direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D80, C80⋊1C6, C48⋊3D5, C15⋊5D16, C240⋊3C2, D40⋊1C6, C6.13D40, C30.27D8, C60.169D4, C12.39D20, C24.70D10, C120.83C22, C5⋊1(C3×D16), C16⋊1(C3×D5), (C3×D40)⋊5C2, C2.3(C3×D40), C8.13(C6×D5), C4.1(C3×D20), C10.1(C3×D8), C40.14(C2×C6), C20.24(C3×D4), SmallGroup(480,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D80
G = < a,b,c | a3=b80=c2=1, ab=ba, ac=ca, cbc=b-1 >
40C2
40C2
20C22
20C22
40C6
40C6
8D5
8D5
10D4
10D4
20C2×C6
20C2×C6
4D10
4D10
5D8
5D8
10C3×D4
10C3×D4
2D20
2D20
5D16
Smallest permutation representation of C3×D80
►On 240 points
Generators in S240
(1 143 185)(2 144 186)(3 145 187)(4 146 188)(5 147 189)(6 148 190)(7 149 191)(8 150 192)(9 151 193)(10 152 194)(11 153 195)(12 154 196)(13 155 197)(14 156 198)(15 157 199)(16 158 200)(17 159 201)(18 160 202)(19 81 203)(20 82 204)(21 83 205)(22 84 206)(23 85 207)(24 86 208)(25 87 209)(26 88 210)(27 89 211)(28 90 212)(29 91 213)(30 92 214)(31 93 215)(32 94 216)(33 95 217)(34 96 218)(35 97 219)(36 98 220)(37 99 221)(38 100 222)(39 101 223)(40 102 224)(41 103 225)(42 104 226)(43 105 227)(44 106 228)(45 107 229)(46 108 230)(47 109 231)(48 110 232)(49 111 233)(50 112 234)(51 113 235)(52 114 236)(53 115 237)(54 116 238)(55 117 239)(56 118 240)(57 119 161)(58 120 162)(59 121 163)(60 122 164)(61 123 165)(62 124 166)(63 125 167)(64 126 168)(65 127 169)(66 128 170)(67 129 171)(68 130 172)(69 131 173)(70 132 174)(71 133 175)(72 134 176)(73 135 177)(74 136 178)(75 137 179)(76 138 180)(77 139 181)(78 140 182)(79 141 183)(80 142 184)
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 130)(91 129)(92 128)(93 127)(94 126)(95 125)(96 124)(97 123)(98 122)(99 121)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(161 223)(162 222)(163 221)(164 220)(165 219)(166 218)(167 217)(168 216)(169 215)(170 214)(171 213)(172 212)(173 211)(174 210)(175 209)(176 208)(177 207)(178 206)(179 205)(180 204)(181 203)(182 202)(183 201)(184 200)(185 199)(186 198)(187 197)(188 196)(189 195)(190 194)(191 193)(224 240)(225 239)(226 238)(227 237)(228 236)(229 235)(230 234)(231 233)
G:=sub<Sym(240)| (1,143,185)(2,144,186)(3,145,187)(4,146,188)(5,147,189)(6,148,190)(7,149,191)(8,150,192)(9,151,193)(10,152,194)(11,153,195)(12,154,196)(13,155,197)(14,156,198)(15,157,199)(16,158,200)(17,159,201)(18,160,202)(19,81,203)(20,82,204)(21,83,205)(22,84,206)(23,85,207)(24,86,208)(25,87,209)(26,88,210)(27,89,211)(28,90,212)(29,91,213)(30,92,214)(31,93,215)(32,94,216)(33,95,217)(34,96,218)(35,97,219)(36,98,220)(37,99,221)(38,100,222)(39,101,223)(40,102,224)(41,103,225)(42,104,226)(43,105,227)(44,106,228)(45,107,229)(46,108,230)(47,109,231)(48,110,232)(49,111,233)(50,112,234)(51,113,235)(52,114,236)(53,115,237)(54,116,238)(55,117,239)(56,118,240)(57,119,161)(58,120,162)(59,121,163)(60,122,164)(61,123,165)(62,124,166)(63,125,167)(64,126,168)(65,127,169)(66,128,170)(67,129,171)(68,130,172)(69,131,173)(70,132,174)(71,133,175)(72,134,176)(73,135,177)(74,136,178)(75,137,179)(76,138,180)(77,139,181)(78,140,182)(79,141,183)(80,142,184), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(161,223)(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216)(169,215)(170,214)(171,213)(172,212)(173,211)(174,210)(175,209)(176,208)(177,207)(178,206)(179,205)(180,204)(181,203)(182,202)(183,201)(184,200)(185,199)(186,198)(187,197)(188,196)(189,195)(190,194)(191,193)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233)>;
G:=Group( (1,143,185)(2,144,186)(3,145,187)(4,146,188)(5,147,189)(6,148,190)(7,149,191)(8,150,192)(9,151,193)(10,152,194)(11,153,195)(12,154,196)(13,155,197)(14,156,198)(15,157,199)(16,158,200)(17,159,201)(18,160,202)(19,81,203)(20,82,204)(21,83,205)(22,84,206)(23,85,207)(24,86,208)(25,87,209)(26,88,210)(27,89,211)(28,90,212)(29,91,213)(30,92,214)(31,93,215)(32,94,216)(33,95,217)(34,96,218)(35,97,219)(36,98,220)(37,99,221)(38,100,222)(39,101,223)(40,102,224)(41,103,225)(42,104,226)(43,105,227)(44,106,228)(45,107,229)(46,108,230)(47,109,231)(48,110,232)(49,111,233)(50,112,234)(51,113,235)(52,114,236)(53,115,237)(54,116,238)(55,117,239)(56,118,240)(57,119,161)(58,120,162)(59,121,163)(60,122,164)(61,123,165)(62,124,166)(63,125,167)(64,126,168)(65,127,169)(66,128,170)(67,129,171)(68,130,172)(69,131,173)(70,132,174)(71,133,175)(72,134,176)(73,135,177)(74,136,178)(75,137,179)(76,138,180)(77,139,181)(78,140,182)(79,141,183)(80,142,184), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(81,139)(82,138)(83,137)(84,136)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,124)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(161,223)(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216)(169,215)(170,214)(171,213)(172,212)(173,211)(174,210)(175,209)(176,208)(177,207)(178,206)(179,205)(180,204)(181,203)(182,202)(183,201)(184,200)(185,199)(186,198)(187,197)(188,196)(189,195)(190,194)(191,193)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233) );
G=PermutationGroup([[(1,143,185),(2,144,186),(3,145,187),(4,146,188),(5,147,189),(6,148,190),(7,149,191),(8,150,192),(9,151,193),(10,152,194),(11,153,195),(12,154,196),(13,155,197),(14,156,198),(15,157,199),(16,158,200),(17,159,201),(18,160,202),(19,81,203),(20,82,204),(21,83,205),(22,84,206),(23,85,207),(24,86,208),(25,87,209),(26,88,210),(27,89,211),(28,90,212),(29,91,213),(30,92,214),(31,93,215),(32,94,216),(33,95,217),(34,96,218),(35,97,219),(36,98,220),(37,99,221),(38,100,222),(39,101,223),(40,102,224),(41,103,225),(42,104,226),(43,105,227),(44,106,228),(45,107,229),(46,108,230),(47,109,231),(48,110,232),(49,111,233),(50,112,234),(51,113,235),(52,114,236),(53,115,237),(54,116,238),(55,117,239),(56,118,240),(57,119,161),(58,120,162),(59,121,163),(60,122,164),(61,123,165),(62,124,166),(63,125,167),(64,126,168),(65,127,169),(66,128,170),(67,129,171),(68,130,172),(69,131,173),(70,132,174),(71,133,175),(72,134,176),(73,135,177),(74,136,178),(75,137,179),(76,138,180),(77,139,181),(78,140,182),(79,141,183),(80,142,184)], [(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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,74),(23,73),(24,72),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(81,139),(82,138),(83,137),(84,136),(85,135),(86,134),(87,133),(88,132),(89,131),(90,130),(91,129),(92,128),(93,127),(94,126),(95,125),(96,124),(97,123),(98,122),(99,121),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(140,160),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151),(161,223),(162,222),(163,221),(164,220),(165,219),(166,218),(167,217),(168,216),(169,215),(170,214),(171,213),(172,212),(173,211),(174,210),(175,209),(176,208),(177,207),(178,206),(179,205),(180,204),(181,203),(182,202),(183,201),(184,200),(185,199),(186,198),(187,197),(188,196),(189,195),(190,194),(191,193),(224,240),(225,239),(226,238),(227,237),(228,236),(229,235),(230,234),(231,233)]])
129 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 40 | 40 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
129 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D5 | D8 | D10 | C3×D4 | C3×D5 | D16 | D20 | C3×D8 | C6×D5 | D40 | C3×D16 | C3×D20 | D80 | C3×D40 | C3×D80 |
| kernel | C3×D80 | C240 | C3×D40 | D80 | C80 | D40 | C60 | C48 | C30 | C24 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C3×D80 ►in GL2(𝔽241) generated by
| 15 | 0 |
| 0 | 15 |
| 160 | 139 |
| 102 | 60 |
| 27 | 185 |
| 13 | 214 |
G:=sub<GL(2,GF(241))| [15,0,0,15],[160,102,139,60],[27,13,185,214] >;
C3×D80 in GAP, Magma, Sage, TeX
C_3\times D_{80} % in TeX
G:=Group("C3xD80"); // GroupNames label
G:=SmallGroup(480,77);
// by ID
G=gap.SmallGroup(480,77);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,197,260,1011,192,2524,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^3=b^80=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export