direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D67, C67⋊3C6, C201⋊2C2, SmallGroup(402,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C67 — C3×D67 |
Generators and relations for C3×D67
G = < a,b,c | a3=b67=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D67
►On 201 points
Generators in S201
(1 148 112)(2 149 113)(3 150 114)(4 151 115)(5 152 116)(6 153 117)(7 154 118)(8 155 119)(9 156 120)(10 157 121)(11 158 122)(12 159 123)(13 160 124)(14 161 125)(15 162 126)(16 163 127)(17 164 128)(18 165 129)(19 166 130)(20 167 131)(21 168 132)(22 169 133)(23 170 134)(24 171 68)(25 172 69)(26 173 70)(27 174 71)(28 175 72)(29 176 73)(30 177 74)(31 178 75)(32 179 76)(33 180 77)(34 181 78)(35 182 79)(36 183 80)(37 184 81)(38 185 82)(39 186 83)(40 187 84)(41 188 85)(42 189 86)(43 190 87)(44 191 88)(45 192 89)(46 193 90)(47 194 91)(48 195 92)(49 196 93)(50 197 94)(51 198 95)(52 199 96)(53 200 97)(54 201 98)(55 135 99)(56 136 100)(57 137 101)(58 138 102)(59 139 103)(60 140 104)(61 141 105)(62 142 106)(63 143 107)(64 144 108)(65 145 109)(66 146 110)(67 147 111)
(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)
(1 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(161 201)(162 200)(163 199)(164 198)(165 197)(166 196)(167 195)(168 194)(169 193)(170 192)(171 191)(172 190)(173 189)(174 188)(175 187)(176 186)(177 185)(178 184)(179 183)(180 182)
G:=sub<Sym(201)| (1,148,112)(2,149,113)(3,150,114)(4,151,115)(5,152,116)(6,153,117)(7,154,118)(8,155,119)(9,156,120)(10,157,121)(11,158,122)(12,159,123)(13,160,124)(14,161,125)(15,162,126)(16,163,127)(17,164,128)(18,165,129)(19,166,130)(20,167,131)(21,168,132)(22,169,133)(23,170,134)(24,171,68)(25,172,69)(26,173,70)(27,174,71)(28,175,72)(29,176,73)(30,177,74)(31,178,75)(32,179,76)(33,180,77)(34,181,78)(35,182,79)(36,183,80)(37,184,81)(38,185,82)(39,186,83)(40,187,84)(41,188,85)(42,189,86)(43,190,87)(44,191,88)(45,192,89)(46,193,90)(47,194,91)(48,195,92)(49,196,93)(50,197,94)(51,198,95)(52,199,96)(53,200,97)(54,201,98)(55,135,99)(56,136,100)(57,137,101)(58,138,102)(59,139,103)(60,140,104)(61,141,105)(62,142,106)(63,143,107)(64,144,108)(65,145,109)(66,146,110)(67,147,111), (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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(161,201)(162,200)(163,199)(164,198)(165,197)(166,196)(167,195)(168,194)(169,193)(170,192)(171,191)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)>;
G:=Group( (1,148,112)(2,149,113)(3,150,114)(4,151,115)(5,152,116)(6,153,117)(7,154,118)(8,155,119)(9,156,120)(10,157,121)(11,158,122)(12,159,123)(13,160,124)(14,161,125)(15,162,126)(16,163,127)(17,164,128)(18,165,129)(19,166,130)(20,167,131)(21,168,132)(22,169,133)(23,170,134)(24,171,68)(25,172,69)(26,173,70)(27,174,71)(28,175,72)(29,176,73)(30,177,74)(31,178,75)(32,179,76)(33,180,77)(34,181,78)(35,182,79)(36,183,80)(37,184,81)(38,185,82)(39,186,83)(40,187,84)(41,188,85)(42,189,86)(43,190,87)(44,191,88)(45,192,89)(46,193,90)(47,194,91)(48,195,92)(49,196,93)(50,197,94)(51,198,95)(52,199,96)(53,200,97)(54,201,98)(55,135,99)(56,136,100)(57,137,101)(58,138,102)(59,139,103)(60,140,104)(61,141,105)(62,142,106)(63,143,107)(64,144,108)(65,145,109)(66,146,110)(67,147,111), (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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(161,201)(162,200)(163,199)(164,198)(165,197)(166,196)(167,195)(168,194)(169,193)(170,192)(171,191)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182) );
G=PermutationGroup([[(1,148,112),(2,149,113),(3,150,114),(4,151,115),(5,152,116),(6,153,117),(7,154,118),(8,155,119),(9,156,120),(10,157,121),(11,158,122),(12,159,123),(13,160,124),(14,161,125),(15,162,126),(16,163,127),(17,164,128),(18,165,129),(19,166,130),(20,167,131),(21,168,132),(22,169,133),(23,170,134),(24,171,68),(25,172,69),(26,173,70),(27,174,71),(28,175,72),(29,176,73),(30,177,74),(31,178,75),(32,179,76),(33,180,77),(34,181,78),(35,182,79),(36,183,80),(37,184,81),(38,185,82),(39,186,83),(40,187,84),(41,188,85),(42,189,86),(43,190,87),(44,191,88),(45,192,89),(46,193,90),(47,194,91),(48,195,92),(49,196,93),(50,197,94),(51,198,95),(52,199,96),(53,200,97),(54,201,98),(55,135,99),(56,136,100),(57,137,101),(58,138,102),(59,139,103),(60,140,104),(61,141,105),(62,142,106),(63,143,107),(64,144,108),(65,145,109),(66,146,110),(67,147,111)], [(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)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,60),(9,59),(10,58),(11,57),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(89,134),(90,133),(91,132),(92,131),(93,130),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123),(101,122),(102,121),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148),(161,201),(162,200),(163,199),(164,198),(165,197),(166,196),(167,195),(168,194),(169,193),(170,192),(171,191),(172,190),(173,189),(174,188),(175,187),(176,186),(177,185),(178,184),(179,183),(180,182)]])
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 67A | ··· | 67AG | 201A | ··· | 201BN |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 67 | ··· | 67 | 201 | ··· | 201 |
| size | 1 | 67 | 1 | 1 | 67 | 67 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D67 | C3×D67 |
| kernel | C3×D67 | C201 | D67 | C67 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 33 | 66 |
Matrix representation of C3×D67 ►in GL2(𝔽1609) generated by
| 250 | 0 |
| 0 | 250 |
| 484 | 1042 |
| 1608 | 932 |
| 484 | 950 |
| 1608 | 1125 |
G:=sub<GL(2,GF(1609))| [250,0,0,250],[484,1608,1042,932],[484,1608,950,1125] >;
C3×D67 in GAP, Magma, Sage, TeX
C_3\times D_{67} % in TeX
G:=Group("C3xD67"); // GroupNames label
G:=SmallGroup(402,4);
// by ID
G=gap.SmallGroup(402,4);
# by ID
G:=PCGroup([3,-2,-3,-67,3566]);
// Polycyclic
G:=Group<a,b,c|a^3=b^67=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export