metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊6D4, D6⋊3D20, C20⋊7D12, C4⋊Dic5⋊9S3, (S3×C10)⋊11D4, (C2×D60)⋊22C2, C5⋊3(C12⋊D4), C12⋊2(C5⋊D4), C4⋊2(C5⋊D12), C3⋊1(C20⋊7D4), C2.26(S3×D20), C6.24(C2×D20), C10.23(S3×D4), C15⋊12(C4⋊D4), C30.154(C2×D4), (C2×C20).303D6, C10.62(C2×D12), D30⋊4C4⋊21C2, C30.92(C4○D4), C6.14(C4○D20), (C2×C12).132D10, (C2×Dic5).46D6, (C2×C30).150C23, (C2×C60).122C22, (C22×S3).76D10, C2.19(D60⋊C2), C10.19(Q8⋊3S3), (C2×Dic3).158D10, (C6×Dic5).90C22, (C22×D15).53C22, (C10×Dic3).192C22, (S3×C2×C4)⋊2D5, (S3×C2×C20)⋊3C2, (C2×C5⋊D12)⋊6C2, (C3×C4⋊Dic5)⋊6C2, C6.17(C2×C5⋊D4), (C2×C4).113(S3×D5), C2.20(C2×C5⋊D12), C22.202(C2×S3×D5), (S3×C2×C10).91C22, (C2×C6).162(C22×D5), (C2×C10).162(C22×S3), SmallGroup(480,536)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊6D4
G = < a,b,c | a60=b4=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >
Subgroups: 1228 in 188 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C4⋊D4, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, C60, S3×C10, S3×C10, D30, C2×C30, C4⋊Dic5, D10⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C12⋊D4, C5⋊D12, C6×Dic5, S3×C20, C10×Dic3, D60, C2×C60, S3×C2×C10, C22×D15, C20⋊7D4, D30⋊4C4, C3×C4⋊Dic5, C2×C5⋊D12, S3×C2×C20, C2×D60, C60⋊6D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C4⋊D4, D20, C5⋊D4, C22×D5, C2×D12, S3×D4, Q8⋊3S3, S3×D5, C2×D20, C4○D20, C2×C5⋊D4, C12⋊D4, C5⋊D12, C2×S3×D5, C20⋊7D4, D60⋊C2, S3×D20, C2×C5⋊D12, C60⋊6D4
►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)
(1 165 114 215)(2 124 115 234)(3 143 116 193)(4 162 117 212)(5 121 118 231)(6 140 119 190)(7 159 120 209)(8 178 61 228)(9 137 62 187)(10 156 63 206)(11 175 64 225)(12 134 65 184)(13 153 66 203)(14 172 67 222)(15 131 68 181)(16 150 69 200)(17 169 70 219)(18 128 71 238)(19 147 72 197)(20 166 73 216)(21 125 74 235)(22 144 75 194)(23 163 76 213)(24 122 77 232)(25 141 78 191)(26 160 79 210)(27 179 80 229)(28 138 81 188)(29 157 82 207)(30 176 83 226)(31 135 84 185)(32 154 85 204)(33 173 86 223)(34 132 87 182)(35 151 88 201)(36 170 89 220)(37 129 90 239)(38 148 91 198)(39 167 92 217)(40 126 93 236)(41 145 94 195)(42 164 95 214)(43 123 96 233)(44 142 97 192)(45 161 98 211)(46 180 99 230)(47 139 100 189)(48 158 101 208)(49 177 102 227)(50 136 103 186)(51 155 104 205)(52 174 105 224)(53 133 106 183)(54 152 107 202)(55 171 108 221)(56 130 109 240)(57 149 110 199)(58 168 111 218)(59 127 112 237)(60 146 113 196)
(1 215)(2 214)(3 213)(4 212)(5 211)(6 210)(7 209)(8 208)(9 207)(10 206)(11 205)(12 204)(13 203)(14 202)(15 201)(16 200)(17 199)(18 198)(19 197)(20 196)(21 195)(22 194)(23 193)(24 192)(25 191)(26 190)(27 189)(28 188)(29 187)(30 186)(31 185)(32 184)(33 183)(34 182)(35 181)(36 240)(37 239)(38 238)(39 237)(40 236)(41 235)(42 234)(43 233)(44 232)(45 231)(46 230)(47 229)(48 228)(49 227)(50 226)(51 225)(52 224)(53 223)(54 222)(55 221)(56 220)(57 219)(58 218)(59 217)(60 216)(61 158)(62 157)(63 156)(64 155)(65 154)(66 153)(67 152)(68 151)(69 150)(70 149)(71 148)(72 147)(73 146)(74 145)(75 144)(76 143)(77 142)(78 141)(79 140)(80 139)(81 138)(82 137)(83 136)(84 135)(85 134)(86 133)(87 132)(88 131)(89 130)(90 129)(91 128)(92 127)(93 126)(94 125)(95 124)(96 123)(97 122)(98 121)(99 180)(100 179)(101 178)(102 177)(103 176)(104 175)(105 174)(106 173)(107 172)(108 171)(109 170)(110 169)(111 168)(112 167)(113 166)(114 165)(115 164)(116 163)(117 162)(118 161)(119 160)(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), (1,165,114,215)(2,124,115,234)(3,143,116,193)(4,162,117,212)(5,121,118,231)(6,140,119,190)(7,159,120,209)(8,178,61,228)(9,137,62,187)(10,156,63,206)(11,175,64,225)(12,134,65,184)(13,153,66,203)(14,172,67,222)(15,131,68,181)(16,150,69,200)(17,169,70,219)(18,128,71,238)(19,147,72,197)(20,166,73,216)(21,125,74,235)(22,144,75,194)(23,163,76,213)(24,122,77,232)(25,141,78,191)(26,160,79,210)(27,179,80,229)(28,138,81,188)(29,157,82,207)(30,176,83,226)(31,135,84,185)(32,154,85,204)(33,173,86,223)(34,132,87,182)(35,151,88,201)(36,170,89,220)(37,129,90,239)(38,148,91,198)(39,167,92,217)(40,126,93,236)(41,145,94,195)(42,164,95,214)(43,123,96,233)(44,142,97,192)(45,161,98,211)(46,180,99,230)(47,139,100,189)(48,158,101,208)(49,177,102,227)(50,136,103,186)(51,155,104,205)(52,174,105,224)(53,133,106,183)(54,152,107,202)(55,171,108,221)(56,130,109,240)(57,149,110,199)(58,168,111,218)(59,127,112,237)(60,146,113,196), (1,215)(2,214)(3,213)(4,212)(5,211)(6,210)(7,209)(8,208)(9,207)(10,206)(11,205)(12,204)(13,203)(14,202)(15,201)(16,200)(17,199)(18,198)(19,197)(20,196)(21,195)(22,194)(23,193)(24,192)(25,191)(26,190)(27,189)(28,188)(29,187)(30,186)(31,185)(32,184)(33,183)(34,182)(35,181)(36,240)(37,239)(38,238)(39,237)(40,236)(41,235)(42,234)(43,233)(44,232)(45,231)(46,230)(47,229)(48,228)(49,227)(50,226)(51,225)(52,224)(53,223)(54,222)(55,221)(56,220)(57,219)(58,218)(59,217)(60,216)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,140)(80,139)(81,138)(82,137)(83,136)(84,135)(85,134)(86,133)(87,132)(88,131)(89,130)(90,129)(91,128)(92,127)(93,126)(94,125)(95,124)(96,123)(97,122)(98,121)(99,180)(100,179)(101,178)(102,177)(103,176)(104,175)(105,174)(106,173)(107,172)(108,171)(109,170)(110,169)(111,168)(112,167)(113,166)(114,165)(115,164)(116,163)(117,162)(118,161)(119,160)(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), (1,165,114,215)(2,124,115,234)(3,143,116,193)(4,162,117,212)(5,121,118,231)(6,140,119,190)(7,159,120,209)(8,178,61,228)(9,137,62,187)(10,156,63,206)(11,175,64,225)(12,134,65,184)(13,153,66,203)(14,172,67,222)(15,131,68,181)(16,150,69,200)(17,169,70,219)(18,128,71,238)(19,147,72,197)(20,166,73,216)(21,125,74,235)(22,144,75,194)(23,163,76,213)(24,122,77,232)(25,141,78,191)(26,160,79,210)(27,179,80,229)(28,138,81,188)(29,157,82,207)(30,176,83,226)(31,135,84,185)(32,154,85,204)(33,173,86,223)(34,132,87,182)(35,151,88,201)(36,170,89,220)(37,129,90,239)(38,148,91,198)(39,167,92,217)(40,126,93,236)(41,145,94,195)(42,164,95,214)(43,123,96,233)(44,142,97,192)(45,161,98,211)(46,180,99,230)(47,139,100,189)(48,158,101,208)(49,177,102,227)(50,136,103,186)(51,155,104,205)(52,174,105,224)(53,133,106,183)(54,152,107,202)(55,171,108,221)(56,130,109,240)(57,149,110,199)(58,168,111,218)(59,127,112,237)(60,146,113,196), (1,215)(2,214)(3,213)(4,212)(5,211)(6,210)(7,209)(8,208)(9,207)(10,206)(11,205)(12,204)(13,203)(14,202)(15,201)(16,200)(17,199)(18,198)(19,197)(20,196)(21,195)(22,194)(23,193)(24,192)(25,191)(26,190)(27,189)(28,188)(29,187)(30,186)(31,185)(32,184)(33,183)(34,182)(35,181)(36,240)(37,239)(38,238)(39,237)(40,236)(41,235)(42,234)(43,233)(44,232)(45,231)(46,230)(47,229)(48,228)(49,227)(50,226)(51,225)(52,224)(53,223)(54,222)(55,221)(56,220)(57,219)(58,218)(59,217)(60,216)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,140)(80,139)(81,138)(82,137)(83,136)(84,135)(85,134)(86,133)(87,132)(88,131)(89,130)(90,129)(91,128)(92,127)(93,126)(94,125)(95,124)(96,123)(97,122)(98,121)(99,180)(100,179)(101,178)(102,177)(103,176)(104,175)(105,174)(106,173)(107,172)(108,171)(109,170)(110,169)(111,168)(112,167)(113,166)(114,165)(115,164)(116,163)(117,162)(118,161)(119,160)(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)], [(1,165,114,215),(2,124,115,234),(3,143,116,193),(4,162,117,212),(5,121,118,231),(6,140,119,190),(7,159,120,209),(8,178,61,228),(9,137,62,187),(10,156,63,206),(11,175,64,225),(12,134,65,184),(13,153,66,203),(14,172,67,222),(15,131,68,181),(16,150,69,200),(17,169,70,219),(18,128,71,238),(19,147,72,197),(20,166,73,216),(21,125,74,235),(22,144,75,194),(23,163,76,213),(24,122,77,232),(25,141,78,191),(26,160,79,210),(27,179,80,229),(28,138,81,188),(29,157,82,207),(30,176,83,226),(31,135,84,185),(32,154,85,204),(33,173,86,223),(34,132,87,182),(35,151,88,201),(36,170,89,220),(37,129,90,239),(38,148,91,198),(39,167,92,217),(40,126,93,236),(41,145,94,195),(42,164,95,214),(43,123,96,233),(44,142,97,192),(45,161,98,211),(46,180,99,230),(47,139,100,189),(48,158,101,208),(49,177,102,227),(50,136,103,186),(51,155,104,205),(52,174,105,224),(53,133,106,183),(54,152,107,202),(55,171,108,221),(56,130,109,240),(57,149,110,199),(58,168,111,218),(59,127,112,237),(60,146,113,196)], [(1,215),(2,214),(3,213),(4,212),(5,211),(6,210),(7,209),(8,208),(9,207),(10,206),(11,205),(12,204),(13,203),(14,202),(15,201),(16,200),(17,199),(18,198),(19,197),(20,196),(21,195),(22,194),(23,193),(24,192),(25,191),(26,190),(27,189),(28,188),(29,187),(30,186),(31,185),(32,184),(33,183),(34,182),(35,181),(36,240),(37,239),(38,238),(39,237),(40,236),(41,235),(42,234),(43,233),(44,232),(45,231),(46,230),(47,229),(48,228),(49,227),(50,226),(51,225),(52,224),(53,223),(54,222),(55,221),(56,220),(57,219),(58,218),(59,217),(60,216),(61,158),(62,157),(63,156),(64,155),(65,154),(66,153),(67,152),(68,151),(69,150),(70,149),(71,148),(72,147),(73,146),(74,145),(75,144),(76,143),(77,142),(78,141),(79,140),(80,139),(81,138),(82,137),(83,136),(84,135),(85,134),(86,133),(87,132),(88,131),(89,130),(90,129),(91,128),(92,127),(93,126),(94,125),(95,124),(96,123),(97,122),(98,121),(99,180),(100,179),(101,178),(102,177),(103,176),(104,175),(105,174),(106,173),(107,172),(108,171),(109,170),(110,169),(111,168),(112,167),(113,166),(114,165),(115,164),(116,163),(117,162),(118,161),(119,160),(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 | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 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 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 60 | 60 | 2 | 2 | 2 | 6 | 6 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 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 | C4○D4 | D10 | D10 | D10 | D12 | C5⋊D4 | D20 | C4○D20 | S3×D4 | Q8⋊3S3 | S3×D5 | C5⋊D12 | C2×S3×D5 | D60⋊C2 | S3×D20 |
| kernel | C60⋊6D4 | D30⋊4C4 | C3×C4⋊Dic5 | C2×C5⋊D12 | S3×C2×C20 | C2×D60 | C4⋊Dic5 | C60 | S3×C10 | S3×C2×C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C12 | D6 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60⋊6D4 ►in GL6(𝔽61)
| 58 | 38 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 2 | 0 | 0 |
| 0 | 0 | 59 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 14 |
| 0 | 0 | 0 | 0 | 39 | 2 |
| 14 | 1 | 0 | 0 | 0 | 0 |
| 47 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 47 | 34 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 39 | 1 |
G:=sub<GL(6,GF(61))| [58,0,0,0,0,0,38,20,0,0,0,0,0,0,59,59,0,0,0,0,2,32,0,0,0,0,0,0,60,39,0,0,0,0,14,2],[14,47,0,0,0,0,1,47,0,0,0,0,0,0,1,17,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[47,14,0,0,0,0,34,14,0,0,0,0,0,0,1,17,0,0,0,0,0,60,0,0,0,0,0,0,60,39,0,0,0,0,0,1] >;
C60⋊6D4 in GAP, Magma, Sage, TeX
C_{60}\rtimes_6D_4 % in TeX
G:=Group("C60:6D4"); // GroupNames label
G:=SmallGroup(480,536);
// by ID
G=gap.SmallGroup(480,536);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations