Copied to
clipboard

G = C3×D83order 498 = 2·3·83

Direct product of C3 and D83

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

Aliases: C3×D83, C83⋊C6, C2492C2, SmallGroup(498,2)

Series: Derived Chief Lower central Upper central

C1C83 — C3×D83
C1C83C249 — C3×D83
C83 — C3×D83
C1C3

Generators and relations for C3×D83
 G = < a,b,c | a3=b83=c2=1, ab=ba, ac=ca, cbc=b-1 >

83C2
83C6

Smallest permutation representation of C3×D83
On 249 points
Generators in S249
(1 241 146)(2 242 147)(3 243 148)(4 244 149)(5 245 150)(6 246 151)(7 247 152)(8 248 153)(9 249 154)(10 167 155)(11 168 156)(12 169 157)(13 170 158)(14 171 159)(15 172 160)(16 173 161)(17 174 162)(18 175 163)(19 176 164)(20 177 165)(21 178 166)(22 179 84)(23 180 85)(24 181 86)(25 182 87)(26 183 88)(27 184 89)(28 185 90)(29 186 91)(30 187 92)(31 188 93)(32 189 94)(33 190 95)(34 191 96)(35 192 97)(36 193 98)(37 194 99)(38 195 100)(39 196 101)(40 197 102)(41 198 103)(42 199 104)(43 200 105)(44 201 106)(45 202 107)(46 203 108)(47 204 109)(48 205 110)(49 206 111)(50 207 112)(51 208 113)(52 209 114)(53 210 115)(54 211 116)(55 212 117)(56 213 118)(57 214 119)(58 215 120)(59 216 121)(60 217 122)(61 218 123)(62 219 124)(63 220 125)(64 221 126)(65 222 127)(66 223 128)(67 224 129)(68 225 130)(69 226 131)(70 227 132)(71 228 133)(72 229 134)(73 230 135)(74 231 136)(75 232 137)(76 233 138)(77 234 139)(78 235 140)(79 236 141)(80 237 142)(81 238 143)(82 239 144)(83 240 145)
(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 83)(2 82)(3 81)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(84 124)(85 123)(86 122)(87 121)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(125 166)(126 165)(127 164)(128 163)(129 162)(130 161)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)(141 150)(142 149)(143 148)(144 147)(145 146)(167 231)(168 230)(169 229)(170 228)(171 227)(172 226)(173 225)(174 224)(175 223)(176 222)(177 221)(178 220)(179 219)(180 218)(181 217)(182 216)(183 215)(184 214)(185 213)(186 212)(187 211)(188 210)(189 209)(190 208)(191 207)(192 206)(193 205)(194 204)(195 203)(196 202)(197 201)(198 200)(232 249)(233 248)(234 247)(235 246)(236 245)(237 244)(238 243)(239 242)(240 241)

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

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

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

129 conjugacy classes

class 1  2 3A3B6A6B83A···83AO249A···249CD
order12336683···83249···249
size1831183832···22···2

129 irreducible representations

dim111122
type+++
imageC1C2C3C6D83C3×D83
kernelC3×D83C249D83C83C3C1
# reps11224182

Matrix representation of C3×D83 in GL3(𝔽499) generated by

35900
010
001
,
100
03131
050378
,
49800
0378498
0169121
G:=sub<GL(3,GF(499))| [359,0,0,0,1,0,0,0,1],[1,0,0,0,313,50,0,1,378],[498,0,0,0,378,169,0,498,121] >;

C3×D83 in GAP, Magma, Sage, TeX

C_3\times D_{83}
% in TeX

G:=Group("C3xD83");
// GroupNames label

G:=SmallGroup(498,2);
// by ID

G=gap.SmallGroup(498,2);
# by ID

G:=PCGroup([3,-2,-3,-83,4430]);
// Polycyclic

G:=Group<a,b,c|a^3=b^83=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D83 in TeX

׿
×
𝔽