direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D73, C73⋊3C6, C219⋊2C2, SmallGroup(438,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C73 — C3×D73 |
Generators and relations for C3×D73
G = < a,b,c | a3=b73=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D73
►On 219 points
Generators in S219
(1 214 104)(2 215 105)(3 216 106)(4 217 107)(5 218 108)(6 219 109)(7 147 110)(8 148 111)(9 149 112)(10 150 113)(11 151 114)(12 152 115)(13 153 116)(14 154 117)(15 155 118)(16 156 119)(17 157 120)(18 158 121)(19 159 122)(20 160 123)(21 161 124)(22 162 125)(23 163 126)(24 164 127)(25 165 128)(26 166 129)(27 167 130)(28 168 131)(29 169 132)(30 170 133)(31 171 134)(32 172 135)(33 173 136)(34 174 137)(35 175 138)(36 176 139)(37 177 140)(38 178 141)(39 179 142)(40 180 143)(41 181 144)(42 182 145)(43 183 146)(44 184 74)(45 185 75)(46 186 76)(47 187 77)(48 188 78)(49 189 79)(50 190 80)(51 191 81)(52 192 82)(53 193 83)(54 194 84)(55 195 85)(56 196 86)(57 197 87)(58 198 88)(59 199 89)(60 200 90)(61 201 91)(62 202 92)(63 203 93)(64 204 94)(65 205 95)(66 206 96)(67 207 97)(68 208 98)(69 209 99)(70 210 100)(71 211 101)(72 212 102)(73 213 103)
(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)
(1 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(74 133)(75 132)(76 131)(77 130)(78 129)(79 128)(80 127)(81 126)(82 125)(83 124)(84 123)(85 122)(86 121)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(147 207)(148 206)(149 205)(150 204)(151 203)(152 202)(153 201)(154 200)(155 199)(156 198)(157 197)(158 196)(159 195)(160 194)(161 193)(162 192)(163 191)(164 190)(165 189)(166 188)(167 187)(168 186)(169 185)(170 184)(171 183)(172 182)(173 181)(174 180)(175 179)(176 178)(208 219)(209 218)(210 217)(211 216)(212 215)(213 214)
G:=sub<Sym(219)| (1,214,104)(2,215,105)(3,216,106)(4,217,107)(5,218,108)(6,219,109)(7,147,110)(8,148,111)(9,149,112)(10,150,113)(11,151,114)(12,152,115)(13,153,116)(14,154,117)(15,155,118)(16,156,119)(17,157,120)(18,158,121)(19,159,122)(20,160,123)(21,161,124)(22,162,125)(23,163,126)(24,164,127)(25,165,128)(26,166,129)(27,167,130)(28,168,131)(29,169,132)(30,170,133)(31,171,134)(32,172,135)(33,173,136)(34,174,137)(35,175,138)(36,176,139)(37,177,140)(38,178,141)(39,179,142)(40,180,143)(41,181,144)(42,182,145)(43,183,146)(44,184,74)(45,185,75)(46,186,76)(47,187,77)(48,188,78)(49,189,79)(50,190,80)(51,191,81)(52,192,82)(53,193,83)(54,194,84)(55,195,85)(56,196,86)(57,197,87)(58,198,88)(59,199,89)(60,200,90)(61,201,91)(62,202,92)(63,203,93)(64,204,94)(65,205,95)(66,206,96)(67,207,97)(68,208,98)(69,209,99)(70,210,100)(71,211,101)(72,212,102)(73,213,103), (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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(147,207)(148,206)(149,205)(150,204)(151,203)(152,202)(153,201)(154,200)(155,199)(156,198)(157,197)(158,196)(159,195)(160,194)(161,193)(162,192)(163,191)(164,190)(165,189)(166,188)(167,187)(168,186)(169,185)(170,184)(171,183)(172,182)(173,181)(174,180)(175,179)(176,178)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)>;
G:=Group( (1,214,104)(2,215,105)(3,216,106)(4,217,107)(5,218,108)(6,219,109)(7,147,110)(8,148,111)(9,149,112)(10,150,113)(11,151,114)(12,152,115)(13,153,116)(14,154,117)(15,155,118)(16,156,119)(17,157,120)(18,158,121)(19,159,122)(20,160,123)(21,161,124)(22,162,125)(23,163,126)(24,164,127)(25,165,128)(26,166,129)(27,167,130)(28,168,131)(29,169,132)(30,170,133)(31,171,134)(32,172,135)(33,173,136)(34,174,137)(35,175,138)(36,176,139)(37,177,140)(38,178,141)(39,179,142)(40,180,143)(41,181,144)(42,182,145)(43,183,146)(44,184,74)(45,185,75)(46,186,76)(47,187,77)(48,188,78)(49,189,79)(50,190,80)(51,191,81)(52,192,82)(53,193,83)(54,194,84)(55,195,85)(56,196,86)(57,197,87)(58,198,88)(59,199,89)(60,200,90)(61,201,91)(62,202,92)(63,203,93)(64,204,94)(65,205,95)(66,206,96)(67,207,97)(68,208,98)(69,209,99)(70,210,100)(71,211,101)(72,212,102)(73,213,103), (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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(134,146)(135,145)(136,144)(137,143)(138,142)(139,141)(147,207)(148,206)(149,205)(150,204)(151,203)(152,202)(153,201)(154,200)(155,199)(156,198)(157,197)(158,196)(159,195)(160,194)(161,193)(162,192)(163,191)(164,190)(165,189)(166,188)(167,187)(168,186)(169,185)(170,184)(171,183)(172,182)(173,181)(174,180)(175,179)(176,178)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214) );
G=PermutationGroup([[(1,214,104),(2,215,105),(3,216,106),(4,217,107),(5,218,108),(6,219,109),(7,147,110),(8,148,111),(9,149,112),(10,150,113),(11,151,114),(12,152,115),(13,153,116),(14,154,117),(15,155,118),(16,156,119),(17,157,120),(18,158,121),(19,159,122),(20,160,123),(21,161,124),(22,162,125),(23,163,126),(24,164,127),(25,165,128),(26,166,129),(27,167,130),(28,168,131),(29,169,132),(30,170,133),(31,171,134),(32,172,135),(33,173,136),(34,174,137),(35,175,138),(36,176,139),(37,177,140),(38,178,141),(39,179,142),(40,180,143),(41,181,144),(42,182,145),(43,183,146),(44,184,74),(45,185,75),(46,186,76),(47,187,77),(48,188,78),(49,189,79),(50,190,80),(51,191,81),(52,192,82),(53,193,83),(54,194,84),(55,195,85),(56,196,86),(57,197,87),(58,198,88),(59,199,89),(60,200,90),(61,201,91),(62,202,92),(63,203,93),(64,204,94),(65,205,95),(66,206,96),(67,207,97),(68,208,98),(69,209,99),(70,210,100),(71,211,101),(72,212,102),(73,213,103)], [(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)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,60),(15,59),(16,58),(17,57),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,49),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(74,133),(75,132),(76,131),(77,130),(78,129),(79,128),(80,127),(81,126),(82,125),(83,124),(84,123),(85,122),(86,121),(87,120),(88,119),(89,118),(90,117),(91,116),(92,115),(93,114),(94,113),(95,112),(96,111),(97,110),(98,109),(99,108),(100,107),(101,106),(102,105),(103,104),(134,146),(135,145),(136,144),(137,143),(138,142),(139,141),(147,207),(148,206),(149,205),(150,204),(151,203),(152,202),(153,201),(154,200),(155,199),(156,198),(157,197),(158,196),(159,195),(160,194),(161,193),(162,192),(163,191),(164,190),(165,189),(166,188),(167,187),(168,186),(169,185),(170,184),(171,183),(172,182),(173,181),(174,180),(175,179),(176,178),(208,219),(209,218),(210,217),(211,216),(212,215),(213,214)]])
114 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 73A | ··· | 73AJ | 219A | ··· | 219BT |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 73 | ··· | 73 | 219 | ··· | 219 |
| size | 1 | 73 | 1 | 1 | 73 | 73 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D73 | C3×D73 |
| kernel | C3×D73 | C219 | D73 | C73 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 36 | 72 |
Matrix representation of C3×D73 ►in GL2(𝔽439) generated by
| 267 | 0 |
| 0 | 267 |
| 27 | 428 |
| 438 | 98 |
| 283 | 316 |
| 180 | 156 |
G:=sub<GL(2,GF(439))| [267,0,0,267],[27,438,428,98],[283,180,316,156] >;
C3×D73 in GAP, Magma, Sage, TeX
C_3\times D_{73} % in TeX
G:=Group("C3xD73"); // GroupNames label
G:=SmallGroup(438,4);
// by ID
G=gap.SmallGroup(438,4);
# by ID
G:=PCGroup([3,-2,-3,-73,3890]);
// Polycyclic
G:=Group<a,b,c|a^3=b^73=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export