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 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×2], C12 [×2], C12 [×3], D6 [×8], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×8], C2×C10, C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], D15 [×4], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, C60 [×2], D30 [×4], D30 [×4], C2×C30, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C4×D12, D30.C2 [×4], C6×Dic5 [×2], C10×Dic3 [×2], D60 [×4], C2×C60, C22×D15 [×2], D20⋊8C4, D30⋊4C4 [×2], C12×Dic5, C5×C4⋊Dic3, C2×D30.C2 [×2], C2×D60, D60⋊17C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], D12 [×2], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, C2×C4×D5, D4×D5, Q8⋊2D5, C4×D12, D30.C2 [×2], 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)
(1 174)(2 173)(3 172)(4 171)(5 170)(6 169)(7 168)(8 167)(9 166)(10 165)(11 164)(12 163)(13 162)(14 161)(15 160)(16 159)(17 158)(18 157)(19 156)(20 155)(21 154)(22 153)(23 152)(24 151)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 140)(36 139)(37 138)(38 137)(39 136)(40 135)(41 134)(42 133)(43 132)(44 131)(45 130)(46 129)(47 128)(48 127)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 180)(56 179)(57 178)(58 177)(59 176)(60 175)(61 239)(62 238)(63 237)(64 236)(65 235)(66 234)(67 233)(68 232)(69 231)(70 230)(71 229)(72 228)(73 227)(74 226)(75 225)(76 224)(77 223)(78 222)(79 221)(80 220)(81 219)(82 218)(83 217)(84 216)(85 215)(86 214)(87 213)(88 212)(89 211)(90 210)(91 209)(92 208)(93 207)(94 206)(95 205)(96 204)(97 203)(98 202)(99 201)(100 200)(101 199)(102 198)(103 197)(104 196)(105 195)(106 194)(107 193)(108 192)(109 191)(110 190)(111 189)(112 188)(113 187)(114 186)(115 185)(116 184)(117 183)(118 182)(119 181)(120 240)
(1 114 160 202)(2 65 161 213)(3 76 162 224)(4 87 163 235)(5 98 164 186)(6 109 165 197)(7 120 166 208)(8 71 167 219)(9 82 168 230)(10 93 169 181)(11 104 170 192)(12 115 171 203)(13 66 172 214)(14 77 173 225)(15 88 174 236)(16 99 175 187)(17 110 176 198)(18 61 177 209)(19 72 178 220)(20 83 179 231)(21 94 180 182)(22 105 121 193)(23 116 122 204)(24 67 123 215)(25 78 124 226)(26 89 125 237)(27 100 126 188)(28 111 127 199)(29 62 128 210)(30 73 129 221)(31 84 130 232)(32 95 131 183)(33 106 132 194)(34 117 133 205)(35 68 134 216)(36 79 135 227)(37 90 136 238)(38 101 137 189)(39 112 138 200)(40 63 139 211)(41 74 140 222)(42 85 141 233)(43 96 142 184)(44 107 143 195)(45 118 144 206)(46 69 145 217)(47 80 146 228)(48 91 147 239)(49 102 148 190)(50 113 149 201)(51 64 150 212)(52 75 151 223)(53 86 152 234)(54 97 153 185)(55 108 154 196)(56 119 155 207)(57 70 156 218)(58 81 157 229)(59 92 158 240)(60 103 159 191)
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), (1,174)(2,173)(3,172)(4,171)(5,170)(6,169)(7,168)(8,167)(9,166)(10,165)(11,164)(12,163)(13,162)(14,161)(15,160)(16,159)(17,158)(18,157)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,140)(36,139)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,239)(62,238)(63,237)(64,236)(65,235)(66,234)(67,233)(68,232)(69,231)(70,230)(71,229)(72,228)(73,227)(74,226)(75,225)(76,224)(77,223)(78,222)(79,221)(80,220)(81,219)(82,218)(83,217)(84,216)(85,215)(86,214)(87,213)(88,212)(89,211)(90,210)(91,209)(92,208)(93,207)(94,206)(95,205)(96,204)(97,203)(98,202)(99,201)(100,200)(101,199)(102,198)(103,197)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,240), (1,114,160,202)(2,65,161,213)(3,76,162,224)(4,87,163,235)(5,98,164,186)(6,109,165,197)(7,120,166,208)(8,71,167,219)(9,82,168,230)(10,93,169,181)(11,104,170,192)(12,115,171,203)(13,66,172,214)(14,77,173,225)(15,88,174,236)(16,99,175,187)(17,110,176,198)(18,61,177,209)(19,72,178,220)(20,83,179,231)(21,94,180,182)(22,105,121,193)(23,116,122,204)(24,67,123,215)(25,78,124,226)(26,89,125,237)(27,100,126,188)(28,111,127,199)(29,62,128,210)(30,73,129,221)(31,84,130,232)(32,95,131,183)(33,106,132,194)(34,117,133,205)(35,68,134,216)(36,79,135,227)(37,90,136,238)(38,101,137,189)(39,112,138,200)(40,63,139,211)(41,74,140,222)(42,85,141,233)(43,96,142,184)(44,107,143,195)(45,118,144,206)(46,69,145,217)(47,80,146,228)(48,91,147,239)(49,102,148,190)(50,113,149,201)(51,64,150,212)(52,75,151,223)(53,86,152,234)(54,97,153,185)(55,108,154,196)(56,119,155,207)(57,70,156,218)(58,81,157,229)(59,92,158,240)(60,103,159,191)>;
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), (1,174)(2,173)(3,172)(4,171)(5,170)(6,169)(7,168)(8,167)(9,166)(10,165)(11,164)(12,163)(13,162)(14,161)(15,160)(16,159)(17,158)(18,157)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,140)(36,139)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,239)(62,238)(63,237)(64,236)(65,235)(66,234)(67,233)(68,232)(69,231)(70,230)(71,229)(72,228)(73,227)(74,226)(75,225)(76,224)(77,223)(78,222)(79,221)(80,220)(81,219)(82,218)(83,217)(84,216)(85,215)(86,214)(87,213)(88,212)(89,211)(90,210)(91,209)(92,208)(93,207)(94,206)(95,205)(96,204)(97,203)(98,202)(99,201)(100,200)(101,199)(102,198)(103,197)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,240), (1,114,160,202)(2,65,161,213)(3,76,162,224)(4,87,163,235)(5,98,164,186)(6,109,165,197)(7,120,166,208)(8,71,167,219)(9,82,168,230)(10,93,169,181)(11,104,170,192)(12,115,171,203)(13,66,172,214)(14,77,173,225)(15,88,174,236)(16,99,175,187)(17,110,176,198)(18,61,177,209)(19,72,178,220)(20,83,179,231)(21,94,180,182)(22,105,121,193)(23,116,122,204)(24,67,123,215)(25,78,124,226)(26,89,125,237)(27,100,126,188)(28,111,127,199)(29,62,128,210)(30,73,129,221)(31,84,130,232)(32,95,131,183)(33,106,132,194)(34,117,133,205)(35,68,134,216)(36,79,135,227)(37,90,136,238)(38,101,137,189)(39,112,138,200)(40,63,139,211)(41,74,140,222)(42,85,141,233)(43,96,142,184)(44,107,143,195)(45,118,144,206)(46,69,145,217)(47,80,146,228)(48,91,147,239)(49,102,148,190)(50,113,149,201)(51,64,150,212)(52,75,151,223)(53,86,152,234)(54,97,153,185)(55,108,154,196)(56,119,155,207)(57,70,156,218)(58,81,157,229)(59,92,158,240)(60,103,159,191) );
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)], [(1,174),(2,173),(3,172),(4,171),(5,170),(6,169),(7,168),(8,167),(9,166),(10,165),(11,164),(12,163),(13,162),(14,161),(15,160),(16,159),(17,158),(18,157),(19,156),(20,155),(21,154),(22,153),(23,152),(24,151),(25,150),(26,149),(27,148),(28,147),(29,146),(30,145),(31,144),(32,143),(33,142),(34,141),(35,140),(36,139),(37,138),(38,137),(39,136),(40,135),(41,134),(42,133),(43,132),(44,131),(45,130),(46,129),(47,128),(48,127),(49,126),(50,125),(51,124),(52,123),(53,122),(54,121),(55,180),(56,179),(57,178),(58,177),(59,176),(60,175),(61,239),(62,238),(63,237),(64,236),(65,235),(66,234),(67,233),(68,232),(69,231),(70,230),(71,229),(72,228),(73,227),(74,226),(75,225),(76,224),(77,223),(78,222),(79,221),(80,220),(81,219),(82,218),(83,217),(84,216),(85,215),(86,214),(87,213),(88,212),(89,211),(90,210),(91,209),(92,208),(93,207),(94,206),(95,205),(96,204),(97,203),(98,202),(99,201),(100,200),(101,199),(102,198),(103,197),(104,196),(105,195),(106,194),(107,193),(108,192),(109,191),(110,190),(111,189),(112,188),(113,187),(114,186),(115,185),(116,184),(117,183),(118,182),(119,181),(120,240)], [(1,114,160,202),(2,65,161,213),(3,76,162,224),(4,87,163,235),(5,98,164,186),(6,109,165,197),(7,120,166,208),(8,71,167,219),(9,82,168,230),(10,93,169,181),(11,104,170,192),(12,115,171,203),(13,66,172,214),(14,77,173,225),(15,88,174,236),(16,99,175,187),(17,110,176,198),(18,61,177,209),(19,72,178,220),(20,83,179,231),(21,94,180,182),(22,105,121,193),(23,116,122,204),(24,67,123,215),(25,78,124,226),(26,89,125,237),(27,100,126,188),(28,111,127,199),(29,62,128,210),(30,73,129,221),(31,84,130,232),(32,95,131,183),(33,106,132,194),(34,117,133,205),(35,68,134,216),(36,79,135,227),(37,90,136,238),(38,101,137,189),(39,112,138,200),(40,63,139,211),(41,74,140,222),(42,85,141,233),(43,96,142,184),(44,107,143,195),(45,118,144,206),(46,69,145,217),(47,80,146,228),(48,91,147,239),(49,102,148,190),(50,113,149,201),(51,64,150,212),(52,75,151,223),(53,86,152,234),(54,97,153,185),(55,108,154,196),(56,119,155,207),(57,70,156,218),(58,81,157,229),(59,92,158,240),(60,103,159,191)])
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