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G = S3×C82order 492 = 22·3·41

Direct product of C82 and S3

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

Aliases: S3×C82, C6⋊C82, C2463C2, C1234C22, C3⋊(C2×C82), SmallGroup(492,10)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C82
C1C3C123S3×C41 — S3×C82
C3 — S3×C82
C1C82

Generators and relations for S3×C82
 G = < a,b,c | a82=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C82
3C82
3C2×C82

Smallest permutation representation of S3×C82
On 246 points
Generators in S246
(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)
(1 189 164)(2 190 83)(3 191 84)(4 192 85)(5 193 86)(6 194 87)(7 195 88)(8 196 89)(9 197 90)(10 198 91)(11 199 92)(12 200 93)(13 201 94)(14 202 95)(15 203 96)(16 204 97)(17 205 98)(18 206 99)(19 207 100)(20 208 101)(21 209 102)(22 210 103)(23 211 104)(24 212 105)(25 213 106)(26 214 107)(27 215 108)(28 216 109)(29 217 110)(30 218 111)(31 219 112)(32 220 113)(33 221 114)(34 222 115)(35 223 116)(36 224 117)(37 225 118)(38 226 119)(39 227 120)(40 228 121)(41 229 122)(42 230 123)(43 231 124)(44 232 125)(45 233 126)(46 234 127)(47 235 128)(48 236 129)(49 237 130)(50 238 131)(51 239 132)(52 240 133)(53 241 134)(54 242 135)(55 243 136)(56 244 137)(57 245 138)(58 246 139)(59 165 140)(60 166 141)(61 167 142)(62 168 143)(63 169 144)(64 170 145)(65 171 146)(66 172 147)(67 173 148)(68 174 149)(69 175 150)(70 176 151)(71 177 152)(72 178 153)(73 179 154)(74 180 155)(75 181 156)(76 182 157)(77 183 158)(78 184 159)(79 185 160)(80 186 161)(81 187 162)(82 188 163)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(83 231)(84 232)(85 233)(86 234)(87 235)(88 236)(89 237)(90 238)(91 239)(92 240)(93 241)(94 242)(95 243)(96 244)(97 245)(98 246)(99 165)(100 166)(101 167)(102 168)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)(109 175)(110 176)(111 177)(112 178)(113 179)(114 180)(115 181)(116 182)(117 183)(118 184)(119 185)(120 186)(121 187)(122 188)(123 189)(124 190)(125 191)(126 192)(127 193)(128 194)(129 195)(130 196)(131 197)(132 198)(133 199)(134 200)(135 201)(136 202)(137 203)(138 204)(139 205)(140 206)(141 207)(142 208)(143 209)(144 210)(145 211)(146 212)(147 213)(148 214)(149 215)(150 216)(151 217)(152 218)(153 219)(154 220)(155 221)(156 222)(157 223)(158 224)(159 225)(160 226)(161 227)(162 228)(163 229)(164 230)

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

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

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

246 conjugacy classes

class 1 2A2B2C 3  6 41A···41AN82A···82AN82AO···82DP123A···123AN246A···246AN
order12223641···4182···8282···82123···123246···246
size1133221···11···13···32···22···2

246 irreducible representations

dim1111112222
type+++++
imageC1C2C2C41C82C82S3D6S3×C41S3×C82
kernelS3×C82S3×C41C246D6S3C6C82C41C2C1
# reps121408040114040

Matrix representation of S3×C82 in GL3(𝔽739) generated by

73800
04380
00438
,
100
00738
01738
,
100
001
010
G:=sub<GL(3,GF(739))| [738,0,0,0,438,0,0,0,438],[1,0,0,0,0,1,0,738,738],[1,0,0,0,0,1,0,1,0] >;

S3×C82 in GAP, Magma, Sage, TeX

S_3\times C_{82}
% in TeX

G:=Group("S3xC82");
// GroupNames label

G:=SmallGroup(492,10);
// by ID

G=gap.SmallGroup(492,10);
# by ID

G:=PCGroup([4,-2,-2,-41,-3,5251]);
// Polycyclic

G:=Group<a,b,c|a^82=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×C82 in TeX

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