direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D79, C79⋊3C6, C237⋊2C2, SmallGroup(474,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C79 — C3×D79 |
Generators and relations for C3×D79
G = < a,b,c | a3=b79=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D79
►On 237 points
Generators in S237
(1 211 144)(2 212 145)(3 213 146)(4 214 147)(5 215 148)(6 216 149)(7 217 150)(8 218 151)(9 219 152)(10 220 153)(11 221 154)(12 222 155)(13 223 156)(14 224 157)(15 225 158)(16 226 80)(17 227 81)(18 228 82)(19 229 83)(20 230 84)(21 231 85)(22 232 86)(23 233 87)(24 234 88)(25 235 89)(26 236 90)(27 237 91)(28 159 92)(29 160 93)(30 161 94)(31 162 95)(32 163 96)(33 164 97)(34 165 98)(35 166 99)(36 167 100)(37 168 101)(38 169 102)(39 170 103)(40 171 104)(41 172 105)(42 173 106)(43 174 107)(44 175 108)(45 176 109)(46 177 110)(47 178 111)(48 179 112)(49 180 113)(50 181 114)(51 182 115)(52 183 116)(53 184 117)(54 185 118)(55 186 119)(56 187 120)(57 188 121)(58 189 122)(59 190 123)(60 191 124)(61 192 125)(62 193 126)(63 194 127)(64 195 128)(65 196 129)(66 197 130)(67 198 131)(68 199 132)(69 200 133)(70 201 134)(71 202 135)(72 203 136)(73 204 137)(74 205 138)(75 206 139)(76 207 140)(77 208 141)(78 209 142)(79 210 143)
(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)
(1 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 72)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(80 128)(81 127)(82 126)(83 125)(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)(129 158)(130 157)(131 156)(132 155)(133 154)(134 153)(135 152)(136 151)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(159 183)(160 182)(161 181)(162 180)(163 179)(164 178)(165 177)(166 176)(167 175)(168 174)(169 173)(170 172)(184 237)(185 236)(186 235)(187 234)(188 233)(189 232)(190 231)(191 230)(192 229)(193 228)(194 227)(195 226)(196 225)(197 224)(198 223)(199 222)(200 221)(201 220)(202 219)(203 218)(204 217)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)
G:=sub<Sym(237)| (1,211,144)(2,212,145)(3,213,146)(4,214,147)(5,215,148)(6,216,149)(7,217,150)(8,218,151)(9,219,152)(10,220,153)(11,221,154)(12,222,155)(13,223,156)(14,224,157)(15,225,158)(16,226,80)(17,227,81)(18,228,82)(19,229,83)(20,230,84)(21,231,85)(22,232,86)(23,233,87)(24,234,88)(25,235,89)(26,236,90)(27,237,91)(28,159,92)(29,160,93)(30,161,94)(31,162,95)(32,163,96)(33,164,97)(34,165,98)(35,166,99)(36,167,100)(37,168,101)(38,169,102)(39,170,103)(40,171,104)(41,172,105)(42,173,106)(43,174,107)(44,175,108)(45,176,109)(46,177,110)(47,178,111)(48,179,112)(49,180,113)(50,181,114)(51,182,115)(52,183,116)(53,184,117)(54,185,118)(55,186,119)(56,187,120)(57,188,121)(58,189,122)(59,190,123)(60,191,124)(61,192,125)(62,193,126)(63,194,127)(64,195,128)(65,196,129)(66,197,130)(67,198,131)(68,199,132)(69,200,133)(70,201,134)(71,202,135)(72,203,136)(73,204,137)(74,205,138)(75,206,139)(76,207,140)(77,208,141)(78,209,142)(79,210,143), (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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(80,128)(81,127)(82,126)(83,125)(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)(129,158)(130,157)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(184,237)(185,236)(186,235)(187,234)(188,233)(189,232)(190,231)(191,230)(192,229)(193,228)(194,227)(195,226)(196,225)(197,224)(198,223)(199,222)(200,221)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)>;
G:=Group( (1,211,144)(2,212,145)(3,213,146)(4,214,147)(5,215,148)(6,216,149)(7,217,150)(8,218,151)(9,219,152)(10,220,153)(11,221,154)(12,222,155)(13,223,156)(14,224,157)(15,225,158)(16,226,80)(17,227,81)(18,228,82)(19,229,83)(20,230,84)(21,231,85)(22,232,86)(23,233,87)(24,234,88)(25,235,89)(26,236,90)(27,237,91)(28,159,92)(29,160,93)(30,161,94)(31,162,95)(32,163,96)(33,164,97)(34,165,98)(35,166,99)(36,167,100)(37,168,101)(38,169,102)(39,170,103)(40,171,104)(41,172,105)(42,173,106)(43,174,107)(44,175,108)(45,176,109)(46,177,110)(47,178,111)(48,179,112)(49,180,113)(50,181,114)(51,182,115)(52,183,116)(53,184,117)(54,185,118)(55,186,119)(56,187,120)(57,188,121)(58,189,122)(59,190,123)(60,191,124)(61,192,125)(62,193,126)(63,194,127)(64,195,128)(65,196,129)(66,197,130)(67,198,131)(68,199,132)(69,200,133)(70,201,134)(71,202,135)(72,203,136)(73,204,137)(74,205,138)(75,206,139)(76,207,140)(77,208,141)(78,209,142)(79,210,143), (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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(80,128)(81,127)(82,126)(83,125)(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)(129,158)(130,157)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,177)(166,176)(167,175)(168,174)(169,173)(170,172)(184,237)(185,236)(186,235)(187,234)(188,233)(189,232)(190,231)(191,230)(192,229)(193,228)(194,227)(195,226)(196,225)(197,224)(198,223)(199,222)(200,221)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211) );
G=PermutationGroup([[(1,211,144),(2,212,145),(3,213,146),(4,214,147),(5,215,148),(6,216,149),(7,217,150),(8,218,151),(9,219,152),(10,220,153),(11,221,154),(12,222,155),(13,223,156),(14,224,157),(15,225,158),(16,226,80),(17,227,81),(18,228,82),(19,229,83),(20,230,84),(21,231,85),(22,232,86),(23,233,87),(24,234,88),(25,235,89),(26,236,90),(27,237,91),(28,159,92),(29,160,93),(30,161,94),(31,162,95),(32,163,96),(33,164,97),(34,165,98),(35,166,99),(36,167,100),(37,168,101),(38,169,102),(39,170,103),(40,171,104),(41,172,105),(42,173,106),(43,174,107),(44,175,108),(45,176,109),(46,177,110),(47,178,111),(48,179,112),(49,180,113),(50,181,114),(51,182,115),(52,183,116),(53,184,117),(54,185,118),(55,186,119),(56,187,120),(57,188,121),(58,189,122),(59,190,123),(60,191,124),(61,192,125),(62,193,126),(63,194,127),(64,195,128),(65,196,129),(66,197,130),(67,198,131),(68,199,132),(69,200,133),(70,201,134),(71,202,135),(72,203,136),(73,204,137),(74,205,138),(75,206,139),(76,207,140),(77,208,141),(78,209,142),(79,210,143)], [(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)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,72),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,56),(25,55),(26,54),(27,53),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(80,128),(81,127),(82,126),(83,125),(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),(129,158),(130,157),(131,156),(132,155),(133,154),(134,153),(135,152),(136,151),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144),(159,183),(160,182),(161,181),(162,180),(163,179),(164,178),(165,177),(166,176),(167,175),(168,174),(169,173),(170,172),(184,237),(185,236),(186,235),(187,234),(188,233),(189,232),(190,231),(191,230),(192,229),(193,228),(194,227),(195,226),(196,225),(197,224),(198,223),(199,222),(200,221),(201,220),(202,219),(203,218),(204,217),(205,216),(206,215),(207,214),(208,213),(209,212),(210,211)]])
123 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 79A | ··· | 79AM | 237A | ··· | 237BZ |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 79 | ··· | 79 | 237 | ··· | 237 |
| size | 1 | 79 | 1 | 1 | 79 | 79 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D79 | C3×D79 |
| kernel | C3×D79 | C237 | D79 | C79 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 39 | 78 |
Matrix representation of C3×D79 ►in GL2(𝔽1423) generated by
| 779 | 0 |
| 0 | 779 |
| 986 | 1 |
| 1422 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1423))| [779,0,0,779],[986,1422,1,0],[0,1,1,0] >;
C3×D79 in GAP, Magma, Sage, TeX
C_3\times D_{79} % in TeX
G:=Group("C3xD79"); // GroupNames label
G:=SmallGroup(474,4);
// by ID
G=gap.SmallGroup(474,4);
# by ID
G:=PCGroup([3,-2,-3,-79,4214]);
// Polycyclic
G:=Group<a,b,c|a^3=b^79=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export