metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊17C4, Dic5⋊5D12, C5⋊2(C4×D12), C20⋊6(C4×S3), C12⋊7(C4×D5), C15⋊10(C4×D4), C60⋊19(C2×C4), C2.4(D5×D12), C6.19(D4×D5), D30⋊10(C2×C4), C4⋊Dic3⋊17D5, (C3×Dic5)⋊8D4, (C4×Dic5)⋊6S3, C30.43(C2×D4), C3⋊1(D20⋊8C4), C4⋊2(D30.C2), (C12×Dic5)⋊6C2, (C2×D60).19C2, (C2×C20).125D6, C10.19(C2×D12), D30⋊4C4⋊13C2, C30.62(C4○D4), (C2×C12).304D10, C10.11(C4○D12), C2.4(C12.28D10), (C2×C60).148C22, C30.126(C22×C4), (C2×C30).108C23, C6.13(Q8⋊2D5), (C2×Dic5).177D6, (C2×Dic3).102D10, (C6×Dic5).202C22, (C10×Dic3).66C22, (C22×D15).35C22, C6.48(C2×C4×D5), C10.80(S3×C2×C4), (C5×C4⋊Dic3)⋊5C2, C22.54(C2×S3×D5), (C2×D30.C2)⋊4C2, (C2×C4).161(S3×D5), C2.12(C2×D30.C2), (C2×C6).120(C22×D5), (C2×C10).120(C22×S3), SmallGroup(480,494)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊17C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >
Subgroups: 1132 in 188 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, C3×Dic5, C60, D30, D30, C2×C30, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C4×D12, D30.C2, C6×Dic5, C10×Dic3, D60, C2×C60, C22×D15, D20⋊8C4, D30⋊4C4, C12×Dic5, C5×C4⋊Dic3, C2×D30.C2, C2×D60, D60⋊17C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, D12, C22×S3, C4×D4, C4×D5, C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, C2×C4×D5, D4×D5, Q8⋊2D5, C4×D12, D30.C2, C2×S3×D5, D20⋊8C4, C12.28D10, D5×D12, C2×D30.C2, D60⋊17C4
►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)
(1 202)(2 201)(3 200)(4 199)(5 198)(6 197)(7 196)(8 195)(9 194)(10 193)(11 192)(12 191)(13 190)(14 189)(15 188)(16 187)(17 186)(18 185)(19 184)(20 183)(21 182)(22 181)(23 240)(24 239)(25 238)(26 237)(27 236)(28 235)(29 234)(30 233)(31 232)(32 231)(33 230)(34 229)(35 228)(36 227)(37 226)(38 225)(39 224)(40 223)(41 222)(42 221)(43 220)(44 219)(45 218)(46 217)(47 216)(48 215)(49 214)(50 213)(51 212)(52 211)(53 210)(54 209)(55 208)(56 207)(57 206)(58 205)(59 204)(60 203)(61 131)(62 130)(63 129)(64 128)(65 127)(66 126)(67 125)(68 124)(69 123)(70 122)(71 121)(72 180)(73 179)(74 178)(75 177)(76 176)(77 175)(78 174)(79 173)(80 172)(81 171)(82 170)(83 169)(84 168)(85 167)(86 166)(87 165)(88 164)(89 163)(90 162)(91 161)(92 160)(93 159)(94 158)(95 157)(96 156)(97 155)(98 154)(99 153)(100 152)(101 151)(102 150)(103 149)(104 148)(105 147)(106 146)(107 145)(108 144)(109 143)(110 142)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 135)(118 134)(119 133)(120 132)
(1 84 188 124)(2 95 189 135)(3 106 190 146)(4 117 191 157)(5 68 192 168)(6 79 193 179)(7 90 194 130)(8 101 195 141)(9 112 196 152)(10 63 197 163)(11 74 198 174)(12 85 199 125)(13 96 200 136)(14 107 201 147)(15 118 202 158)(16 69 203 169)(17 80 204 180)(18 91 205 131)(19 102 206 142)(20 113 207 153)(21 64 208 164)(22 75 209 175)(23 86 210 126)(24 97 211 137)(25 108 212 148)(26 119 213 159)(27 70 214 170)(28 81 215 121)(29 92 216 132)(30 103 217 143)(31 114 218 154)(32 65 219 165)(33 76 220 176)(34 87 221 127)(35 98 222 138)(36 109 223 149)(37 120 224 160)(38 71 225 171)(39 82 226 122)(40 93 227 133)(41 104 228 144)(42 115 229 155)(43 66 230 166)(44 77 231 177)(45 88 232 128)(46 99 233 139)(47 110 234 150)(48 61 235 161)(49 72 236 172)(50 83 237 123)(51 94 238 134)(52 105 239 145)(53 116 240 156)(54 67 181 167)(55 78 182 178)(56 89 183 129)(57 100 184 140)(58 111 185 151)(59 62 186 162)(60 73 187 173)
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), (1,202)(2,201)(3,200)(4,199)(5,198)(6,197)(7,196)(8,195)(9,194)(10,193)(11,192)(12,191)(13,190)(14,189)(15,188)(16,187)(17,186)(18,185)(19,184)(20,183)(21,182)(22,181)(23,240)(24,239)(25,238)(26,237)(27,236)(28,235)(29,234)(30,233)(31,232)(32,231)(33,230)(34,229)(35,228)(36,227)(37,226)(38,225)(39,224)(40,223)(41,222)(42,221)(43,220)(44,219)(45,218)(46,217)(47,216)(48,215)(49,214)(50,213)(51,212)(52,211)(53,210)(54,209)(55,208)(56,207)(57,206)(58,205)(59,204)(60,203)(61,131)(62,130)(63,129)(64,128)(65,127)(66,126)(67,125)(68,124)(69,123)(70,122)(71,121)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(81,171)(82,170)(83,169)(84,168)(85,167)(86,166)(87,165)(88,164)(89,163)(90,162)(91,161)(92,160)(93,159)(94,158)(95,157)(96,156)(97,155)(98,154)(99,153)(100,152)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132), (1,84,188,124)(2,95,189,135)(3,106,190,146)(4,117,191,157)(5,68,192,168)(6,79,193,179)(7,90,194,130)(8,101,195,141)(9,112,196,152)(10,63,197,163)(11,74,198,174)(12,85,199,125)(13,96,200,136)(14,107,201,147)(15,118,202,158)(16,69,203,169)(17,80,204,180)(18,91,205,131)(19,102,206,142)(20,113,207,153)(21,64,208,164)(22,75,209,175)(23,86,210,126)(24,97,211,137)(25,108,212,148)(26,119,213,159)(27,70,214,170)(28,81,215,121)(29,92,216,132)(30,103,217,143)(31,114,218,154)(32,65,219,165)(33,76,220,176)(34,87,221,127)(35,98,222,138)(36,109,223,149)(37,120,224,160)(38,71,225,171)(39,82,226,122)(40,93,227,133)(41,104,228,144)(42,115,229,155)(43,66,230,166)(44,77,231,177)(45,88,232,128)(46,99,233,139)(47,110,234,150)(48,61,235,161)(49,72,236,172)(50,83,237,123)(51,94,238,134)(52,105,239,145)(53,116,240,156)(54,67,181,167)(55,78,182,178)(56,89,183,129)(57,100,184,140)(58,111,185,151)(59,62,186,162)(60,73,187,173)>;
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), (1,202)(2,201)(3,200)(4,199)(5,198)(6,197)(7,196)(8,195)(9,194)(10,193)(11,192)(12,191)(13,190)(14,189)(15,188)(16,187)(17,186)(18,185)(19,184)(20,183)(21,182)(22,181)(23,240)(24,239)(25,238)(26,237)(27,236)(28,235)(29,234)(30,233)(31,232)(32,231)(33,230)(34,229)(35,228)(36,227)(37,226)(38,225)(39,224)(40,223)(41,222)(42,221)(43,220)(44,219)(45,218)(46,217)(47,216)(48,215)(49,214)(50,213)(51,212)(52,211)(53,210)(54,209)(55,208)(56,207)(57,206)(58,205)(59,204)(60,203)(61,131)(62,130)(63,129)(64,128)(65,127)(66,126)(67,125)(68,124)(69,123)(70,122)(71,121)(72,180)(73,179)(74,178)(75,177)(76,176)(77,175)(78,174)(79,173)(80,172)(81,171)(82,170)(83,169)(84,168)(85,167)(86,166)(87,165)(88,164)(89,163)(90,162)(91,161)(92,160)(93,159)(94,158)(95,157)(96,156)(97,155)(98,154)(99,153)(100,152)(101,151)(102,150)(103,149)(104,148)(105,147)(106,146)(107,145)(108,144)(109,143)(110,142)(111,141)(112,140)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132), (1,84,188,124)(2,95,189,135)(3,106,190,146)(4,117,191,157)(5,68,192,168)(6,79,193,179)(7,90,194,130)(8,101,195,141)(9,112,196,152)(10,63,197,163)(11,74,198,174)(12,85,199,125)(13,96,200,136)(14,107,201,147)(15,118,202,158)(16,69,203,169)(17,80,204,180)(18,91,205,131)(19,102,206,142)(20,113,207,153)(21,64,208,164)(22,75,209,175)(23,86,210,126)(24,97,211,137)(25,108,212,148)(26,119,213,159)(27,70,214,170)(28,81,215,121)(29,92,216,132)(30,103,217,143)(31,114,218,154)(32,65,219,165)(33,76,220,176)(34,87,221,127)(35,98,222,138)(36,109,223,149)(37,120,224,160)(38,71,225,171)(39,82,226,122)(40,93,227,133)(41,104,228,144)(42,115,229,155)(43,66,230,166)(44,77,231,177)(45,88,232,128)(46,99,233,139)(47,110,234,150)(48,61,235,161)(49,72,236,172)(50,83,237,123)(51,94,238,134)(52,105,239,145)(53,116,240,156)(54,67,181,167)(55,78,182,178)(56,89,183,129)(57,100,184,140)(58,111,185,151)(59,62,186,162)(60,73,187,173) );
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)], [(1,202),(2,201),(3,200),(4,199),(5,198),(6,197),(7,196),(8,195),(9,194),(10,193),(11,192),(12,191),(13,190),(14,189),(15,188),(16,187),(17,186),(18,185),(19,184),(20,183),(21,182),(22,181),(23,240),(24,239),(25,238),(26,237),(27,236),(28,235),(29,234),(30,233),(31,232),(32,231),(33,230),(34,229),(35,228),(36,227),(37,226),(38,225),(39,224),(40,223),(41,222),(42,221),(43,220),(44,219),(45,218),(46,217),(47,216),(48,215),(49,214),(50,213),(51,212),(52,211),(53,210),(54,209),(55,208),(56,207),(57,206),(58,205),(59,204),(60,203),(61,131),(62,130),(63,129),(64,128),(65,127),(66,126),(67,125),(68,124),(69,123),(70,122),(71,121),(72,180),(73,179),(74,178),(75,177),(76,176),(77,175),(78,174),(79,173),(80,172),(81,171),(82,170),(83,169),(84,168),(85,167),(86,166),(87,165),(88,164),(89,163),(90,162),(91,161),(92,160),(93,159),(94,158),(95,157),(96,156),(97,155),(98,154),(99,153),(100,152),(101,151),(102,150),(103,149),(104,148),(105,147),(106,146),(107,145),(108,144),(109,143),(110,142),(111,141),(112,140),(113,139),(114,138),(115,137),(116,136),(117,135),(118,134),(119,133),(120,132)], [(1,84,188,124),(2,95,189,135),(3,106,190,146),(4,117,191,157),(5,68,192,168),(6,79,193,179),(7,90,194,130),(8,101,195,141),(9,112,196,152),(10,63,197,163),(11,74,198,174),(12,85,199,125),(13,96,200,136),(14,107,201,147),(15,118,202,158),(16,69,203,169),(17,80,204,180),(18,91,205,131),(19,102,206,142),(20,113,207,153),(21,64,208,164),(22,75,209,175),(23,86,210,126),(24,97,211,137),(25,108,212,148),(26,119,213,159),(27,70,214,170),(28,81,215,121),(29,92,216,132),(30,103,217,143),(31,114,218,154),(32,65,219,165),(33,76,220,176),(34,87,221,127),(35,98,222,138),(36,109,223,149),(37,120,224,160),(38,71,225,171),(39,82,226,122),(40,93,227,133),(41,104,228,144),(42,115,229,155),(43,66,230,166),(44,77,231,177),(45,88,232,128),(46,99,233,139),(47,110,234,150),(48,61,235,161),(49,72,236,172),(50,83,237,123),(51,94,238,134),(52,105,239,145),(53,116,240,156),(54,67,181,167),(55,78,182,178),(56,89,183,129),(57,100,184,140),(58,111,185,151),(59,62,186,162),(60,73,187,173)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D12 | C4×S3 | C4×D5 | C4○D12 | S3×D5 | D4×D5 | Q8⋊2D5 | D30.C2 | C2×S3×D5 | C12.28D10 | D5×D12 |
| kernel | D60⋊17C4 | D30⋊4C4 | C12×Dic5 | C5×C4⋊Dic3 | C2×D30.C2 | C2×D60 | D60 | C4×Dic5 | C3×Dic5 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic5 | C20 | C12 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of D60⋊17C4 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 38 | 23 | 0 | 0 |
| 0 | 38 | 15 | 0 | 0 |
| 0 | 0 | 0 | 44 | 18 |
| 0 | 0 | 0 | 44 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 60 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 52 | 43 | 0 | 0 |
| 0 | 52 | 9 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,38,38,0,0,0,23,15,0,0,0,0,0,44,44,0,0,0,18,0],[1,0,0,0,0,0,60,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,60,60],[11,0,0,0,0,0,52,52,0,0,0,43,9,0,0,0,0,0,60,0,0,0,0,0,60] >;
D60⋊17C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{17}C_4 % in TeX
G:=Group("D60:17C4"); // GroupNames label
G:=SmallGroup(480,494);
// by ID
G=gap.SmallGroup(480,494);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations