direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D120, C30⋊4D8, C6⋊1D40, C8⋊7D30, C10⋊1D24, C40⋊29D6, C4.7D60, C24⋊29D10, C60.163D4, C12.32D20, C20.32D12, C120⋊36C22, D60⋊20C22, C22.13D60, C60.245C23, C3⋊2(C2×D40), C5⋊2(C2×D24), (C2×C40)⋊5S3, (C2×C24)⋊5D5, (C2×C8)⋊3D15, C15⋊10(C2×D8), (C2×C120)⋊9C2, (C2×D60)⋊8C2, C2.12(C2×D60), C6.38(C2×D20), (C2×C4).81D30, (C2×C6).19D20, C10.39(C2×D12), (C2×C20).391D6, (C2×C10).19D12, (C2×C30).104D4, C30.266(C2×D4), (C2×C12).397D10, C4.26(C22×D15), C20.216(C22×S3), (C2×C60).478C22, C12.218(C22×D5), SmallGroup(480,868)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D120
G = < a,b,c | a2=b120=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 1460 in 152 conjugacy classes, 55 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C15, C2×C8, D8, C2×D4, C20, D10, C2×C10, C24, D12, C2×C12, C22×S3, D15, C30, C30, C2×D8, C40, D20, C2×C20, C22×D5, D24, C2×C24, C2×D12, C60, D30, C2×C30, D40, C2×C40, C2×D20, C2×D24, C120, D60, D60, C2×C60, C22×D15, C2×D40, D120, C2×C120, C2×D60, C2×D120
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, D12, C22×S3, D15, C2×D8, D20, C22×D5, D24, C2×D12, D30, D40, C2×D20, C2×D24, D60, C22×D15, C2×D40, D120, C2×D60, C2×D120
►On 240 points
Generators in S240
(1 239)(2 240)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 160)(43 161)(44 162)(45 163)(46 164)(47 165)(48 166)(49 167)(50 168)(51 169)(52 170)(53 171)(54 172)(55 173)(56 174)(57 175)(58 176)(59 177)(60 178)(61 179)(62 180)(63 181)(64 182)(65 183)(66 184)(67 185)(68 186)(69 187)(70 188)(71 189)(72 190)(73 191)(74 192)(75 193)(76 194)(77 195)(78 196)(79 197)(80 198)(81 199)(82 200)(83 201)(84 202)(85 203)(86 204)(87 205)(88 206)(89 207)(90 208)(91 209)(92 210)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(99 217)(100 218)(101 219)(102 220)(103 221)(104 222)(105 223)(106 224)(107 225)(108 226)(109 227)(110 228)(111 229)(112 230)(113 231)(114 232)(115 233)(116 234)(117 235)(118 236)(119 237)(120 238)
(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)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(121 131)(122 130)(123 129)(124 128)(125 127)(132 240)(133 239)(134 238)(135 237)(136 236)(137 235)(138 234)(139 233)(140 232)(141 231)(142 230)(143 229)(144 228)(145 227)(146 226)(147 225)(148 224)(149 223)(150 222)(151 221)(152 220)(153 219)(154 218)(155 217)(156 216)(157 215)(158 214)(159 213)(160 212)(161 211)(162 210)(163 209)(164 208)(165 207)(166 206)(167 205)(168 204)(169 203)(170 202)(171 201)(172 200)(173 199)(174 198)(175 197)(176 196)(177 195)(178 194)(179 193)(180 192)(181 191)(182 190)(183 189)(184 188)(185 187)
G:=sub<Sym(240)| (1,239)(2,240)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(65,183)(66,184)(67,185)(68,186)(69,187)(70,188)(71,189)(72,190)(73,191)(74,192)(75,193)(76,194)(77,195)(78,196)(79,197)(80,198)(81,199)(82,200)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(101,219)(102,220)(103,221)(104,222)(105,223)(106,224)(107,225)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238), (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)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(121,131)(122,130)(123,129)(124,128)(125,127)(132,240)(133,239)(134,238)(135,237)(136,236)(137,235)(138,234)(139,233)(140,232)(141,231)(142,230)(143,229)(144,228)(145,227)(146,226)(147,225)(148,224)(149,223)(150,222)(151,221)(152,220)(153,219)(154,218)(155,217)(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,210)(163,209)(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,202)(171,201)(172,200)(173,199)(174,198)(175,197)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)>;
G:=Group( (1,239)(2,240)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(65,183)(66,184)(67,185)(68,186)(69,187)(70,188)(71,189)(72,190)(73,191)(74,192)(75,193)(76,194)(77,195)(78,196)(79,197)(80,198)(81,199)(82,200)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,209)(92,210)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(99,217)(100,218)(101,219)(102,220)(103,221)(104,222)(105,223)(106,224)(107,225)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238), (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)(16,120)(17,119)(18,118)(19,117)(20,116)(21,115)(22,114)(23,113)(24,112)(25,111)(26,110)(27,109)(28,108)(29,107)(30,106)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(121,131)(122,130)(123,129)(124,128)(125,127)(132,240)(133,239)(134,238)(135,237)(136,236)(137,235)(138,234)(139,233)(140,232)(141,231)(142,230)(143,229)(144,228)(145,227)(146,226)(147,225)(148,224)(149,223)(150,222)(151,221)(152,220)(153,219)(154,218)(155,217)(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,210)(163,209)(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,202)(171,201)(172,200)(173,199)(174,198)(175,197)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187) );
G=PermutationGroup([[(1,239),(2,240),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,159),(42,160),(43,161),(44,162),(45,163),(46,164),(47,165),(48,166),(49,167),(50,168),(51,169),(52,170),(53,171),(54,172),(55,173),(56,174),(57,175),(58,176),(59,177),(60,178),(61,179),(62,180),(63,181),(64,182),(65,183),(66,184),(67,185),(68,186),(69,187),(70,188),(71,189),(72,190),(73,191),(74,192),(75,193),(76,194),(77,195),(78,196),(79,197),(80,198),(81,199),(82,200),(83,201),(84,202),(85,203),(86,204),(87,205),(88,206),(89,207),(90,208),(91,209),(92,210),(93,211),(94,212),(95,213),(96,214),(97,215),(98,216),(99,217),(100,218),(101,219),(102,220),(103,221),(104,222),(105,223),(106,224),(107,225),(108,226),(109,227),(110,228),(111,229),(112,230),(113,231),(114,232),(115,233),(116,234),(117,235),(118,236),(119,237),(120,238)], [(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),(16,120),(17,119),(18,118),(19,117),(20,116),(21,115),(22,114),(23,113),(24,112),(25,111),(26,110),(27,109),(28,108),(29,107),(30,106),(31,105),(32,104),(33,103),(34,102),(35,101),(36,100),(37,99),(38,98),(39,97),(40,96),(41,95),(42,94),(43,93),(44,92),(45,91),(46,90),(47,89),(48,88),(49,87),(50,86),(51,85),(52,84),(53,83),(54,82),(55,81),(56,80),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(121,131),(122,130),(123,129),(124,128),(125,127),(132,240),(133,239),(134,238),(135,237),(136,236),(137,235),(138,234),(139,233),(140,232),(141,231),(142,230),(143,229),(144,228),(145,227),(146,226),(147,225),(148,224),(149,223),(150,222),(151,221),(152,220),(153,219),(154,218),(155,217),(156,216),(157,215),(158,214),(159,213),(160,212),(161,211),(162,210),(163,209),(164,208),(165,207),(166,206),(167,205),(168,204),(169,203),(170,202),(171,201),(172,200),(173,199),(174,198),(175,197),(176,196),(177,195),(178,194),(179,193),(180,192),(181,191),(182,190),(183,189),(184,188),(185,187)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 60 | 60 | 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 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D8 | D10 | D10 | D12 | D12 | D15 | D20 | D20 | D24 | D30 | D30 | D40 | D60 | D60 | D120 |
| kernel | C2×D120 | D120 | C2×C120 | C2×D60 | C2×C40 | C60 | C2×C30 | C2×C24 | C40 | C2×C20 | C30 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C2×D120 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 39 | 138 |
| 0 | 0 | 157 | 123 |
| 240 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 |
| 0 | 0 | 113 | 122 |
| 0 | 0 | 164 | 128 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,240,0,0,0,0,0,39,157,0,0,138,123],[240,240,0,0,0,1,0,0,0,0,113,164,0,0,122,128] >;
C2×D120 in GAP, Magma, Sage, TeX
C_2\times D_{120} % in TeX
G:=Group("C2xD120"); // GroupNames label
G:=SmallGroup(480,868);
// by ID
G=gap.SmallGroup(480,868);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,142,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^120=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations