 94 223)(70 95 224)(71 96 225)(72 97 226)(73 98 227)(74 99 228)(75 100 229)(76 101 230)(77 102 231)(78 103 232)(79 104 233)(80 105 234)(81 106 235)(82 107 236)(83 108 237)
(84 213)(85 214)(86 215)(87 216)(88 217)(89 218)(90 219)(91 220)(92 221)(93 222)(94 223)(95 224)(96 225)(97 226)(98 227)(99 228)(100 229)(101 230)(102 231)(103 232)(104 233)(105 234)(106 235)(107 236)(108 237)(109 238)(110 239)(111 240)(112 241)(113 242)(114 243)(115 244)(116 245)(117 246)(118 247)(119 248)(120 249)(121 167)(122 168)(123 169)(124 170)(125 171)(126 172)(127 173)(128 174)(129 175)(130 176)(131 177)(132 178)(133 179)(134 180)(135 181)(136 182)(137 183)(138 184)(139 185)(140 186)(141 187)(142 188)(143 189)(144 190)(145 191)(146 192)(147 193)(148 194)(149 195)(150 196)(151 197)(152 198)(153 199)(154 200)(155 201)(156 202)(157 203)(158 204)(159 205)(160 206)(161 207)(162 208)(163 209)(164 210)(165 211)(166 212)

G:=sub<Sym(249)| (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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,109,238)(2,110,239)(3,111,240)(4,112,241)(5,113,242)(6,114,243)(7,115,244)(8,116,245)(9,117,246)(10,118,247)(11,119,248)(12,120,249)(13,121,167)(14,122,168)(15,123,169)(16,124,170)(17,125,171)(18,126,172)(19,127,173)(20,128,174)(21,129,175)(22,130,176)(23,131,177)(24,132,178)(25,133,179)(26,134,180)(27,135,181)(28,136,182)(29,137,183)(30,138,184)(31,139,185)(32,140,186)(33,141,187)(34,142,188)(35,143,189)(36,144,190)(37,145,191)(38,146,192)(39,147,193)(40,148,194)(41,149,195)(42,150,196)(43,151,197)(44,152,198)(45,153,199)(46,154,200)(47,155,201)(48,156,202)(49,157,203)(50,158,204)(51,159,205)(52,160,206)(53,161,207)(54,162,208)(55,163,209)(56,164,210)(57,165,211)(58,166,212)(59,84,213)(60,85,214)(61,86,215)(62,87,216)(63,88,217)(64,89,218)(65,90,219)(66,91,220)(67,92,221)(68,93,222)(69,94,223)(70,95,224)(71,96,225)(72,97,226)(73,98,227)(74,99,228)(75,100,229)(76,101,230)(77,102,231)(78,103,232)(79,104,233)(80,105,234)(81,106,235)(82,107,236)(83,108,237), (84,213)(85,214)(86,215)(87,216)(88,217)(89,218)(90,219)(91,220)(92,221)(93,222)(94,223)(95,224)(96,225)(97,226)(98,227)(99,228)(100,229)(101,230)(102,231)(103,232)(104,233)(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)(112,241)(113,242)(114,243)(115,244)(116,245)(117,246)(118,247)(119,248)(120,249)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184)(139,185)(140,186)(141,187)(142,188)(143,189)(144,190)(145,191)(146,192)(147,193)(148,194)(149,195)(150,196)(151,197)(152,198)(153,199)(154,200)(155,201)(156,202)(157,203)(158,204)(159,205)(160,206)(161,207)(162,208)(163,209)(164,210)(165,211)(166,212)>;

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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,109,238)(2,110,239)(3,111,240)(4,112,241)(5,113,242)(6,114,243)(7,115,244)(8,116,245)(9,117,246)(10,118,247)(11,119,248)(12,120,249)(13,121,167)(14,122,168)(15,123,169)(16,124,170)(17,125,171)(18,126,172)(19,127,173)(20,128,174)(21,129,175)(22,130,176)(23,131,177)(24,132,178)(25,133,179)(26,134,180)(27,135,181)(28,136,182)(29,137,183)(30,138,184)(31,139,185)(32,140,186)(33,141,187)(34,142,188)(35,143,189)(36,144,190)(37,145,191)(38,146,192)(39,147,193)(40,148,194)(41,149,195)(42,150,196)(43,151,197)(44,152,198)(45,153,199)(46,154,200)(47,155,201)(48,156,202)(49,157,203)(50,158,204)(51,159,205)(52,160,206)(53,161,207)(54,162,208)(55,163,209)(56,164,210)(57,165,211)(58,166,212)(59,84,213)(60,85,214)(61,86,215)(62,87,216)(63,88,217)(64,89,218)(65,90,219)(66,91,220)(67,92,221)(68,93,222)(69,94,223)(70,95,224)(71,96,225)(72,97,226)(73,98,227)(74,99,228)(75,100,229)(76,101,230)(77,102,231)(78,103,232)(79,104,233)(80,105,234)(81,106,235)(82,107,236)(83,108,237), (84,213)(85,214)(86,215)(87,216)(88,217)(89,218)(90,219)(91,220)(92,221)(93,222)(94,223)(95,224)(96,225)(97,226)(98,227)(99,228)(100,229)(101,230)(102,231)(103,232)(104,233)(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)(112,241)(113,242)(114,243)(115,244)(116,245)(117,246)(118,247)(119,248)(120,249)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184)(139,185)(140,186)(141,187)(142,188)(143,189)(144,190)(145,191)(146,192)(147,193)(148,194)(149,195)(150,196)(151,197)(152,198)(153,199)(154,200)(155,201)(156,202)(157,203)(158,204)(159,205)(160,206)(161,207)(162,208)(163,209)(164,210)(165,211)(166,212) );

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),(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249)], [(1,109,238),(2,110,239),(3,111,240),(4,112,241),(5,113,242),(6,114,243),(7,115,244),(8,116,245),(9,117,246),(10,118,247),(11,119,248),(12,120,249),(13,121,167),(14,122,168),(15,123,169),(16,124,170),(17,125,171),(18,126,172),(19,127,173),(20,128,174),(21,129,175),(22,130,176),(23,131,177),(24,132,178),(25,133,179),(26,134,180),(27,135,181),(28,136,182),(29,137,183),(30,138,184),(31,139,185),(32,140,186),(33,141,187),(34,142,188),(35,143,189),(36,144,190),(37,145,191),(38,146,192),(39,147,193),(40,148,194),(41,149,195),(42,150,196),(43,151,197),(44,152,198),(45,153,199),(46,154,200),(47,155,201),(48,156,202),(49,157,203),(50,158,204),(51,159,205),(52,160,206),(53,161,207),(54,162,208),(55,163,209),(56,164,210),(57,165,211),(58,166,212),(59,84,213),(60,85,214),(61,86,215),(62,87,216),(63,88,217),(64,89,218),(65,90,219),(66,91,220),(67,92,221),(68,93,222),(69,94,223),(70,95,224),(71,96,225),(72,97,226),(73,98,227),(74,99,228),(75,100,229),(76,101,230),(77,102,231),(78,103,232),(79,104,233),(80,105,234),(81,106,235),(82,107,236),(83,108,237)], [(84,213),(85,214),(86,215),(87,216),(88,217),(89,218),(90,219),(91,220),(92,221),(93,222),(94,223),(95,224),(96,225),(97,226),(98,227),(99,228),(100,229),(101,230),(102,231),(103,232),(104,233),(105,234),(106,235),(107,236),(108,237),(109,238),(110,239),(111,240),(112,241),(113,242),(114,243),(115,244),(116,245),(117,246),(118,247),(119,248),(120,249),(121,167),(122,168),(123,169),(124,170),(125,171),(126,172),(127,173),(128,174),(129,175),(130,176),(131,177),(132,178),(133,179),(134,180),(135,181),(136,182),(137,183),(138,184),(139,185),(140,186),(141,187),(142,188),(143,189),(144,190),(145,191),(146,192),(147,193),(148,194),(149,195),(150,196),(151,197),(152,198),(153,199),(154,200),(155,201),(156,202),(157,203),(158,204),(159,205),(160,206),(161,207),(162,208),(163,209),(164,210),(165,211),(166,212)])

249 conjugacy classes

class 1  2  3 83A···83CD166A···166CD249A···249CD
order12383···83166···166249···249
size1321···13···32···2

249 irreducible representations

dim111122
type+++
imageC1C2C83C166S3S3×C83
kernelS3×C83C249S3C3C83C1
# reps118282182

Matrix representation of S3×C83 in GL2(𝔽499) generated by

2680
0268
,
498498
10
,
10
498498
G:=sub<GL(2,GF(499))| [268,0,0,268],[498,1,498,0],[1,498,0,498] >;

S3×C83 in GAP, Magma, Sage, TeX

S_3\times C_{83}
% in TeX

G:=Group("S3xC83");
// GroupNames label

G:=SmallGroup(498,1);
// by ID

G=gap.SmallGroup(498,1);
# by ID

G:=PCGroup([3,-2,-83,-3,2990]);
// Polycyclic

G:=Group<a,b,c|a^83=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×C83 in TeX

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