direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D65, C65⋊5C6, C39⋊2D5, C195⋊2C2, C15⋊2D13, C5⋊(C3×D13), C13⋊3(C3×D5), SmallGroup(390,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — C3×D65 |
Generators and relations for C3×D65
G = < a,b,c | a3=b65=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D65
►On 195 points
Generators in S195
(1 175 74)(2 176 75)(3 177 76)(4 178 77)(5 179 78)(6 180 79)(7 181 80)(8 182 81)(9 183 82)(10 184 83)(11 185 84)(12 186 85)(13 187 86)(14 188 87)(15 189 88)(16 190 89)(17 191 90)(18 192 91)(19 193 92)(20 194 93)(21 195 94)(22 131 95)(23 132 96)(24 133 97)(25 134 98)(26 135 99)(27 136 100)(28 137 101)(29 138 102)(30 139 103)(31 140 104)(32 141 105)(33 142 106)(34 143 107)(35 144 108)(36 145 109)(37 146 110)(38 147 111)(39 148 112)(40 149 113)(41 150 114)(42 151 115)(43 152 116)(44 153 117)(45 154 118)(46 155 119)(47 156 120)(48 157 121)(49 158 122)(50 159 123)(51 160 124)(52 161 125)(53 162 126)(54 163 127)(55 164 128)(56 165 129)(57 166 130)(58 167 66)(59 168 67)(60 169 68)(61 170 69)(62 171 70)(63 172 71)(64 173 72)(65 174 73)
(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)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(82 130)(83 129)(84 128)(85 127)(86 126)(87 125)(88 124)(89 123)(90 122)(91 121)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(131 153)(132 152)(133 151)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(141 143)(154 195)(155 194)(156 193)(157 192)(158 191)(159 190)(160 189)(161 188)(162 187)(163 186)(164 185)(165 184)(166 183)(167 182)(168 181)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)
G:=sub<Sym(195)| (1,175,74)(2,176,75)(3,177,76)(4,178,77)(5,179,78)(6,180,79)(7,181,80)(8,182,81)(9,183,82)(10,184,83)(11,185,84)(12,186,85)(13,187,86)(14,188,87)(15,189,88)(16,190,89)(17,191,90)(18,192,91)(19,193,92)(20,194,93)(21,195,94)(22,131,95)(23,132,96)(24,133,97)(25,134,98)(26,135,99)(27,136,100)(28,137,101)(29,138,102)(30,139,103)(31,140,104)(32,141,105)(33,142,106)(34,143,107)(35,144,108)(36,145,109)(37,146,110)(38,147,111)(39,148,112)(40,149,113)(41,150,114)(42,151,115)(43,152,116)(44,153,117)(45,154,118)(46,155,119)(47,156,120)(48,157,121)(49,158,122)(50,159,123)(51,160,124)(52,161,125)(53,162,126)(54,163,127)(55,164,128)(56,165,129)(57,166,130)(58,167,66)(59,168,67)(60,169,68)(61,170,69)(62,171,70)(63,172,71)(64,173,72)(65,174,73), (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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)(154,195)(155,194)(156,193)(157,192)(158,191)(159,190)(160,189)(161,188)(162,187)(163,186)(164,185)(165,184)(166,183)(167,182)(168,181)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)>;
G:=Group( (1,175,74)(2,176,75)(3,177,76)(4,178,77)(5,179,78)(6,180,79)(7,181,80)(8,182,81)(9,183,82)(10,184,83)(11,185,84)(12,186,85)(13,187,86)(14,188,87)(15,189,88)(16,190,89)(17,191,90)(18,192,91)(19,193,92)(20,194,93)(21,195,94)(22,131,95)(23,132,96)(24,133,97)(25,134,98)(26,135,99)(27,136,100)(28,137,101)(29,138,102)(30,139,103)(31,140,104)(32,141,105)(33,142,106)(34,143,107)(35,144,108)(36,145,109)(37,146,110)(38,147,111)(39,148,112)(40,149,113)(41,150,114)(42,151,115)(43,152,116)(44,153,117)(45,154,118)(46,155,119)(47,156,120)(48,157,121)(49,158,122)(50,159,123)(51,160,124)(52,161,125)(53,162,126)(54,163,127)(55,164,128)(56,165,129)(57,166,130)(58,167,66)(59,168,67)(60,169,68)(61,170,69)(62,171,70)(63,172,71)(64,173,72)(65,174,73), (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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)(154,195)(155,194)(156,193)(157,192)(158,191)(159,190)(160,189)(161,188)(162,187)(163,186)(164,185)(165,184)(166,183)(167,182)(168,181)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175) );
G=PermutationGroup([[(1,175,74),(2,176,75),(3,177,76),(4,178,77),(5,179,78),(6,180,79),(7,181,80),(8,182,81),(9,183,82),(10,184,83),(11,185,84),(12,186,85),(13,187,86),(14,188,87),(15,189,88),(16,190,89),(17,191,90),(18,192,91),(19,193,92),(20,194,93),(21,195,94),(22,131,95),(23,132,96),(24,133,97),(25,134,98),(26,135,99),(27,136,100),(28,137,101),(29,138,102),(30,139,103),(31,140,104),(32,141,105),(33,142,106),(34,143,107),(35,144,108),(36,145,109),(37,146,110),(38,147,111),(39,148,112),(40,149,113),(41,150,114),(42,151,115),(43,152,116),(44,153,117),(45,154,118),(46,155,119),(47,156,120),(48,157,121),(49,158,122),(50,159,123),(51,160,124),(52,161,125),(53,162,126),(54,163,127),(55,164,128),(56,165,129),(57,166,130),(58,167,66),(59,168,67),(60,169,68),(61,170,69),(62,171,70),(63,172,71),(64,173,72),(65,174,73)], [(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)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(82,130),(83,129),(84,128),(85,127),(86,126),(87,125),(88,124),(89,123),(90,122),(91,121),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(131,153),(132,152),(133,151),(134,150),(135,149),(136,148),(137,147),(138,146),(139,145),(140,144),(141,143),(154,195),(155,194),(156,193),(157,192),(158,191),(159,190),(160,189),(161,188),(162,187),(163,186),(164,185),(165,184),(166,183),(167,182),(168,181),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175)]])
102 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 13A | ··· | 13F | 15A | 15B | 15C | 15D | 39A | ··· | 39L | 65A | ··· | 65X | 195A | ··· | 195AV |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 13 | ··· | 13 | 15 | 15 | 15 | 15 | 39 | ··· | 39 | 65 | ··· | 65 | 195 | ··· | 195 |
| size | 1 | 65 | 1 | 1 | 2 | 2 | 65 | 65 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | D5 | D13 | C3×D5 | C3×D13 | D65 | C3×D65 |
| kernel | C3×D65 | C195 | D65 | C65 | C39 | C15 | C13 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 12 | 24 | 48 |
Matrix representation of C3×D65 ►in GL2(𝔽1171) generated by
| 420 | 0 |
| 0 | 420 |
| 866 | 646 |
| 699 | 228 |
| 618 | 951 |
| 70 | 553 |
G:=sub<GL(2,GF(1171))| [420,0,0,420],[866,699,646,228],[618,70,951,553] >;
C3×D65 in GAP, Magma, Sage, TeX
C_3\times D_{65} % in TeX
G:=Group("C3xD65"); // GroupNames label
G:=SmallGroup(390,7);
// by ID
G=gap.SmallGroup(390,7);
# by ID
G:=PCGroup([4,-2,-3,-5,-13,290,5763]);
// Polycyclic
G:=Group<a,b,c|a^3=b^65=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export