metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊7D4, D10⋊2D12, (C2×D12)⋊3D5, (C6×D5)⋊10D4, C6.22(D4×D5), (C10×D12)⋊5C2, C3⋊3(C20⋊2D4), C20⋊1(C3⋊D4), C4⋊2(C15⋊D4), C5⋊3(C12⋊7D4), C15⋊6(C4⋊D4), C60⋊5C4⋊33C2, C2.24(D5×D12), C30.59(C2×D4), C12⋊10(C5⋊D4), D6⋊Dic5⋊16C2, (C2×C20).129D6, C10.23(C2×D12), C30.84(C4○D4), (C2×C12).305D10, (C22×D5).91D6, C10.60(C4○D12), C6.29(D4⋊2D5), (C2×C30).139C23, (C2×C60).149C22, (C2×Dic5).183D6, (C22×S3).16D10, C2.16(D12⋊5D5), (C6×Dic5).210C22, (C2×Dic15).108C22, (C2×C4×D5)⋊1S3, (D5×C2×C12)⋊2C2, (C2×C15⋊D4)⋊3C2, C6.86(C2×C5⋊D4), (C2×C4).162(S3×D5), C10.87(C2×C3⋊D4), C2.19(C2×C15⋊D4), C22.191(C2×S3×D5), (S3×C2×C10).31C22, (D5×C2×C6).107C22, (C2×C6).151(C22×D5), (C2×C10).151(C22×S3), SmallGroup(480,525)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊D4
G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a49, cbc=b-1 >
Subgroups: 956 in 188 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C4⋊D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C3×Dic5, Dic15, C60, C6×D5, C6×D5, S3×C10, C2×C30, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C5⋊D4, D4×C10, C12⋊7D4, C15⋊D4, D5×C12, C6×Dic5, C5×D12, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C20⋊2D4, D6⋊Dic5, C60⋊5C4, C2×C15⋊D4, D5×C2×C12, C10×D12, C60⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C3⋊D4, C22×S3, C4⋊D4, C5⋊D4, C22×D5, C2×D12, C4○D12, C2×C3⋊D4, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C12⋊7D4, C15⋊D4, C2×S3×D5, C20⋊2D4, D12⋊5D5, D5×D12, C2×C15⋊D4, C60⋊D4
►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)
(1 162 102 183)(2 161 103 182)(3 160 104 181)(4 159 105 240)(5 158 106 239)(6 157 107 238)(7 156 108 237)(8 155 109 236)(9 154 110 235)(10 153 111 234)(11 152 112 233)(12 151 113 232)(13 150 114 231)(14 149 115 230)(15 148 116 229)(16 147 117 228)(17 146 118 227)(18 145 119 226)(19 144 120 225)(20 143 61 224)(21 142 62 223)(22 141 63 222)(23 140 64 221)(24 139 65 220)(25 138 66 219)(26 137 67 218)(27 136 68 217)(28 135 69 216)(29 134 70 215)(30 133 71 214)(31 132 72 213)(32 131 73 212)(33 130 74 211)(34 129 75 210)(35 128 76 209)(36 127 77 208)(37 126 78 207)(38 125 79 206)(39 124 80 205)(40 123 81 204)(41 122 82 203)(42 121 83 202)(43 180 84 201)(44 179 85 200)(45 178 86 199)(46 177 87 198)(47 176 88 197)(48 175 89 196)(49 174 90 195)(50 173 91 194)(51 172 92 193)(52 171 93 192)(53 170 94 191)(54 169 95 190)(55 168 96 189)(56 167 97 188)(57 166 98 187)(58 165 99 186)(59 164 100 185)(60 163 101 184)
(2 50)(3 39)(4 28)(5 17)(7 55)(8 44)(9 33)(10 22)(12 60)(13 49)(14 38)(15 27)(18 54)(19 43)(20 32)(23 59)(24 48)(25 37)(29 53)(30 42)(34 58)(35 47)(40 52)(45 57)(61 73)(63 111)(64 100)(65 89)(66 78)(68 116)(69 105)(70 94)(71 83)(74 110)(75 99)(76 88)(79 115)(80 104)(81 93)(84 120)(85 109)(86 98)(90 114)(91 103)(95 119)(96 108)(101 113)(106 118)(121 214)(122 203)(123 192)(124 181)(125 230)(126 219)(127 208)(128 197)(129 186)(130 235)(131 224)(132 213)(133 202)(134 191)(135 240)(136 229)(137 218)(138 207)(139 196)(140 185)(141 234)(142 223)(143 212)(144 201)(145 190)(146 239)(147 228)(148 217)(149 206)(150 195)(151 184)(152 233)(153 222)(154 211)(155 200)(156 189)(157 238)(158 227)(159 216)(160 205)(161 194)(162 183)(163 232)(164 221)(165 210)(166 199)(167 188)(168 237)(169 226)(170 215)(171 204)(172 193)(173 182)(174 231)(175 220)(176 209)(177 198)(178 187)(179 236)(180 225)
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), (1,162,102,183)(2,161,103,182)(3,160,104,181)(4,159,105,240)(5,158,106,239)(6,157,107,238)(7,156,108,237)(8,155,109,236)(9,154,110,235)(10,153,111,234)(11,152,112,233)(12,151,113,232)(13,150,114,231)(14,149,115,230)(15,148,116,229)(16,147,117,228)(17,146,118,227)(18,145,119,226)(19,144,120,225)(20,143,61,224)(21,142,62,223)(22,141,63,222)(23,140,64,221)(24,139,65,220)(25,138,66,219)(26,137,67,218)(27,136,68,217)(28,135,69,216)(29,134,70,215)(30,133,71,214)(31,132,72,213)(32,131,73,212)(33,130,74,211)(34,129,75,210)(35,128,76,209)(36,127,77,208)(37,126,78,207)(38,125,79,206)(39,124,80,205)(40,123,81,204)(41,122,82,203)(42,121,83,202)(43,180,84,201)(44,179,85,200)(45,178,86,199)(46,177,87,198)(47,176,88,197)(48,175,89,196)(49,174,90,195)(50,173,91,194)(51,172,92,193)(52,171,93,192)(53,170,94,191)(54,169,95,190)(55,168,96,189)(56,167,97,188)(57,166,98,187)(58,165,99,186)(59,164,100,185)(60,163,101,184), (2,50)(3,39)(4,28)(5,17)(7,55)(8,44)(9,33)(10,22)(12,60)(13,49)(14,38)(15,27)(18,54)(19,43)(20,32)(23,59)(24,48)(25,37)(29,53)(30,42)(34,58)(35,47)(40,52)(45,57)(61,73)(63,111)(64,100)(65,89)(66,78)(68,116)(69,105)(70,94)(71,83)(74,110)(75,99)(76,88)(79,115)(80,104)(81,93)(84,120)(85,109)(86,98)(90,114)(91,103)(95,119)(96,108)(101,113)(106,118)(121,214)(122,203)(123,192)(124,181)(125,230)(126,219)(127,208)(128,197)(129,186)(130,235)(131,224)(132,213)(133,202)(134,191)(135,240)(136,229)(137,218)(138,207)(139,196)(140,185)(141,234)(142,223)(143,212)(144,201)(145,190)(146,239)(147,228)(148,217)(149,206)(150,195)(151,184)(152,233)(153,222)(154,211)(155,200)(156,189)(157,238)(158,227)(159,216)(160,205)(161,194)(162,183)(163,232)(164,221)(165,210)(166,199)(167,188)(168,237)(169,226)(170,215)(171,204)(172,193)(173,182)(174,231)(175,220)(176,209)(177,198)(178,187)(179,236)(180,225)>;
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), (1,162,102,183)(2,161,103,182)(3,160,104,181)(4,159,105,240)(5,158,106,239)(6,157,107,238)(7,156,108,237)(8,155,109,236)(9,154,110,235)(10,153,111,234)(11,152,112,233)(12,151,113,232)(13,150,114,231)(14,149,115,230)(15,148,116,229)(16,147,117,228)(17,146,118,227)(18,145,119,226)(19,144,120,225)(20,143,61,224)(21,142,62,223)(22,141,63,222)(23,140,64,221)(24,139,65,220)(25,138,66,219)(26,137,67,218)(27,136,68,217)(28,135,69,216)(29,134,70,215)(30,133,71,214)(31,132,72,213)(32,131,73,212)(33,130,74,211)(34,129,75,210)(35,128,76,209)(36,127,77,208)(37,126,78,207)(38,125,79,206)(39,124,80,205)(40,123,81,204)(41,122,82,203)(42,121,83,202)(43,180,84,201)(44,179,85,200)(45,178,86,199)(46,177,87,198)(47,176,88,197)(48,175,89,196)(49,174,90,195)(50,173,91,194)(51,172,92,193)(52,171,93,192)(53,170,94,191)(54,169,95,190)(55,168,96,189)(56,167,97,188)(57,166,98,187)(58,165,99,186)(59,164,100,185)(60,163,101,184), (2,50)(3,39)(4,28)(5,17)(7,55)(8,44)(9,33)(10,22)(12,60)(13,49)(14,38)(15,27)(18,54)(19,43)(20,32)(23,59)(24,48)(25,37)(29,53)(30,42)(34,58)(35,47)(40,52)(45,57)(61,73)(63,111)(64,100)(65,89)(66,78)(68,116)(69,105)(70,94)(71,83)(74,110)(75,99)(76,88)(79,115)(80,104)(81,93)(84,120)(85,109)(86,98)(90,114)(91,103)(95,119)(96,108)(101,113)(106,118)(121,214)(122,203)(123,192)(124,181)(125,230)(126,219)(127,208)(128,197)(129,186)(130,235)(131,224)(132,213)(133,202)(134,191)(135,240)(136,229)(137,218)(138,207)(139,196)(140,185)(141,234)(142,223)(143,212)(144,201)(145,190)(146,239)(147,228)(148,217)(149,206)(150,195)(151,184)(152,233)(153,222)(154,211)(155,200)(156,189)(157,238)(158,227)(159,216)(160,205)(161,194)(162,183)(163,232)(164,221)(165,210)(166,199)(167,188)(168,237)(169,226)(170,215)(171,204)(172,193)(173,182)(174,231)(175,220)(176,209)(177,198)(178,187)(179,236)(180,225) );
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)], [(1,162,102,183),(2,161,103,182),(3,160,104,181),(4,159,105,240),(5,158,106,239),(6,157,107,238),(7,156,108,237),(8,155,109,236),(9,154,110,235),(10,153,111,234),(11,152,112,233),(12,151,113,232),(13,150,114,231),(14,149,115,230),(15,148,116,229),(16,147,117,228),(17,146,118,227),(18,145,119,226),(19,144,120,225),(20,143,61,224),(21,142,62,223),(22,141,63,222),(23,140,64,221),(24,139,65,220),(25,138,66,219),(26,137,67,218),(27,136,68,217),(28,135,69,216),(29,134,70,215),(30,133,71,214),(31,132,72,213),(32,131,73,212),(33,130,74,211),(34,129,75,210),(35,128,76,209),(36,127,77,208),(37,126,78,207),(38,125,79,206),(39,124,80,205),(40,123,81,204),(41,122,82,203),(42,121,83,202),(43,180,84,201),(44,179,85,200),(45,178,86,199),(46,177,87,198),(47,176,88,197),(48,175,89,196),(49,174,90,195),(50,173,91,194),(51,172,92,193),(52,171,93,192),(53,170,94,191),(54,169,95,190),(55,168,96,189),(56,167,97,188),(57,166,98,187),(58,165,99,186),(59,164,100,185),(60,163,101,184)], [(2,50),(3,39),(4,28),(5,17),(7,55),(8,44),(9,33),(10,22),(12,60),(13,49),(14,38),(15,27),(18,54),(19,43),(20,32),(23,59),(24,48),(25,37),(29,53),(30,42),(34,58),(35,47),(40,52),(45,57),(61,73),(63,111),(64,100),(65,89),(66,78),(68,116),(69,105),(70,94),(71,83),(74,110),(75,99),(76,88),(79,115),(80,104),(81,93),(84,120),(85,109),(86,98),(90,114),(91,103),(95,119),(96,108),(101,113),(106,118),(121,214),(122,203),(123,192),(124,181),(125,230),(126,219),(127,208),(128,197),(129,186),(130,235),(131,224),(132,213),(133,202),(134,191),(135,240),(136,229),(137,218),(138,207),(139,196),(140,185),(141,234),(142,223),(143,212),(144,201),(145,190),(146,239),(147,228),(148,217),(149,206),(150,195),(151,184),(152,233),(153,222),(154,211),(155,200),(156,189),(157,238),(158,227),(159,216),(160,205),(161,194),(162,183),(163,232),(164,221),(165,210),(166,199),(167,188),(168,237),(169,226),(170,215),(171,204),(172,193),(173,182),(174,231),(175,220),(176,209),(177,198),(178,187),(179,236),(180,225)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 12 | 12 | 2 | 2 | 2 | 10 | 10 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
66 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 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D12 | C5⋊D4 | C4○D12 | S3×D5 | D4×D5 | D4⋊2D5 | C15⋊D4 | C2×S3×D5 | D12⋊5D5 | D5×D12 |
| kernel | C60⋊D4 | D6⋊Dic5 | C60⋊5C4 | C2×C15⋊D4 | D5×C2×C12 | C10×D12 | C2×C4×D5 | C60 | C6×D5 | C2×D12 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×C12 | C22×S3 | C20 | D10 | C12 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60⋊D4 ►in GL4(𝔽61) generated by
| 21 | 0 | 0 | 0 |
| 30 | 32 | 0 | 0 |
| 0 | 0 | 18 | 18 |
| 0 | 0 | 43 | 60 |
| 52 | 15 | 0 | 0 |
| 23 | 9 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 8 | 30 |
| 1 | 0 | 0 | 0 |
| 50 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 43 | 60 |
G:=sub<GL(4,GF(61))| [21,30,0,0,0,32,0,0,0,0,18,43,0,0,18,60],[52,23,0,0,15,9,0,0,0,0,31,8,0,0,17,30],[1,50,0,0,0,60,0,0,0,0,1,43,0,0,0,60] >;
C60⋊D4 in GAP, Magma, Sage, TeX
C_{60}\rtimes D_4 % in TeX
G:=Group("C60:D4"); // GroupNames label
G:=SmallGroup(480,525);
// by ID
G=gap.SmallGroup(480,525);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^49,c*b*c=b^-1>;
// generators/relations