metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60:4D4, D6:2D20, (C6xD20):5C2, (C2xD20):4S3, (S3xC10):10D4, C5:3(D6:3D4), C3:3(C20:7D4), C4:3(C15:D4), C12:1(C5:D4), C60:5C4:27C2, C6.21(C2xD20), C2.22(S3xD20), C10.20(S3xD4), C15:10(C4:D4), C20:10(C3:D4), (C2xC20).301D6, C30.151(C2xD4), C6.58(C4oD20), C30.90(C4oD4), (C2xC12).130D10, (C22xD5).17D6, D10:Dic3:17C2, (C2xC30).146C23, (C2xC60).120C22, (C22xS3).75D10, C10.33(D4:2S3), C2.18(D20:5S3), (C2xDic3).156D10, (C10xDic3).190C22, (C2xDic15).113C22, (S3xC2xC4):1D5, (S3xC2xC20):2C2, (C2xC15:D4):6C2, C6.87(C2xC5:D4), (C2xC4).111(S3xD5), C10.88(C2xC3:D4), C2.20(C2xC15:D4), (D5xC2xC6).31C22, C22.198(C2xS3xD5), (S3xC2xC10).90C22, (C2xC6).158(C22xD5), (C2xC10).158(C22xS3), SmallGroup(480,532)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60:4D4
G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a19, cbc=b-1 >
Subgroups: 988 in 188 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2xC4, C2xC4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2xC6, C2xC6, C15, C22:C4, C4:C4, C22xC4, C2xD4, Dic5, C20, C20, D10, C2xC10, C2xC10, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C5xS3, C3xD5, C30, C4:D4, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C22xD5, C22xC10, C4:Dic3, C6.D4, S3xC2xC4, C2xC3:D4, C6xD4, C5xDic3, Dic15, C60, C6xD5, S3xC10, S3xC10, C2xC30, C4:Dic5, D10:C4, C2xD20, C2xC5:D4, C22xC20, D6:3D4, C15:D4, C3xD20, S3xC20, C10xDic3, C2xDic15, C2xC60, D5xC2xC6, S3xC2xC10, C20:7D4, D10:Dic3, C60:5C4, C2xC15:D4, C6xD20, S3xC2xC20, C60:4D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, C4oD4, D10, C3:D4, C22xS3, C4:D4, D20, C5:D4, C22xD5, S3xD4, D4:2S3, C2xC3:D4, S3xD5, C2xD20, C4oD20, C2xC5:D4, D6:3D4, C15:D4, C2xS3xD5, C20:7D4, D20:5S3, S3xD20, C2xC15:D4, C60:4D4
►On 240 points
Generators in S240
(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)
(1 120 159 221)(2 119 160 220)(3 118 161 219)(4 117 162 218)(5 116 163 217)(6 115 164 216)(7 114 165 215)(8 113 166 214)(9 112 167 213)(10 111 168 212)(11 110 169 211)(12 109 170 210)(13 108 171 209)(14 107 172 208)(15 106 173 207)(16 105 174 206)(17 104 175 205)(18 103 176 204)(19 102 177 203)(20 101 178 202)(21 100 179 201)(22 99 180 200)(23 98 121 199)(24 97 122 198)(25 96 123 197)(26 95 124 196)(27 94 125 195)(28 93 126 194)(29 92 127 193)(30 91 128 192)(31 90 129 191)(32 89 130 190)(33 88 131 189)(34 87 132 188)(35 86 133 187)(36 85 134 186)(37 84 135 185)(38 83 136 184)(39 82 137 183)(40 81 138 182)(41 80 139 181)(42 79 140 240)(43 78 141 239)(44 77 142 238)(45 76 143 237)(46 75 144 236)(47 74 145 235)(48 73 146 234)(49 72 147 233)(50 71 148 232)(51 70 149 231)(52 69 150 230)(53 68 151 229)(54 67 152 228)(55 66 153 227)(56 65 154 226)(57 64 155 225)(58 63 156 224)(59 62 157 223)(60 61 158 222)
(1 221)(2 240)(3 199)(4 218)(5 237)(6 196)(7 215)(8 234)(9 193)(10 212)(11 231)(12 190)(13 209)(14 228)(15 187)(16 206)(17 225)(18 184)(19 203)(20 222)(21 181)(22 200)(23 219)(24 238)(25 197)(26 216)(27 235)(28 194)(29 213)(30 232)(31 191)(32 210)(33 229)(34 188)(35 207)(36 226)(37 185)(38 204)(39 223)(40 182)(41 201)(42 220)(43 239)(44 198)(45 217)(46 236)(47 195)(48 214)(49 233)(50 192)(51 211)(52 230)(53 189)(54 208)(55 227)(56 186)(57 205)(58 224)(59 183)(60 202)(61 178)(62 137)(63 156)(64 175)(65 134)(66 153)(67 172)(68 131)(69 150)(70 169)(71 128)(72 147)(73 166)(74 125)(75 144)(76 163)(77 122)(78 141)(79 160)(80 179)(81 138)(82 157)(83 176)(84 135)(85 154)(86 173)(87 132)(88 151)(89 170)(90 129)(91 148)(92 167)(93 126)(94 145)(95 164)(96 123)(97 142)(98 161)(99 180)(100 139)(101 158)(102 177)(103 136)(104 155)(105 174)(106 133)(107 152)(108 171)(109 130)(110 149)(111 168)(112 127)(113 146)(114 165)(115 124)(116 143)(117 162)(118 121)(119 140)(120 159)
G:=sub<Sym(240)| (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), (1,120,159,221)(2,119,160,220)(3,118,161,219)(4,117,162,218)(5,116,163,217)(6,115,164,216)(7,114,165,215)(8,113,166,214)(9,112,167,213)(10,111,168,212)(11,110,169,211)(12,109,170,210)(13,108,171,209)(14,107,172,208)(15,106,173,207)(16,105,174,206)(17,104,175,205)(18,103,176,204)(19,102,177,203)(20,101,178,202)(21,100,179,201)(22,99,180,200)(23,98,121,199)(24,97,122,198)(25,96,123,197)(26,95,124,196)(27,94,125,195)(28,93,126,194)(29,92,127,193)(30,91,128,192)(31,90,129,191)(32,89,130,190)(33,88,131,189)(34,87,132,188)(35,86,133,187)(36,85,134,186)(37,84,135,185)(38,83,136,184)(39,82,137,183)(40,81,138,182)(41,80,139,181)(42,79,140,240)(43,78,141,239)(44,77,142,238)(45,76,143,237)(46,75,144,236)(47,74,145,235)(48,73,146,234)(49,72,147,233)(50,71,148,232)(51,70,149,231)(52,69,150,230)(53,68,151,229)(54,67,152,228)(55,66,153,227)(56,65,154,226)(57,64,155,225)(58,63,156,224)(59,62,157,223)(60,61,158,222), (1,221)(2,240)(3,199)(4,218)(5,237)(6,196)(7,215)(8,234)(9,193)(10,212)(11,231)(12,190)(13,209)(14,228)(15,187)(16,206)(17,225)(18,184)(19,203)(20,222)(21,181)(22,200)(23,219)(24,238)(25,197)(26,216)(27,235)(28,194)(29,213)(30,232)(31,191)(32,210)(33,229)(34,188)(35,207)(36,226)(37,185)(38,204)(39,223)(40,182)(41,201)(42,220)(43,239)(44,198)(45,217)(46,236)(47,195)(48,214)(49,233)(50,192)(51,211)(52,230)(53,189)(54,208)(55,227)(56,186)(57,205)(58,224)(59,183)(60,202)(61,178)(62,137)(63,156)(64,175)(65,134)(66,153)(67,172)(68,131)(69,150)(70,169)(71,128)(72,147)(73,166)(74,125)(75,144)(76,163)(77,122)(78,141)(79,160)(80,179)(81,138)(82,157)(83,176)(84,135)(85,154)(86,173)(87,132)(88,151)(89,170)(90,129)(91,148)(92,167)(93,126)(94,145)(95,164)(96,123)(97,142)(98,161)(99,180)(100,139)(101,158)(102,177)(103,136)(104,155)(105,174)(106,133)(107,152)(108,171)(109,130)(110,149)(111,168)(112,127)(113,146)(114,165)(115,124)(116,143)(117,162)(118,121)(119,140)(120,159)>;
G:=Group( (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), (1,120,159,221)(2,119,160,220)(3,118,161,219)(4,117,162,218)(5,116,163,217)(6,115,164,216)(7,114,165,215)(8,113,166,214)(9,112,167,213)(10,111,168,212)(11,110,169,211)(12,109,170,210)(13,108,171,209)(14,107,172,208)(15,106,173,207)(16,105,174,206)(17,104,175,205)(18,103,176,204)(19,102,177,203)(20,101,178,202)(21,100,179,201)(22,99,180,200)(23,98,121,199)(24,97,122,198)(25,96,123,197)(26,95,124,196)(27,94,125,195)(28,93,126,194)(29,92,127,193)(30,91,128,192)(31,90,129,191)(32,89,130,190)(33,88,131,189)(34,87,132,188)(35,86,133,187)(36,85,134,186)(37,84,135,185)(38,83,136,184)(39,82,137,183)(40,81,138,182)(41,80,139,181)(42,79,140,240)(43,78,141,239)(44,77,142,238)(45,76,143,237)(46,75,144,236)(47,74,145,235)(48,73,146,234)(49,72,147,233)(50,71,148,232)(51,70,149,231)(52,69,150,230)(53,68,151,229)(54,67,152,228)(55,66,153,227)(56,65,154,226)(57,64,155,225)(58,63,156,224)(59,62,157,223)(60,61,158,222), (1,221)(2,240)(3,199)(4,218)(5,237)(6,196)(7,215)(8,234)(9,193)(10,212)(11,231)(12,190)(13,209)(14,228)(15,187)(16,206)(17,225)(18,184)(19,203)(20,222)(21,181)(22,200)(23,219)(24,238)(25,197)(26,216)(27,235)(28,194)(29,213)(30,232)(31,191)(32,210)(33,229)(34,188)(35,207)(36,226)(37,185)(38,204)(39,223)(40,182)(41,201)(42,220)(43,239)(44,198)(45,217)(46,236)(47,195)(48,214)(49,233)(50,192)(51,211)(52,230)(53,189)(54,208)(55,227)(56,186)(57,205)(58,224)(59,183)(60,202)(61,178)(62,137)(63,156)(64,175)(65,134)(66,153)(67,172)(68,131)(69,150)(70,169)(71,128)(72,147)(73,166)(74,125)(75,144)(76,163)(77,122)(78,141)(79,160)(80,179)(81,138)(82,157)(83,176)(84,135)(85,154)(86,173)(87,132)(88,151)(89,170)(90,129)(91,148)(92,167)(93,126)(94,145)(95,164)(96,123)(97,142)(98,161)(99,180)(100,139)(101,158)(102,177)(103,136)(104,155)(105,174)(106,133)(107,152)(108,171)(109,130)(110,149)(111,168)(112,127)(113,146)(114,165)(115,124)(116,143)(117,162)(118,121)(119,140)(120,159) );
G=PermutationGroup([[(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)], [(1,120,159,221),(2,119,160,220),(3,118,161,219),(4,117,162,218),(5,116,163,217),(6,115,164,216),(7,114,165,215),(8,113,166,214),(9,112,167,213),(10,111,168,212),(11,110,169,211),(12,109,170,210),(13,108,171,209),(14,107,172,208),(15,106,173,207),(16,105,174,206),(17,104,175,205),(18,103,176,204),(19,102,177,203),(20,101,178,202),(21,100,179,201),(22,99,180,200),(23,98,121,199),(24,97,122,198),(25,96,123,197),(26,95,124,196),(27,94,125,195),(28,93,126,194),(29,92,127,193),(30,91,128,192),(31,90,129,191),(32,89,130,190),(33,88,131,189),(34,87,132,188),(35,86,133,187),(36,85,134,186),(37,84,135,185),(38,83,136,184),(39,82,137,183),(40,81,138,182),(41,80,139,181),(42,79,140,240),(43,78,141,239),(44,77,142,238),(45,76,143,237),(46,75,144,236),(47,74,145,235),(48,73,146,234),(49,72,147,233),(50,71,148,232),(51,70,149,231),(52,69,150,230),(53,68,151,229),(54,67,152,228),(55,66,153,227),(56,65,154,226),(57,64,155,225),(58,63,156,224),(59,62,157,223),(60,61,158,222)], [(1,221),(2,240),(3,199),(4,218),(5,237),(6,196),(7,215),(8,234),(9,193),(10,212),(11,231),(12,190),(13,209),(14,228),(15,187),(16,206),(17,225),(18,184),(19,203),(20,222),(21,181),(22,200),(23,219),(24,238),(25,197),(26,216),(27,235),(28,194),(29,213),(30,232),(31,191),(32,210),(33,229),(34,188),(35,207),(36,226),(37,185),(38,204),(39,223),(40,182),(41,201),(42,220),(43,239),(44,198),(45,217),(46,236),(47,195),(48,214),(49,233),(50,192),(51,211),(52,230),(53,189),(54,208),(55,227),(56,186),(57,205),(58,224),(59,183),(60,202),(61,178),(62,137),(63,156),(64,175),(65,134),(66,153),(67,172),(68,131),(69,150),(70,169),(71,128),(72,147),(73,166),(74,125),(75,144),(76,163),(77,122),(78,141),(79,160),(80,179),(81,138),(82,157),(83,176),(84,135),(85,154),(86,173),(87,132),(88,151),(89,170),(90,129),(91,148),(92,167),(93,126),(94,145),(95,164),(96,123),(97,142),(98,161),(99,180),(100,139),(101,158),(102,177),(103,136),(104,155),(105,174),(106,133),(107,152),(108,171),(109,130),(110,149),(111,168),(112,127),(113,146),(114,165),(115,124),(116,143),(117,162),(118,121),(119,140),(120,159)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 20 | 20 | 2 | 2 | 2 | 6 | 6 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4oD4 | D10 | D10 | D10 | C3:D4 | C5:D4 | D20 | C4oD20 | S3xD4 | D4:2S3 | S3xD5 | C15:D4 | C2xS3xD5 | D20:5S3 | S3xD20 |
| kernel | C60:4D4 | D10:Dic3 | C60:5C4 | C2xC15:D4 | C6xD20 | S3xC2xC20 | C2xD20 | C60 | S3xC10 | S3xC2xC4 | C2xC20 | C22xD5 | C30 | C2xDic3 | C2xC12 | C22xS3 | C20 | C12 | D6 | C6 | C10 | C10 | C2xC4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60:4D4 ►in GL6(F61)
| 33 | 0 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 18 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 58 |
| 0 | 0 | 0 | 0 | 60 | 22 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 5 |
| 0 | 0 | 0 | 0 | 1 | 39 |
G:=sub<GL(6,GF(61))| [33,0,0,0,0,0,0,37,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,41,0,0,0,0,0,18,3],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,39,60,0,0,0,0,58,22],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,1,0,0,0,0,5,39] >;
C60:4D4 in GAP, Magma, Sage, TeX
C_{60}\rtimes_4D_4 % in TeX
G:=Group("C60:4D4"); // GroupNames label
G:=SmallGroup(480,532);
// by ID
G=gap.SmallGroup(480,532);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations