metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊8C4, C6.5D40, C2.2D120, C10.5D24, C30.26D8, C60.202D4, C30.18SD16, C22.10D60, (C2×C40)⋊4S3, (C2×C8)⋊2D15, (C2×C24)⋊4D5, (C2×C120)⋊6C2, C60⋊5C4⋊1C2, C4.8(C4×D15), C20.83(C4×S3), (C2×D60).1C2, (C2×C6).16D20, C12.51(C4×D5), (C2×C4).74D30, C5⋊3(C2.D24), C3⋊2(D20⋊5C4), C60.188(C2×C4), (C2×C10).16D12, (C2×C20).387D6, (C2×C30).102D4, C6.3(C40⋊C2), C15⋊14(D4⋊C4), C10.3(C24⋊C2), C2.3(C24⋊D5), C10.33(D6⋊C4), (C2×C12).389D10, C20.99(C3⋊D4), C4.20(C15⋊7D4), C12.99(C5⋊D4), C2.8(D30⋊3C4), C30.75(C22⋊C4), (C2×C60).474C22, C6.18(D10⋊C4), SmallGroup(480,181)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊8C4
G = < a,b,c | a60=b2=c4=1, bab=cac-1=a-1, cbc-1=a43b >
Subgroups: 884 in 100 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, D10, C2×C10, C24, D12, C2×Dic3, C2×C12, C22×S3, D15, C30, D4⋊C4, C40, D20, C2×Dic5, C2×C20, C22×D5, C4⋊Dic3, C2×C24, C2×D12, Dic15, C60, D30, C2×C30, C4⋊Dic5, C2×C40, C2×D20, C2.D24, C120, D60, D60, C2×Dic15, C2×C60, C22×D15, D20⋊5C4, C60⋊5C4, C2×C120, C2×D60, D60⋊8C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D8, SD16, D10, C4×S3, D12, C3⋊D4, D15, D4⋊C4, C4×D5, D20, C5⋊D4, C24⋊C2, D24, D6⋊C4, D30, C40⋊C2, D40, D10⋊C4, C2.D24, C4×D15, D60, C15⋊7D4, D20⋊5C4, C24⋊D5, D120, D30⋊3C4, D60⋊8C4
►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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(121 145)(122 144)(123 143)(124 142)(125 141)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(146 180)(147 179)(148 178)(149 177)(150 176)(151 175)(152 174)(153 173)(154 172)(155 171)(156 170)(157 169)(158 168)(159 167)(160 166)(161 165)(162 164)(181 229)(182 228)(183 227)(184 226)(185 225)(186 224)(187 223)(188 222)(189 221)(190 220)(191 219)(192 218)(193 217)(194 216)(195 215)(196 214)(197 213)(198 212)(199 211)(200 210)(201 209)(202 208)(203 207)(204 206)(230 240)(231 239)(232 238)(233 237)(234 236)
(1 183 104 141)(2 182 105 140)(3 181 106 139)(4 240 107 138)(5 239 108 137)(6 238 109 136)(7 237 110 135)(8 236 111 134)(9 235 112 133)(10 234 113 132)(11 233 114 131)(12 232 115 130)(13 231 116 129)(14 230 117 128)(15 229 118 127)(16 228 119 126)(17 227 120 125)(18 226 61 124)(19 225 62 123)(20 224 63 122)(21 223 64 121)(22 222 65 180)(23 221 66 179)(24 220 67 178)(25 219 68 177)(26 218 69 176)(27 217 70 175)(28 216 71 174)(29 215 72 173)(30 214 73 172)(31 213 74 171)(32 212 75 170)(33 211 76 169)(34 210 77 168)(35 209 78 167)(36 208 79 166)(37 207 80 165)(38 206 81 164)(39 205 82 163)(40 204 83 162)(41 203 84 161)(42 202 85 160)(43 201 86 159)(44 200 87 158)(45 199 88 157)(46 198 89 156)(47 197 90 155)(48 196 91 154)(49 195 92 153)(50 194 93 152)(51 193 94 151)(52 192 95 150)(53 191 96 149)(54 190 97 148)(55 189 98 147)(56 188 99 146)(57 187 100 145)(58 186 101 144)(59 185 102 143)(60 184 103 142)
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,180)(147,179)(148,178)(149,177)(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(181,229)(182,228)(183,227)(184,226)(185,225)(186,224)(187,223)(188,222)(189,221)(190,220)(191,219)(192,218)(193,217)(194,216)(195,215)(196,214)(197,213)(198,212)(199,211)(200,210)(201,209)(202,208)(203,207)(204,206)(230,240)(231,239)(232,238)(233,237)(234,236), (1,183,104,141)(2,182,105,140)(3,181,106,139)(4,240,107,138)(5,239,108,137)(6,238,109,136)(7,237,110,135)(8,236,111,134)(9,235,112,133)(10,234,113,132)(11,233,114,131)(12,232,115,130)(13,231,116,129)(14,230,117,128)(15,229,118,127)(16,228,119,126)(17,227,120,125)(18,226,61,124)(19,225,62,123)(20,224,63,122)(21,223,64,121)(22,222,65,180)(23,221,66,179)(24,220,67,178)(25,219,68,177)(26,218,69,176)(27,217,70,175)(28,216,71,174)(29,215,72,173)(30,214,73,172)(31,213,74,171)(32,212,75,170)(33,211,76,169)(34,210,77,168)(35,209,78,167)(36,208,79,166)(37,207,80,165)(38,206,81,164)(39,205,82,163)(40,204,83,162)(41,203,84,161)(42,202,85,160)(43,201,86,159)(44,200,87,158)(45,199,88,157)(46,198,89,156)(47,197,90,155)(48,196,91,154)(49,195,92,153)(50,194,93,152)(51,193,94,151)(52,192,95,150)(53,191,96,149)(54,190,97,148)(55,189,98,147)(56,188,99,146)(57,187,100,145)(58,186,101,144)(59,185,102,143)(60,184,103,142)>;
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,180)(147,179)(148,178)(149,177)(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(181,229)(182,228)(183,227)(184,226)(185,225)(186,224)(187,223)(188,222)(189,221)(190,220)(191,219)(192,218)(193,217)(194,216)(195,215)(196,214)(197,213)(198,212)(199,211)(200,210)(201,209)(202,208)(203,207)(204,206)(230,240)(231,239)(232,238)(233,237)(234,236), (1,183,104,141)(2,182,105,140)(3,181,106,139)(4,240,107,138)(5,239,108,137)(6,238,109,136)(7,237,110,135)(8,236,111,134)(9,235,112,133)(10,234,113,132)(11,233,114,131)(12,232,115,130)(13,231,116,129)(14,230,117,128)(15,229,118,127)(16,228,119,126)(17,227,120,125)(18,226,61,124)(19,225,62,123)(20,224,63,122)(21,223,64,121)(22,222,65,180)(23,221,66,179)(24,220,67,178)(25,219,68,177)(26,218,69,176)(27,217,70,175)(28,216,71,174)(29,215,72,173)(30,214,73,172)(31,213,74,171)(32,212,75,170)(33,211,76,169)(34,210,77,168)(35,209,78,167)(36,208,79,166)(37,207,80,165)(38,206,81,164)(39,205,82,163)(40,204,83,162)(41,203,84,161)(42,202,85,160)(43,201,86,159)(44,200,87,158)(45,199,88,157)(46,198,89,156)(47,197,90,155)(48,196,91,154)(49,195,92,153)(50,194,93,152)(51,193,94,151)(52,192,95,150)(53,191,96,149)(54,190,97,148)(55,189,98,147)(56,188,99,146)(57,187,100,145)(58,186,101,144)(59,185,102,143)(60,184,103,142) );
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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(87,120),(88,119),(89,118),(90,117),(91,116),(92,115),(93,114),(94,113),(95,112),(96,111),(97,110),(98,109),(99,108),(100,107),(101,106),(102,105),(103,104),(121,145),(122,144),(123,143),(124,142),(125,141),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(146,180),(147,179),(148,178),(149,177),(150,176),(151,175),(152,174),(153,173),(154,172),(155,171),(156,170),(157,169),(158,168),(159,167),(160,166),(161,165),(162,164),(181,229),(182,228),(183,227),(184,226),(185,225),(186,224),(187,223),(188,222),(189,221),(190,220),(191,219),(192,218),(193,217),(194,216),(195,215),(196,214),(197,213),(198,212),(199,211),(200,210),(201,209),(202,208),(203,207),(204,206),(230,240),(231,239),(232,238),(233,237),(234,236)], [(1,183,104,141),(2,182,105,140),(3,181,106,139),(4,240,107,138),(5,239,108,137),(6,238,109,136),(7,237,110,135),(8,236,111,134),(9,235,112,133),(10,234,113,132),(11,233,114,131),(12,232,115,130),(13,231,116,129),(14,230,117,128),(15,229,118,127),(16,228,119,126),(17,227,120,125),(18,226,61,124),(19,225,62,123),(20,224,63,122),(21,223,64,121),(22,222,65,180),(23,221,66,179),(24,220,67,178),(25,219,68,177),(26,218,69,176),(27,217,70,175),(28,216,71,174),(29,215,72,173),(30,214,73,172),(31,213,74,171),(32,212,75,170),(33,211,76,169),(34,210,77,168),(35,209,78,167),(36,208,79,166),(37,207,80,165),(38,206,81,164),(39,205,82,163),(40,204,83,162),(41,203,84,161),(42,202,85,160),(43,201,86,159),(44,200,87,158),(45,199,88,157),(46,198,89,156),(47,197,90,155),(48,196,91,154),(49,195,92,153),(50,194,93,152),(51,193,94,151),(52,192,95,150),(53,191,96,149),(54,190,97,148),(55,189,98,147),(56,188,99,146),(57,187,100,145),(58,186,101,144),(59,185,102,143),(60,184,103,142)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 60 | 60 | 2 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 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 | C4 | S3 | D4 | D4 | D5 | D6 | D8 | SD16 | D10 | C4×S3 | C3⋊D4 | D12 | D15 | C4×D5 | C5⋊D4 | D20 | C24⋊C2 | D24 | D30 | C40⋊C2 | D40 | C4×D15 | C15⋊7D4 | D60 | C24⋊D5 | D120 |
| kernel | D60⋊8C4 | C60⋊5C4 | C2×C120 | C2×D60 | D60 | C2×C40 | C60 | C2×C30 | C2×C24 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C2×C8 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of D60⋊8C4 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 142 | 99 | 0 | 0 |
| 0 | 142 | 43 | 0 | 0 |
| 0 | 0 | 0 | 80 | 131 |
| 0 | 0 | 0 | 110 | 147 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 142 | 99 | 0 | 0 |
| 0 | 198 | 99 | 0 | 0 |
| 0 | 0 | 0 | 157 | 148 |
| 0 | 0 | 0 | 94 | 84 |
| 177 | 0 | 0 | 0 | 0 |
| 0 | 66 | 94 | 0 | 0 |
| 0 | 28 | 175 | 0 | 0 |
| 0 | 0 | 0 | 78 | 78 |
| 0 | 0 | 0 | 197 | 163 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,142,142,0,0,0,99,43,0,0,0,0,0,80,110,0,0,0,131,147],[240,0,0,0,0,0,142,198,0,0,0,99,99,0,0,0,0,0,157,94,0,0,0,148,84],[177,0,0,0,0,0,66,28,0,0,0,94,175,0,0,0,0,0,78,197,0,0,0,78,163] >;
D60⋊8C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_8C_4 % in TeX
G:=Group("D60:8C4"); // GroupNames label
G:=SmallGroup(480,181);
// by ID
G=gap.SmallGroup(480,181);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,422,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^43*b>;
// generators/relations