direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D71, C71⋊C6, C213⋊2C2, SmallGroup(426,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C71 — C3×D71 |
Generators and relations for C3×D71
G = < a,b,c | a3=b71=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D71
►On 213 points
Generators in S213
(1 166 115)(2 167 116)(3 168 117)(4 169 118)(5 170 119)(6 171 120)(7 172 121)(8 173 122)(9 174 123)(10 175 124)(11 176 125)(12 177 126)(13 178 127)(14 179 128)(15 180 129)(16 181 130)(17 182 131)(18 183 132)(19 184 133)(20 185 134)(21 186 135)(22 187 136)(23 188 137)(24 189 138)(25 190 139)(26 191 140)(27 192 141)(28 193 142)(29 194 72)(30 195 73)(31 196 74)(32 197 75)(33 198 76)(34 199 77)(35 200 78)(36 201 79)(37 202 80)(38 203 81)(39 204 82)(40 205 83)(41 206 84)(42 207 85)(43 208 86)(44 209 87)(45 210 88)(46 211 89)(47 212 90)(48 213 91)(49 143 92)(50 144 93)(51 145 94)(52 146 95)(53 147 96)(54 148 97)(55 149 98)(56 150 99)(57 151 100)(58 152 101)(59 153 102)(60 154 103)(61 155 104)(62 156 105)(63 157 106)(64 158 107)(65 159 108)(66 160 109)(67 161 110)(68 162 111)(69 163 112)(70 164 113)(71 165 114)
(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)
(1 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(87 142)(88 141)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(143 188)(144 187)(145 186)(146 185)(147 184)(148 183)(149 182)(150 181)(151 180)(152 179)(153 178)(154 177)(155 176)(156 175)(157 174)(158 173)(159 172)(160 171)(161 170)(162 169)(163 168)(164 167)(165 166)(189 213)(190 212)(191 211)(192 210)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)
G:=sub<Sym(213)| (1,166,115)(2,167,116)(3,168,117)(4,169,118)(5,170,119)(6,171,120)(7,172,121)(8,173,122)(9,174,123)(10,175,124)(11,176,125)(12,177,126)(13,178,127)(14,179,128)(15,180,129)(16,181,130)(17,182,131)(18,183,132)(19,184,133)(20,185,134)(21,186,135)(22,187,136)(23,188,137)(24,189,138)(25,190,139)(26,191,140)(27,192,141)(28,193,142)(29,194,72)(30,195,73)(31,196,74)(32,197,75)(33,198,76)(34,199,77)(35,200,78)(36,201,79)(37,202,80)(38,203,81)(39,204,82)(40,205,83)(41,206,84)(42,207,85)(43,208,86)(44,209,87)(45,210,88)(46,211,89)(47,212,90)(48,213,91)(49,143,92)(50,144,93)(51,145,94)(52,146,95)(53,147,96)(54,148,97)(55,149,98)(56,150,99)(57,151,100)(58,152,101)(59,153,102)(60,154,103)(61,155,104)(62,156,105)(63,157,106)(64,158,107)(65,159,108)(66,160,109)(67,161,110)(68,162,111)(69,163,112)(70,164,113)(71,165,114), (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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)(149,182)(150,181)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)>;
G:=Group( (1,166,115)(2,167,116)(3,168,117)(4,169,118)(5,170,119)(6,171,120)(7,172,121)(8,173,122)(9,174,123)(10,175,124)(11,176,125)(12,177,126)(13,178,127)(14,179,128)(15,180,129)(16,181,130)(17,182,131)(18,183,132)(19,184,133)(20,185,134)(21,186,135)(22,187,136)(23,188,137)(24,189,138)(25,190,139)(26,191,140)(27,192,141)(28,193,142)(29,194,72)(30,195,73)(31,196,74)(32,197,75)(33,198,76)(34,199,77)(35,200,78)(36,201,79)(37,202,80)(38,203,81)(39,204,82)(40,205,83)(41,206,84)(42,207,85)(43,208,86)(44,209,87)(45,210,88)(46,211,89)(47,212,90)(48,213,91)(49,143,92)(50,144,93)(51,145,94)(52,146,95)(53,147,96)(54,148,97)(55,149,98)(56,150,99)(57,151,100)(58,152,101)(59,153,102)(60,154,103)(61,155,104)(62,156,105)(63,157,106)(64,158,107)(65,159,108)(66,160,109)(67,161,110)(68,162,111)(69,163,112)(70,164,113)(71,165,114), (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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)(149,182)(150,181)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202) );
G=PermutationGroup([[(1,166,115),(2,167,116),(3,168,117),(4,169,118),(5,170,119),(6,171,120),(7,172,121),(8,173,122),(9,174,123),(10,175,124),(11,176,125),(12,177,126),(13,178,127),(14,179,128),(15,180,129),(16,181,130),(17,182,131),(18,183,132),(19,184,133),(20,185,134),(21,186,135),(22,187,136),(23,188,137),(24,189,138),(25,190,139),(26,191,140),(27,192,141),(28,193,142),(29,194,72),(30,195,73),(31,196,74),(32,197,75),(33,198,76),(34,199,77),(35,200,78),(36,201,79),(37,202,80),(38,203,81),(39,204,82),(40,205,83),(41,206,84),(42,207,85),(43,208,86),(44,209,87),(45,210,88),(46,211,89),(47,212,90),(48,213,91),(49,143,92),(50,144,93),(51,145,94),(52,146,95),(53,147,96),(54,148,97),(55,149,98),(56,150,99),(57,151,100),(58,152,101),(59,153,102),(60,154,103),(61,155,104),(62,156,105),(63,157,106),(64,158,107),(65,159,108),(66,160,109),(67,161,110),(68,162,111),(69,163,112),(70,164,113),(71,165,114)], [(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)], [(1,71),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,60),(13,59),(14,58),(15,57),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80),(87,142),(88,141),(89,140),(90,139),(91,138),(92,137),(93,136),(94,135),(95,134),(96,133),(97,132),(98,131),(99,130),(100,129),(101,128),(102,127),(103,126),(104,125),(105,124),(106,123),(107,122),(108,121),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(143,188),(144,187),(145,186),(146,185),(147,184),(148,183),(149,182),(150,181),(151,180),(152,179),(153,178),(154,177),(155,176),(156,175),(157,174),(158,173),(159,172),(160,171),(161,170),(162,169),(163,168),(164,167),(165,166),(189,213),(190,212),(191,211),(192,210),(193,209),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202)]])
111 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 71A | ··· | 71AI | 213A | ··· | 213BR |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 71 | ··· | 71 | 213 | ··· | 213 |
| size | 1 | 71 | 1 | 1 | 71 | 71 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D71 | C3×D71 |
| kernel | C3×D71 | C213 | D71 | C71 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 35 | 70 |
Matrix representation of C3×D71 ►in GL2(𝔽853) generated by
| 632 | 0 |
| 0 | 632 |
| 0 | 1 |
| 852 | 418 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(853))| [632,0,0,632],[0,852,1,418],[0,1,1,0] >;
C3×D71 in GAP, Magma, Sage, TeX
C_3\times D_{71} % in TeX
G:=Group("C3xD71"); // GroupNames label
G:=SmallGroup(426,2);
// by ID
G=gap.SmallGroup(426,2);
# by ID
G:=PCGroup([3,-2,-3,-71,3782]);
// Polycyclic
G:=Group<a,b,c|a^3=b^71=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export