metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊2D60, C60⋊1D4, C20⋊3D12, C12⋊3D20, D30⋊10D4, C4⋊C4⋊3D15, (C2×D60)⋊7C2, C2.9(C2×D60), C5⋊2(C12⋊D4), C3⋊2(C4⋊D20), C6.35(C2×D20), (C2×C4).43D30, C2.13(D4×D15), C6.106(D4×D5), D30⋊3C4⋊8C2, C15⋊28(C4⋊D4), C30.263(C2×D4), (C2×C20).138D6, C10.108(S3×D4), C10.36(C2×D12), (C2×C12).39D10, (C2×C60).20C22, C30.259(C4○D4), C2.6(Q8⋊3D15), (C2×C30).292C23, C6.42(Q8⋊2D5), C10.42(Q8⋊3S3), (C22×D15).8C22, C22.50(C22×D15), (C2×Dic15).165C22, (C5×C4⋊C4)⋊6S3, (C3×C4⋊C4)⋊6D5, (C2×C4×D15)⋊1C2, (C15×C4⋊C4)⋊6C2, (C2×C6).288(C22×D5), (C2×C10).287(C22×S3), SmallGroup(480,860)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D60
G = < a,b,c | a4=b60=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 1476 in 188 conjugacy classes, 57 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, 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, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, Dic15, C60, C60, D30, D30, C2×C30, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C12⋊D4, C4×D15, D60, C2×Dic15, C2×C60, C2×C60, C22×D15, C22×D15, C4⋊D20, D30⋊3C4, C15×C4⋊C4, C2×C4×D15, C2×D60, C2×D60, C4⋊D60
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, D15, C4⋊D4, D20, C22×D5, C2×D12, S3×D4, Q8⋊3S3, D30, C2×D20, D4×D5, Q8⋊2D5, C12⋊D4, D60, C22×D15, C4⋊D20, C2×D60, D4×D15, Q8⋊3D15, C4⋊D60
►On 240 points
Generators in S240
(1 100 140 218)(2 219 141 101)(3 102 142 220)(4 221 143 103)(5 104 144 222)(6 223 145 105)(7 106 146 224)(8 225 147 107)(9 108 148 226)(10 227 149 109)(11 110 150 228)(12 229 151 111)(13 112 152 230)(14 231 153 113)(15 114 154 232)(16 233 155 115)(17 116 156 234)(18 235 157 117)(19 118 158 236)(20 237 159 119)(21 120 160 238)(22 239 161 61)(23 62 162 240)(24 181 163 63)(25 64 164 182)(26 183 165 65)(27 66 166 184)(28 185 167 67)(29 68 168 186)(30 187 169 69)(31 70 170 188)(32 189 171 71)(33 72 172 190)(34 191 173 73)(35 74 174 192)(36 193 175 75)(37 76 176 194)(38 195 177 77)(39 78 178 196)(40 197 179 79)(41 80 180 198)(42 199 121 81)(43 82 122 200)(44 201 123 83)(45 84 124 202)(46 203 125 85)(47 86 126 204)(48 205 127 87)(49 88 128 206)(50 207 129 89)(51 90 130 208)(52 209 131 91)(53 92 132 210)(54 211 133 93)(55 94 134 212)(56 213 135 95)(57 96 136 214)(58 215 137 97)(59 98 138 216)(60 217 139 99)
(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 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)(121 158)(122 157)(123 156)(124 155)(125 154)(126 153)(127 152)(128 151)(129 150)(130 149)(131 148)(132 147)(133 146)(134 145)(135 144)(136 143)(137 142)(138 141)(139 140)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(181 194)(182 193)(183 192)(184 191)(185 190)(186 189)(187 188)(195 240)(196 239)(197 238)(198 237)(199 236)(200 235)(201 234)(202 233)(203 232)(204 231)(205 230)(206 229)(207 228)(208 227)(209 226)(210 225)(211 224)(212 223)(213 222)(214 221)(215 220)(216 219)(217 218)
G:=sub<Sym(240)| (1,100,140,218)(2,219,141,101)(3,102,142,220)(4,221,143,103)(5,104,144,222)(6,223,145,105)(7,106,146,224)(8,225,147,107)(9,108,148,226)(10,227,149,109)(11,110,150,228)(12,229,151,111)(13,112,152,230)(14,231,153,113)(15,114,154,232)(16,233,155,115)(17,116,156,234)(18,235,157,117)(19,118,158,236)(20,237,159,119)(21,120,160,238)(22,239,161,61)(23,62,162,240)(24,181,163,63)(25,64,164,182)(26,183,165,65)(27,66,166,184)(28,185,167,67)(29,68,168,186)(30,187,169,69)(31,70,170,188)(32,189,171,71)(33,72,172,190)(34,191,173,73)(35,74,174,192)(36,193,175,75)(37,76,176,194)(38,195,177,77)(39,78,178,196)(40,197,179,79)(41,80,180,198)(42,199,121,81)(43,82,122,200)(44,201,123,83)(45,84,124,202)(46,203,125,85)(47,86,126,204)(48,205,127,87)(49,88,128,206)(50,207,129,89)(51,90,130,208)(52,209,131,91)(53,92,132,210)(54,211,133,93)(55,94,134,212)(56,213,135,95)(57,96,136,214)(58,215,137,97)(59,98,138,216)(60,217,139,99), (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)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(121,158)(122,157)(123,156)(124,155)(125,154)(126,153)(127,152)(128,151)(129,150)(130,149)(131,148)(132,147)(133,146)(134,145)(135,144)(136,143)(137,142)(138,141)(139,140)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,240)(196,239)(197,238)(198,237)(199,236)(200,235)(201,234)(202,233)(203,232)(204,231)(205,230)(206,229)(207,228)(208,227)(209,226)(210,225)(211,224)(212,223)(213,222)(214,221)(215,220)(216,219)(217,218)>;
G:=Group( (1,100,140,218)(2,219,141,101)(3,102,142,220)(4,221,143,103)(5,104,144,222)(6,223,145,105)(7,106,146,224)(8,225,147,107)(9,108,148,226)(10,227,149,109)(11,110,150,228)(12,229,151,111)(13,112,152,230)(14,231,153,113)(15,114,154,232)(16,233,155,115)(17,116,156,234)(18,235,157,117)(19,118,158,236)(20,237,159,119)(21,120,160,238)(22,239,161,61)(23,62,162,240)(24,181,163,63)(25,64,164,182)(26,183,165,65)(27,66,166,184)(28,185,167,67)(29,68,168,186)(30,187,169,69)(31,70,170,188)(32,189,171,71)(33,72,172,190)(34,191,173,73)(35,74,174,192)(36,193,175,75)(37,76,176,194)(38,195,177,77)(39,78,178,196)(40,197,179,79)(41,80,180,198)(42,199,121,81)(43,82,122,200)(44,201,123,83)(45,84,124,202)(46,203,125,85)(47,86,126,204)(48,205,127,87)(49,88,128,206)(50,207,129,89)(51,90,130,208)(52,209,131,91)(53,92,132,210)(54,211,133,93)(55,94,134,212)(56,213,135,95)(57,96,136,214)(58,215,137,97)(59,98,138,216)(60,217,139,99), (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)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(121,158)(122,157)(123,156)(124,155)(125,154)(126,153)(127,152)(128,151)(129,150)(130,149)(131,148)(132,147)(133,146)(134,145)(135,144)(136,143)(137,142)(138,141)(139,140)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,240)(196,239)(197,238)(198,237)(199,236)(200,235)(201,234)(202,233)(203,232)(204,231)(205,230)(206,229)(207,228)(208,227)(209,226)(210,225)(211,224)(212,223)(213,222)(214,221)(215,220)(216,219)(217,218) );
G=PermutationGroup([[(1,100,140,218),(2,219,141,101),(3,102,142,220),(4,221,143,103),(5,104,144,222),(6,223,145,105),(7,106,146,224),(8,225,147,107),(9,108,148,226),(10,227,149,109),(11,110,150,228),(12,229,151,111),(13,112,152,230),(14,231,153,113),(15,114,154,232),(16,233,155,115),(17,116,156,234),(18,235,157,117),(19,118,158,236),(20,237,159,119),(21,120,160,238),(22,239,161,61),(23,62,162,240),(24,181,163,63),(25,64,164,182),(26,183,165,65),(27,66,166,184),(28,185,167,67),(29,68,168,186),(30,187,169,69),(31,70,170,188),(32,189,171,71),(33,72,172,190),(34,191,173,73),(35,74,174,192),(36,193,175,75),(37,76,176,194),(38,195,177,77),(39,78,178,196),(40,197,179,79),(41,80,180,198),(42,199,121,81),(43,82,122,200),(44,201,123,83),(45,84,124,202),(46,203,125,85),(47,86,126,204),(48,205,127,87),(49,88,128,206),(50,207,129,89),(51,90,130,208),(52,209,131,91),(53,92,132,210),(54,211,133,93),(55,94,134,212),(56,213,135,95),(57,96,136,214),(58,215,137,97),(59,98,138,216),(60,217,139,99)], [(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),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,120),(80,119),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(91,108),(92,107),(93,106),(94,105),(95,104),(96,103),(97,102),(98,101),(99,100),(121,158),(122,157),(123,156),(124,155),(125,154),(126,153),(127,152),(128,151),(129,150),(130,149),(131,148),(132,147),(133,146),(134,145),(135,144),(136,143),(137,142),(138,141),(139,140),(159,180),(160,179),(161,178),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,171),(169,170),(181,194),(182,193),(183,192),(184,191),(185,190),(186,189),(187,188),(195,240),(196,239),(197,238),(198,237),(199,236),(200,235),(201,234),(202,233),(203,232),(204,231),(205,230),(206,229),(207,228),(208,227),(209,226),(210,225),(211,224),(212,223),(213,222),(214,221),(215,220),(216,219),(217,218)]])
84 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 | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 4 | 4 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | C4○D4 | D10 | D12 | D15 | D20 | D30 | D60 | S3×D4 | Q8⋊3S3 | D4×D5 | Q8⋊2D5 | D4×D15 | Q8⋊3D15 |
| kernel | C4⋊D60 | D30⋊3C4 | C15×C4⋊C4 | C2×C4×D15 | C2×D60 | C5×C4⋊C4 | C60 | D30 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C20 | C4⋊C4 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 1 | 2 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C4⋊D60 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 34 |
| 0 | 0 | 0 | 0 | 27 | 39 |
| 27 | 2 | 0 | 0 | 0 | 0 |
| 23 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 23 | 0 | 0 |
| 0 | 0 | 38 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 36 | 32 | 0 | 0 | 0 | 0 |
| 11 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 23 | 0 | 0 |
| 0 | 0 | 45 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,22,27,0,0,0,0,34,39],[27,23,0,0,0,0,2,4,0,0,0,0,0,0,53,38,0,0,0,0,23,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[36,11,0,0,0,0,32,25,0,0,0,0,0,0,53,45,0,0,0,0,23,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C4⋊D60 in GAP, Magma, Sage, TeX
C_4\rtimes D_{60} % in TeX
G:=Group("C4:D60"); // GroupNames label
G:=SmallGroup(480,860);
// by ID
G=gap.SmallGroup(480,860);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^60=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations