direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D77, C77⋊5C6, C33⋊2D7, C231⋊2C2, C21⋊2D11, C11⋊(C3×D7), C7⋊3(C3×D11), SmallGroup(462,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C77 — C3×D77 |
Generators and relations for C3×D77
G = < a,b,c | a3=b77=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D77
►On 231 points
Generators in S231
(1 180 144)(2 181 145)(3 182 146)(4 183 147)(5 184 148)(6 185 149)(7 186 150)(8 187 151)(9 188 152)(10 189 153)(11 190 154)(12 191 78)(13 192 79)(14 193 80)(15 194 81)(16 195 82)(17 196 83)(18 197 84)(19 198 85)(20 199 86)(21 200 87)(22 201 88)(23 202 89)(24 203 90)(25 204 91)(26 205 92)(27 206 93)(28 207 94)(29 208 95)(30 209 96)(31 210 97)(32 211 98)(33 212 99)(34 213 100)(35 214 101)(36 215 102)(37 216 103)(38 217 104)(39 218 105)(40 219 106)(41 220 107)(42 221 108)(43 222 109)(44 223 110)(45 224 111)(46 225 112)(47 226 113)(48 227 114)(49 228 115)(50 229 116)(51 230 117)(52 231 118)(53 155 119)(54 156 120)(55 157 121)(56 158 122)(57 159 123)(58 160 124)(59 161 125)(60 162 126)(61 163 127)(62 164 128)(63 165 129)(64 166 130)(65 167 131)(66 168 132)(67 169 133)(68 170 134)(69 171 135)(70 172 136)(71 173 137)(72 174 138)(73 175 139)(74 176 140)(75 177 141)(76 178 142)(77 179 143)
(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)
(1 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(78 132)(79 131)(80 130)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 120)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(133 154)(134 153)(135 152)(136 151)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(155 204)(156 203)(157 202)(158 201)(159 200)(160 199)(161 198)(162 197)(163 196)(164 195)(165 194)(166 193)(167 192)(168 191)(169 190)(170 189)(171 188)(172 187)(173 186)(174 185)(175 184)(176 183)(177 182)(178 181)(179 180)(205 231)(206 230)(207 229)(208 228)(209 227)(210 226)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
G:=sub<Sym(231)| (1,180,144)(2,181,145)(3,182,146)(4,183,147)(5,184,148)(6,185,149)(7,186,150)(8,187,151)(9,188,152)(10,189,153)(11,190,154)(12,191,78)(13,192,79)(14,193,80)(15,194,81)(16,195,82)(17,196,83)(18,197,84)(19,198,85)(20,199,86)(21,200,87)(22,201,88)(23,202,89)(24,203,90)(25,204,91)(26,205,92)(27,206,93)(28,207,94)(29,208,95)(30,209,96)(31,210,97)(32,211,98)(33,212,99)(34,213,100)(35,214,101)(36,215,102)(37,216,103)(38,217,104)(39,218,105)(40,219,106)(41,220,107)(42,221,108)(43,222,109)(44,223,110)(45,224,111)(46,225,112)(47,226,113)(48,227,114)(49,228,115)(50,229,116)(51,230,117)(52,231,118)(53,155,119)(54,156,120)(55,157,121)(56,158,122)(57,159,123)(58,160,124)(59,161,125)(60,162,126)(61,163,127)(62,164,128)(63,165,129)(64,166,130)(65,167,131)(66,168,132)(67,169,133)(68,170,134)(69,171,135)(70,172,136)(71,173,137)(72,174,138)(73,175,139)(74,176,140)(75,177,141)(76,178,142)(77,179,143), (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,120)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(133,154)(134,153)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(155,204)(156,203)(157,202)(158,201)(159,200)(160,199)(161,198)(162,197)(163,196)(164,195)(165,194)(166,193)(167,192)(168,191)(169,190)(170,189)(171,188)(172,187)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)(205,231)(206,230)(207,229)(208,228)(209,227)(210,226)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219)>;
G:=Group( (1,180,144)(2,181,145)(3,182,146)(4,183,147)(5,184,148)(6,185,149)(7,186,150)(8,187,151)(9,188,152)(10,189,153)(11,190,154)(12,191,78)(13,192,79)(14,193,80)(15,194,81)(16,195,82)(17,196,83)(18,197,84)(19,198,85)(20,199,86)(21,200,87)(22,201,88)(23,202,89)(24,203,90)(25,204,91)(26,205,92)(27,206,93)(28,207,94)(29,208,95)(30,209,96)(31,210,97)(32,211,98)(33,212,99)(34,213,100)(35,214,101)(36,215,102)(37,216,103)(38,217,104)(39,218,105)(40,219,106)(41,220,107)(42,221,108)(43,222,109)(44,223,110)(45,224,111)(46,225,112)(47,226,113)(48,227,114)(49,228,115)(50,229,116)(51,230,117)(52,231,118)(53,155,119)(54,156,120)(55,157,121)(56,158,122)(57,159,123)(58,160,124)(59,161,125)(60,162,126)(61,163,127)(62,164,128)(63,165,129)(64,166,130)(65,167,131)(66,168,132)(67,169,133)(68,170,134)(69,171,135)(70,172,136)(71,173,137)(72,174,138)(73,175,139)(74,176,140)(75,177,141)(76,178,142)(77,179,143), (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,120)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(133,154)(134,153)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(155,204)(156,203)(157,202)(158,201)(159,200)(160,199)(161,198)(162,197)(163,196)(164,195)(165,194)(166,193)(167,192)(168,191)(169,190)(170,189)(171,188)(172,187)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)(205,231)(206,230)(207,229)(208,228)(209,227)(210,226)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219) );
G=PermutationGroup([[(1,180,144),(2,181,145),(3,182,146),(4,183,147),(5,184,148),(6,185,149),(7,186,150),(8,187,151),(9,188,152),(10,189,153),(11,190,154),(12,191,78),(13,192,79),(14,193,80),(15,194,81),(16,195,82),(17,196,83),(18,197,84),(19,198,85),(20,199,86),(21,200,87),(22,201,88),(23,202,89),(24,203,90),(25,204,91),(26,205,92),(27,206,93),(28,207,94),(29,208,95),(30,209,96),(31,210,97),(32,211,98),(33,212,99),(34,213,100),(35,214,101),(36,215,102),(37,216,103),(38,217,104),(39,218,105),(40,219,106),(41,220,107),(42,221,108),(43,222,109),(44,223,110),(45,224,111),(46,225,112),(47,226,113),(48,227,114),(49,228,115),(50,229,116),(51,230,117),(52,231,118),(53,155,119),(54,156,120),(55,157,121),(56,158,122),(57,159,123),(58,160,124),(59,161,125),(60,162,126),(61,163,127),(62,164,128),(63,165,129),(64,166,130),(65,167,131),(66,168,132),(67,169,133),(68,170,134),(69,171,135),(70,172,136),(71,173,137),(72,174,138),(73,175,139),(74,176,140),(75,177,141),(76,178,142),(77,179,143)], [(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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(78,132),(79,131),(80,130),(81,129),(82,128),(83,127),(84,126),(85,125),(86,124),(87,123),(88,122),(89,121),(90,120),(91,119),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(133,154),(134,153),(135,152),(136,151),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144),(155,204),(156,203),(157,202),(158,201),(159,200),(160,199),(161,198),(162,197),(163,196),(164,195),(165,194),(166,193),(167,192),(168,191),(169,190),(170,189),(171,188),(172,187),(173,186),(174,185),(175,184),(176,183),(177,182),(178,181),(179,180),(205,231),(206,230),(207,229),(208,228),(209,227),(210,226),(211,225),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219)]])
120 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 11A | ··· | 11E | 21A | ··· | 21F | 33A | ··· | 33J | 77A | ··· | 77AD | 231A | ··· | 231BH |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 11 | ··· | 11 | 21 | ··· | 21 | 33 | ··· | 33 | 77 | ··· | 77 | 231 | ··· | 231 |
| size | 1 | 77 | 1 | 1 | 77 | 77 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | D7 | D11 | C3×D7 | C3×D11 | D77 | C3×D77 |
| kernel | C3×D77 | C231 | D77 | C77 | C33 | C21 | C11 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 5 | 6 | 10 | 30 | 60 |
Matrix representation of C3×D77 ►in GL3(𝔽463) generated by
| 441 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 15 | 223 |
| 0 | 331 | 44 |
| 462 | 0 | 0 |
| 0 | 202 | 373 |
| 0 | 191 | 261 |
G:=sub<GL(3,GF(463))| [441,0,0,0,1,0,0,0,1],[1,0,0,0,15,331,0,223,44],[462,0,0,0,202,191,0,373,261] >;
C3×D77 in GAP, Magma, Sage, TeX
C_3\times D_{77} % in TeX
G:=Group("C3xD77"); // GroupNames label
G:=SmallGroup(462,7);
// by ID
G=gap.SmallGroup(462,7);
# by ID
G:=PCGroup([4,-2,-3,-7,-11,434,6723]);
// Polycyclic
G:=Group<a,b,c|a^3=b^77=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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