metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊29D4, C22⋊2D60, C23.29D30, (C2×C6)⋊5D20, (C2×D60)⋊9C2, (C2×C10)⋊8D12, (C2×C30)⋊23D4, C4⋊3(C15⋊7D4), C3⋊5(C20⋊7D4), C5⋊5(C12⋊7D4), C60⋊5C4⋊15C2, (C2×C4).86D30, C6.45(C2×D20), C2.17(C2×D60), (C22×C4)⋊6D15, (C22×C12)⋊6D5, D30⋊3C4⋊3C2, C20⋊13(C3⋊D4), C12⋊13(C5⋊D4), C15⋊31(C4⋊D4), (C22×C60)⋊10C2, (C22×C20)⋊10S3, C30.273(C2×D4), (C2×C20).395D6, C10.46(C2×D12), (C2×C12).385D10, C6.107(C4○D20), C30.179(C4○D4), (C2×C60).467C22, (C2×C30).304C23, (C22×C6).121D10, (C22×C10).139D6, C10.107(C4○D12), C2.19(D60⋊11C2), C22.56(C22×D15), (C22×C30).144C22, (C2×Dic15).15C22, (C22×D15).11C22, (C2×C15⋊7D4)⋊3C2, C2.7(C2×C15⋊7D4), C6.102(C2×C5⋊D4), C10.102(C2×C3⋊D4), (C2×C6).300(C22×D5), (C2×C10).299(C22×S3), SmallGroup(480,895)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊29D4
G = < a,b,c | a60=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 1236 in 188 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, D15, C30, C30, C4⋊D4, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, Dic15, C60, C60, D30, C2×C30, C2×C30, C2×C30, C4⋊Dic5, D10⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C12⋊7D4, D60, C2×Dic15, C15⋊7D4, C2×C60, C2×C60, C22×D15, C22×C30, C20⋊7D4, C60⋊5C4, D30⋊3C4, C2×D60, C2×C15⋊7D4, C22×C60, C60⋊29D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C3⋊D4, C22×S3, D15, C4⋊D4, D20, C5⋊D4, C22×D5, C2×D12, C4○D12, C2×C3⋊D4, D30, C2×D20, C4○D20, C2×C5⋊D4, C12⋊7D4, D60, C15⋊7D4, C22×D15, C20⋊7D4, C2×D60, D60⋊11C2, C2×C15⋊7D4, C60⋊29D4
►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)
(1 161 112 225)(2 160 113 224)(3 159 114 223)(4 158 115 222)(5 157 116 221)(6 156 117 220)(7 155 118 219)(8 154 119 218)(9 153 120 217)(10 152 61 216)(11 151 62 215)(12 150 63 214)(13 149 64 213)(14 148 65 212)(15 147 66 211)(16 146 67 210)(17 145 68 209)(18 144 69 208)(19 143 70 207)(20 142 71 206)(21 141 72 205)(22 140 73 204)(23 139 74 203)(24 138 75 202)(25 137 76 201)(26 136 77 200)(27 135 78 199)(28 134 79 198)(29 133 80 197)(30 132 81 196)(31 131 82 195)(32 130 83 194)(33 129 84 193)(34 128 85 192)(35 127 86 191)(36 126 87 190)(37 125 88 189)(38 124 89 188)(39 123 90 187)(40 122 91 186)(41 121 92 185)(42 180 93 184)(43 179 94 183)(44 178 95 182)(45 177 96 181)(46 176 97 240)(47 175 98 239)(48 174 99 238)(49 173 100 237)(50 172 101 236)(51 171 102 235)(52 170 103 234)(53 169 104 233)(54 168 105 232)(55 167 106 231)(56 166 107 230)(57 165 108 229)(58 164 109 228)(59 163 110 227)(60 162 111 226)
(1 225)(2 224)(3 223)(4 222)(5 221)(6 220)(7 219)(8 218)(9 217)(10 216)(11 215)(12 214)(13 213)(14 212)(15 211)(16 210)(17 209)(18 208)(19 207)(20 206)(21 205)(22 204)(23 203)(24 202)(25 201)(26 200)(27 199)(28 198)(29 197)(30 196)(31 195)(32 194)(33 193)(34 192)(35 191)(36 190)(37 189)(38 188)(39 187)(40 186)(41 185)(42 184)(43 183)(44 182)(45 181)(46 240)(47 239)(48 238)(49 237)(50 236)(51 235)(52 234)(53 233)(54 232)(55 231)(56 230)(57 229)(58 228)(59 227)(60 226)(61 152)(62 151)(63 150)(64 149)(65 148)(66 147)(67 146)(68 145)(69 144)(70 143)(71 142)(72 141)(73 140)(74 139)(75 138)(76 137)(77 136)(78 135)(79 134)(80 133)(81 132)(82 131)(83 130)(84 129)(85 128)(86 127)(87 126)(88 125)(89 124)(90 123)(91 122)(92 121)(93 180)(94 179)(95 178)(96 177)(97 176)(98 175)(99 174)(100 173)(101 172)(102 171)(103 170)(104 169)(105 168)(106 167)(107 166)(108 165)(109 164)(110 163)(111 162)(112 161)(113 160)(114 159)(115 158)(116 157)(117 156)(118 155)(119 154)(120 153)
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), (1,161,112,225)(2,160,113,224)(3,159,114,223)(4,158,115,222)(5,157,116,221)(6,156,117,220)(7,155,118,219)(8,154,119,218)(9,153,120,217)(10,152,61,216)(11,151,62,215)(12,150,63,214)(13,149,64,213)(14,148,65,212)(15,147,66,211)(16,146,67,210)(17,145,68,209)(18,144,69,208)(19,143,70,207)(20,142,71,206)(21,141,72,205)(22,140,73,204)(23,139,74,203)(24,138,75,202)(25,137,76,201)(26,136,77,200)(27,135,78,199)(28,134,79,198)(29,133,80,197)(30,132,81,196)(31,131,82,195)(32,130,83,194)(33,129,84,193)(34,128,85,192)(35,127,86,191)(36,126,87,190)(37,125,88,189)(38,124,89,188)(39,123,90,187)(40,122,91,186)(41,121,92,185)(42,180,93,184)(43,179,94,183)(44,178,95,182)(45,177,96,181)(46,176,97,240)(47,175,98,239)(48,174,99,238)(49,173,100,237)(50,172,101,236)(51,171,102,235)(52,170,103,234)(53,169,104,233)(54,168,105,232)(55,167,106,231)(56,166,107,230)(57,165,108,229)(58,164,109,228)(59,163,110,227)(60,162,111,226), (1,225)(2,224)(3,223)(4,222)(5,221)(6,220)(7,219)(8,218)(9,217)(10,216)(11,215)(12,214)(13,213)(14,212)(15,211)(16,210)(17,209)(18,208)(19,207)(20,206)(21,205)(22,204)(23,203)(24,202)(25,201)(26,200)(27,199)(28,198)(29,197)(30,196)(31,195)(32,194)(33,193)(34,192)(35,191)(36,190)(37,189)(38,188)(39,187)(40,186)(41,185)(42,184)(43,183)(44,182)(45,181)(46,240)(47,239)(48,238)(49,237)(50,236)(51,235)(52,234)(53,233)(54,232)(55,231)(56,230)(57,229)(58,228)(59,227)(60,226)(61,152)(62,151)(63,150)(64,149)(65,148)(66,147)(67,146)(68,145)(69,144)(70,143)(71,142)(72,141)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,168)(106,167)(107,166)(108,165)(109,164)(110,163)(111,162)(112,161)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153)>;
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), (1,161,112,225)(2,160,113,224)(3,159,114,223)(4,158,115,222)(5,157,116,221)(6,156,117,220)(7,155,118,219)(8,154,119,218)(9,153,120,217)(10,152,61,216)(11,151,62,215)(12,150,63,214)(13,149,64,213)(14,148,65,212)(15,147,66,211)(16,146,67,210)(17,145,68,209)(18,144,69,208)(19,143,70,207)(20,142,71,206)(21,141,72,205)(22,140,73,204)(23,139,74,203)(24,138,75,202)(25,137,76,201)(26,136,77,200)(27,135,78,199)(28,134,79,198)(29,133,80,197)(30,132,81,196)(31,131,82,195)(32,130,83,194)(33,129,84,193)(34,128,85,192)(35,127,86,191)(36,126,87,190)(37,125,88,189)(38,124,89,188)(39,123,90,187)(40,122,91,186)(41,121,92,185)(42,180,93,184)(43,179,94,183)(44,178,95,182)(45,177,96,181)(46,176,97,240)(47,175,98,239)(48,174,99,238)(49,173,100,237)(50,172,101,236)(51,171,102,235)(52,170,103,234)(53,169,104,233)(54,168,105,232)(55,167,106,231)(56,166,107,230)(57,165,108,229)(58,164,109,228)(59,163,110,227)(60,162,111,226), (1,225)(2,224)(3,223)(4,222)(5,221)(6,220)(7,219)(8,218)(9,217)(10,216)(11,215)(12,214)(13,213)(14,212)(15,211)(16,210)(17,209)(18,208)(19,207)(20,206)(21,205)(22,204)(23,203)(24,202)(25,201)(26,200)(27,199)(28,198)(29,197)(30,196)(31,195)(32,194)(33,193)(34,192)(35,191)(36,190)(37,189)(38,188)(39,187)(40,186)(41,185)(42,184)(43,183)(44,182)(45,181)(46,240)(47,239)(48,238)(49,237)(50,236)(51,235)(52,234)(53,233)(54,232)(55,231)(56,230)(57,229)(58,228)(59,227)(60,226)(61,152)(62,151)(63,150)(64,149)(65,148)(66,147)(67,146)(68,145)(69,144)(70,143)(71,142)(72,141)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,168)(106,167)(107,166)(108,165)(109,164)(110,163)(111,162)(112,161)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153) );
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)], [(1,161,112,225),(2,160,113,224),(3,159,114,223),(4,158,115,222),(5,157,116,221),(6,156,117,220),(7,155,118,219),(8,154,119,218),(9,153,120,217),(10,152,61,216),(11,151,62,215),(12,150,63,214),(13,149,64,213),(14,148,65,212),(15,147,66,211),(16,146,67,210),(17,145,68,209),(18,144,69,208),(19,143,70,207),(20,142,71,206),(21,141,72,205),(22,140,73,204),(23,139,74,203),(24,138,75,202),(25,137,76,201),(26,136,77,200),(27,135,78,199),(28,134,79,198),(29,133,80,197),(30,132,81,196),(31,131,82,195),(32,130,83,194),(33,129,84,193),(34,128,85,192),(35,127,86,191),(36,126,87,190),(37,125,88,189),(38,124,89,188),(39,123,90,187),(40,122,91,186),(41,121,92,185),(42,180,93,184),(43,179,94,183),(44,178,95,182),(45,177,96,181),(46,176,97,240),(47,175,98,239),(48,174,99,238),(49,173,100,237),(50,172,101,236),(51,171,102,235),(52,170,103,234),(53,169,104,233),(54,168,105,232),(55,167,106,231),(56,166,107,230),(57,165,108,229),(58,164,109,228),(59,163,110,227),(60,162,111,226)], [(1,225),(2,224),(3,223),(4,222),(5,221),(6,220),(7,219),(8,218),(9,217),(10,216),(11,215),(12,214),(13,213),(14,212),(15,211),(16,210),(17,209),(18,208),(19,207),(20,206),(21,205),(22,204),(23,203),(24,202),(25,201),(26,200),(27,199),(28,198),(29,197),(30,196),(31,195),(32,194),(33,193),(34,192),(35,191),(36,190),(37,189),(38,188),(39,187),(40,186),(41,185),(42,184),(43,183),(44,182),(45,181),(46,240),(47,239),(48,238),(49,237),(50,236),(51,235),(52,234),(53,233),(54,232),(55,231),(56,230),(57,229),(58,228),(59,227),(60,226),(61,152),(62,151),(63,150),(64,149),(65,148),(66,147),(67,146),(68,145),(69,144),(70,143),(71,142),(72,141),(73,140),(74,139),(75,138),(76,137),(77,136),(78,135),(79,134),(80,133),(81,132),(82,131),(83,130),(84,129),(85,128),(86,127),(87,126),(88,125),(89,124),(90,123),(91,122),(92,121),(93,180),(94,179),(95,178),(96,177),(97,176),(98,175),(99,174),(100,173),(101,172),(102,171),(103,170),(104,169),(105,168),(106,167),(107,166),(108,165),(109,164),(110,163),(111,162),(112,161),(113,160),(114,159),(115,158),(116,157),(117,156),(118,155),(119,154),(120,153)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 60 | 60 | 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 | 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 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D12 | D15 | C5⋊D4 | D20 | C4○D12 | D30 | D30 | C4○D20 | C15⋊7D4 | D60 | D60⋊11C2 |
| kernel | C60⋊29D4 | C60⋊5C4 | D30⋊3C4 | C2×D60 | C2×C15⋊7D4 | C22×C60 | C22×C20 | C60 | C2×C30 | C22×C12 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C20 | C2×C10 | C22×C4 | C12 | C2×C6 | C10 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C60⋊29D4 ►in GL4(𝔽61) generated by
| 46 | 23 | 0 | 0 |
| 38 | 23 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 51 | 37 |
| 9 | 22 | 0 | 0 |
| 13 | 52 | 0 | 0 |
| 0 | 0 | 41 | 8 |
| 0 | 0 | 49 | 20 |
| 9 | 22 | 0 | 0 |
| 13 | 52 | 0 | 0 |
| 0 | 0 | 41 | 8 |
| 0 | 0 | 34 | 20 |
G:=sub<GL(4,GF(61))| [46,38,0,0,23,23,0,0,0,0,33,51,0,0,0,37],[9,13,0,0,22,52,0,0,0,0,41,49,0,0,8,20],[9,13,0,0,22,52,0,0,0,0,41,34,0,0,8,20] >;
C60⋊29D4 in GAP, Magma, Sage, TeX
C_{60}\rtimes_{29}D_4 % in TeX
G:=Group("C60:29D4"); // GroupNames label
G:=SmallGroup(480,895);
// by ID
G=gap.SmallGroup(480,895);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations