direct product, abelian, monomial, 2-elementary
Aliases: C2×C120, SmallGroup(240,84)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C120 |
| C1 — C2×C120 |
| C1 — C2×C120 |
Generators and relations for C2×C120
G = < a,b | a2=b120=1, ab=ba >
Smallest permutation representation of C2×C120
►Regular action on 240 points
Generators in S240
(1 180)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 190)(12 191)(13 192)(14 193)(15 194)(16 195)(17 196)(18 197)(19 198)(20 199)(21 200)(22 201)(23 202)(24 203)(25 204)(26 205)(27 206)(28 207)(29 208)(30 209)(31 210)(32 211)(33 212)(34 213)(35 214)(36 215)(37 216)(38 217)(39 218)(40 219)(41 220)(42 221)(43 222)(44 223)(45 224)(46 225)(47 226)(48 227)(49 228)(50 229)(51 230)(52 231)(53 232)(54 233)(55 234)(56 235)(57 236)(58 237)(59 238)(60 239)(61 240)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 169)(111 170)(112 171)(113 172)(114 173)(115 174)(116 175)(117 176)(118 177)(119 178)(120 179)
(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)
G:=sub<Sym(240)| (1,180)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,192)(14,193)(15,194)(16,195)(17,196)(18,197)(19,198)(20,199)(21,200)(22,201)(23,202)(24,203)(25,204)(26,205)(27,206)(28,207)(29,208)(30,209)(31,210)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,220)(42,221)(43,222)(44,223)(45,224)(46,225)(47,226)(48,227)(49,228)(50,229)(51,230)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,240)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179), (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)>;
G:=Group( (1,180)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,192)(14,193)(15,194)(16,195)(17,196)(18,197)(19,198)(20,199)(21,200)(22,201)(23,202)(24,203)(25,204)(26,205)(27,206)(28,207)(29,208)(30,209)(31,210)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(41,220)(42,221)(43,222)(44,223)(45,224)(46,225)(47,226)(48,227)(49,228)(50,229)(51,230)(52,231)(53,232)(54,233)(55,234)(56,235)(57,236)(58,237)(59,238)(60,239)(61,240)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179), (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) );
G=PermutationGroup([[(1,180),(2,181),(3,182),(4,183),(5,184),(6,185),(7,186),(8,187),(9,188),(10,189),(11,190),(12,191),(13,192),(14,193),(15,194),(16,195),(17,196),(18,197),(19,198),(20,199),(21,200),(22,201),(23,202),(24,203),(25,204),(26,205),(27,206),(28,207),(29,208),(30,209),(31,210),(32,211),(33,212),(34,213),(35,214),(36,215),(37,216),(38,217),(39,218),(40,219),(41,220),(42,221),(43,222),(44,223),(45,224),(46,225),(47,226),(48,227),(49,228),(50,229),(51,230),(52,231),(53,232),(54,233),(55,234),(56,235),(57,236),(58,237),(59,238),(60,239),(61,240),(62,121),(63,122),(64,123),(65,124),(66,125),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,168),(110,169),(111,170),(112,171),(113,172),(114,173),(115,174),(116,175),(117,176),(118,177),(119,178),(120,179)], [(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)]])
C2×C120 is a maximal subgroup of
C60.7C8 C60.26Q8 C120⋊13C4 Dic30⋊8C4 C120⋊10C4 C120⋊9C4 C4.18D60 D30⋊3C8 D60⋊8C4 D60.6C4 C40.69D6
240 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 8A | ··· | 8H | 10A | ··· | 10L | 12A | ··· | 12H | 15A | ··· | 15H | 20A | ··· | 20P | 24A | ··· | 24P | 30A | ··· | 30X | 40A | ··· | 40AF | 60A | ··· | 60AF | 120A | ··· | 120BL |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
240 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C5 | C6 | C6 | C8 | C10 | C10 | C12 | C12 | C15 | C20 | C20 | C24 | C30 | C30 | C40 | C60 | C60 | C120 |
| kernel | C2×C120 | C120 | C2×C60 | 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 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 8 | 32 | 16 | 16 | 64 |
Matrix representation of C2×C120 ►in GL2(𝔽241) generated by
| 240 | 0 |
| 0 | 240 |
| 9 | 0 |
| 0 | 77 |
G:=sub<GL(2,GF(241))| [240,0,0,240],[9,0,0,77] >;
C2×C120 in GAP, Magma, Sage, TeX
C_2\times C_{120} % in TeX
G:=Group("C2xC120"); // GroupNames label
G:=SmallGroup(240,84);
// by ID
G=gap.SmallGroup(240,84);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-2,360,88]);
// Polycyclic
G:=Group<a,b|a^2=b^120=1,a*b=b*a>;
// generators/relations
Export