metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊2D4, D30⋊11D4, C23.13D30, (C6×D4)⋊3D5, (D4×C30)⋊3C2, (D4×C10)⋊3S3, (C2×D4)⋊4D15, C5⋊5(D6⋊3D4), C3⋊5(C20⋊2D4), C20⋊5(C3⋊D4), C12⋊5(C5⋊D4), C4⋊2(C15⋊7D4), C60⋊5C4⋊20C2, C2.26(D4×D15), C6.120(D4×D5), (C2×C4).51D30, C15⋊34(C4⋊D4), (C2×C20).149D6, C30.383(C2×D4), C10.122(S3×D4), (C2×C12).148D10, (C2×C60).75C22, (C22×C10).80D6, (C22×C6).65D10, C30.225(C4○D4), C30.38D4⋊11C2, (C2×C30).309C23, C6.104(D4⋊2D5), C2.17(D4⋊2D15), (C22×C30).21C22, C10.104(D4⋊2S3), C22.60(C22×D15), (C2×Dic15).18C22, (C22×D15).88C22, (C2×C4×D15)⋊2C2, (C2×C15⋊7D4)⋊5C2, C6.109(C2×C5⋊D4), C2.14(C2×C15⋊7D4), C10.109(C2×C3⋊D4), (C2×C6).305(C22×D5), (C2×C10).304(C22×S3), SmallGroup(480,903)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊2D4
G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a29, cbc=b-1 >
Subgroups: 1076 in 188 conjugacy classes, 57 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D15, C30, C30, C4⋊D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4⋊Dic3, C6.D4, S3×C2×C4, C2×C3⋊D4, C6×D4, Dic15, C60, D30, D30, C2×C30, C2×C30, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C5⋊D4, D4×C10, D6⋊3D4, C4×D15, C2×Dic15, C2×Dic15, C15⋊7D4, C2×C60, D4×C15, C22×D15, C22×C30, C20⋊2D4, C60⋊5C4, C30.38D4, C2×C4×D15, C2×C15⋊7D4, D4×C30, C60⋊2D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, D15, C4⋊D4, C5⋊D4, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, D30, D4×D5, D4⋊2D5, C2×C5⋊D4, D6⋊3D4, C15⋊7D4, C22×D15, C20⋊2D4, D4×D15, D4⋊2D15, C2×C15⋊7D4, C60⋊2D4
►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)
(1 144 116 181)(2 143 117 240)(3 142 118 239)(4 141 119 238)(5 140 120 237)(6 139 61 236)(7 138 62 235)(8 137 63 234)(9 136 64 233)(10 135 65 232)(11 134 66 231)(12 133 67 230)(13 132 68 229)(14 131 69 228)(15 130 70 227)(16 129 71 226)(17 128 72 225)(18 127 73 224)(19 126 74 223)(20 125 75 222)(21 124 76 221)(22 123 77 220)(23 122 78 219)(24 121 79 218)(25 180 80 217)(26 179 81 216)(27 178 82 215)(28 177 83 214)(29 176 84 213)(30 175 85 212)(31 174 86 211)(32 173 87 210)(33 172 88 209)(34 171 89 208)(35 170 90 207)(36 169 91 206)(37 168 92 205)(38 167 93 204)(39 166 94 203)(40 165 95 202)(41 164 96 201)(42 163 97 200)(43 162 98 199)(44 161 99 198)(45 160 100 197)(46 159 101 196)(47 158 102 195)(48 157 103 194)(49 156 104 193)(50 155 105 192)(51 154 106 191)(52 153 107 190)(53 152 108 189)(54 151 109 188)(55 150 110 187)(56 149 111 186)(57 148 112 185)(58 147 113 184)(59 146 114 183)(60 145 115 182)
(2 30)(3 59)(4 28)(5 57)(6 26)(7 55)(8 24)(9 53)(10 22)(11 51)(12 20)(13 49)(14 18)(15 47)(17 45)(19 43)(21 41)(23 39)(25 37)(27 35)(29 33)(32 60)(34 58)(36 56)(38 54)(40 52)(42 50)(44 48)(61 81)(62 110)(63 79)(64 108)(65 77)(66 106)(67 75)(68 104)(69 73)(70 102)(72 100)(74 98)(76 96)(78 94)(80 92)(82 90)(83 119)(84 88)(85 117)(87 115)(89 113)(91 111)(93 109)(95 107)(97 105)(99 103)(112 120)(114 118)(121 234)(122 203)(123 232)(124 201)(125 230)(126 199)(127 228)(128 197)(129 226)(130 195)(131 224)(132 193)(133 222)(134 191)(135 220)(136 189)(137 218)(138 187)(139 216)(140 185)(141 214)(142 183)(143 212)(144 181)(145 210)(146 239)(147 208)(148 237)(149 206)(150 235)(151 204)(152 233)(153 202)(154 231)(155 200)(156 229)(157 198)(158 227)(159 196)(160 225)(161 194)(162 223)(163 192)(164 221)(165 190)(166 219)(167 188)(168 217)(169 186)(170 215)(171 184)(172 213)(173 182)(174 211)(175 240)(176 209)(177 238)(178 207)(179 236)(180 205)
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), (1,144,116,181)(2,143,117,240)(3,142,118,239)(4,141,119,238)(5,140,120,237)(6,139,61,236)(7,138,62,235)(8,137,63,234)(9,136,64,233)(10,135,65,232)(11,134,66,231)(12,133,67,230)(13,132,68,229)(14,131,69,228)(15,130,70,227)(16,129,71,226)(17,128,72,225)(18,127,73,224)(19,126,74,223)(20,125,75,222)(21,124,76,221)(22,123,77,220)(23,122,78,219)(24,121,79,218)(25,180,80,217)(26,179,81,216)(27,178,82,215)(28,177,83,214)(29,176,84,213)(30,175,85,212)(31,174,86,211)(32,173,87,210)(33,172,88,209)(34,171,89,208)(35,170,90,207)(36,169,91,206)(37,168,92,205)(38,167,93,204)(39,166,94,203)(40,165,95,202)(41,164,96,201)(42,163,97,200)(43,162,98,199)(44,161,99,198)(45,160,100,197)(46,159,101,196)(47,158,102,195)(48,157,103,194)(49,156,104,193)(50,155,105,192)(51,154,106,191)(52,153,107,190)(53,152,108,189)(54,151,109,188)(55,150,110,187)(56,149,111,186)(57,148,112,185)(58,147,113,184)(59,146,114,183)(60,145,115,182), (2,30)(3,59)(4,28)(5,57)(6,26)(7,55)(8,24)(9,53)(10,22)(11,51)(12,20)(13,49)(14,18)(15,47)(17,45)(19,43)(21,41)(23,39)(25,37)(27,35)(29,33)(32,60)(34,58)(36,56)(38,54)(40,52)(42,50)(44,48)(61,81)(62,110)(63,79)(64,108)(65,77)(66,106)(67,75)(68,104)(69,73)(70,102)(72,100)(74,98)(76,96)(78,94)(80,92)(82,90)(83,119)(84,88)(85,117)(87,115)(89,113)(91,111)(93,109)(95,107)(97,105)(99,103)(112,120)(114,118)(121,234)(122,203)(123,232)(124,201)(125,230)(126,199)(127,228)(128,197)(129,226)(130,195)(131,224)(132,193)(133,222)(134,191)(135,220)(136,189)(137,218)(138,187)(139,216)(140,185)(141,214)(142,183)(143,212)(144,181)(145,210)(146,239)(147,208)(148,237)(149,206)(150,235)(151,204)(152,233)(153,202)(154,231)(155,200)(156,229)(157,198)(158,227)(159,196)(160,225)(161,194)(162,223)(163,192)(164,221)(165,190)(166,219)(167,188)(168,217)(169,186)(170,215)(171,184)(172,213)(173,182)(174,211)(175,240)(176,209)(177,238)(178,207)(179,236)(180,205)>;
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), (1,144,116,181)(2,143,117,240)(3,142,118,239)(4,141,119,238)(5,140,120,237)(6,139,61,236)(7,138,62,235)(8,137,63,234)(9,136,64,233)(10,135,65,232)(11,134,66,231)(12,133,67,230)(13,132,68,229)(14,131,69,228)(15,130,70,227)(16,129,71,226)(17,128,72,225)(18,127,73,224)(19,126,74,223)(20,125,75,222)(21,124,76,221)(22,123,77,220)(23,122,78,219)(24,121,79,218)(25,180,80,217)(26,179,81,216)(27,178,82,215)(28,177,83,214)(29,176,84,213)(30,175,85,212)(31,174,86,211)(32,173,87,210)(33,172,88,209)(34,171,89,208)(35,170,90,207)(36,169,91,206)(37,168,92,205)(38,167,93,204)(39,166,94,203)(40,165,95,202)(41,164,96,201)(42,163,97,200)(43,162,98,199)(44,161,99,198)(45,160,100,197)(46,159,101,196)(47,158,102,195)(48,157,103,194)(49,156,104,193)(50,155,105,192)(51,154,106,191)(52,153,107,190)(53,152,108,189)(54,151,109,188)(55,150,110,187)(56,149,111,186)(57,148,112,185)(58,147,113,184)(59,146,114,183)(60,145,115,182), (2,30)(3,59)(4,28)(5,57)(6,26)(7,55)(8,24)(9,53)(10,22)(11,51)(12,20)(13,49)(14,18)(15,47)(17,45)(19,43)(21,41)(23,39)(25,37)(27,35)(29,33)(32,60)(34,58)(36,56)(38,54)(40,52)(42,50)(44,48)(61,81)(62,110)(63,79)(64,108)(65,77)(66,106)(67,75)(68,104)(69,73)(70,102)(72,100)(74,98)(76,96)(78,94)(80,92)(82,90)(83,119)(84,88)(85,117)(87,115)(89,113)(91,111)(93,109)(95,107)(97,105)(99,103)(112,120)(114,118)(121,234)(122,203)(123,232)(124,201)(125,230)(126,199)(127,228)(128,197)(129,226)(130,195)(131,224)(132,193)(133,222)(134,191)(135,220)(136,189)(137,218)(138,187)(139,216)(140,185)(141,214)(142,183)(143,212)(144,181)(145,210)(146,239)(147,208)(148,237)(149,206)(150,235)(151,204)(152,233)(153,202)(154,231)(155,200)(156,229)(157,198)(158,227)(159,196)(160,225)(161,194)(162,223)(163,192)(164,221)(165,190)(166,219)(167,188)(168,217)(169,186)(170,215)(171,184)(172,213)(173,182)(174,211)(175,240)(176,209)(177,238)(178,207)(179,236)(180,205) );
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)], [(1,144,116,181),(2,143,117,240),(3,142,118,239),(4,141,119,238),(5,140,120,237),(6,139,61,236),(7,138,62,235),(8,137,63,234),(9,136,64,233),(10,135,65,232),(11,134,66,231),(12,133,67,230),(13,132,68,229),(14,131,69,228),(15,130,70,227),(16,129,71,226),(17,128,72,225),(18,127,73,224),(19,126,74,223),(20,125,75,222),(21,124,76,221),(22,123,77,220),(23,122,78,219),(24,121,79,218),(25,180,80,217),(26,179,81,216),(27,178,82,215),(28,177,83,214),(29,176,84,213),(30,175,85,212),(31,174,86,211),(32,173,87,210),(33,172,88,209),(34,171,89,208),(35,170,90,207),(36,169,91,206),(37,168,92,205),(38,167,93,204),(39,166,94,203),(40,165,95,202),(41,164,96,201),(42,163,97,200),(43,162,98,199),(44,161,99,198),(45,160,100,197),(46,159,101,196),(47,158,102,195),(48,157,103,194),(49,156,104,193),(50,155,105,192),(51,154,106,191),(52,153,107,190),(53,152,108,189),(54,151,109,188),(55,150,110,187),(56,149,111,186),(57,148,112,185),(58,147,113,184),(59,146,114,183),(60,145,115,182)], [(2,30),(3,59),(4,28),(5,57),(6,26),(7,55),(8,24),(9,53),(10,22),(11,51),(12,20),(13,49),(14,18),(15,47),(17,45),(19,43),(21,41),(23,39),(25,37),(27,35),(29,33),(32,60),(34,58),(36,56),(38,54),(40,52),(42,50),(44,48),(61,81),(62,110),(63,79),(64,108),(65,77),(66,106),(67,75),(68,104),(69,73),(70,102),(72,100),(74,98),(76,96),(78,94),(80,92),(82,90),(83,119),(84,88),(85,117),(87,115),(89,113),(91,111),(93,109),(95,107),(97,105),(99,103),(112,120),(114,118),(121,234),(122,203),(123,232),(124,201),(125,230),(126,199),(127,228),(128,197),(129,226),(130,195),(131,224),(132,193),(133,222),(134,191),(135,220),(136,189),(137,218),(138,187),(139,216),(140,185),(141,214),(142,183),(143,212),(144,181),(145,210),(146,239),(147,208),(148,237),(149,206),(150,235),(151,204),(152,233),(153,202),(154,231),(155,200),(156,229),(157,198),(158,227),(159,196),(160,225),(161,194),(162,223),(163,192),(164,221),(165,190),(166,219),(167,188),(168,217),(169,186),(170,215),(171,184),(172,213),(173,182),(174,211),(175,240),(176,209),(177,238),(178,207),(179,236),(180,205)]])
84 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 | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 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 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 30 | 30 | 2 | 2 | 2 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 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 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | D30 | C15⋊7D4 | S3×D4 | D4⋊2S3 | D4×D5 | D4⋊2D5 | D4×D15 | D4⋊2D15 |
| kernel | C60⋊2D4 | C60⋊5C4 | C30.38D4 | C2×C4×D15 | C2×C15⋊7D4 | D4×C30 | D4×C10 | C60 | D30 | C6×D4 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C20 | C2×D4 | C12 | C2×C4 | C23 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C60⋊2D4 ►in GL6(𝔽61)
| 44 | 44 | 0 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 6 |
| 0 | 0 | 0 | 0 | 12 | 7 |
| 24 | 53 | 0 | 0 | 0 | 0 |
| 34 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 52 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 36 | 28 | 0 | 0 | 0 | 0 |
| 30 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [44,17,0,0,0,0,44,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,54,12,0,0,0,0,6,7],[24,34,0,0,0,0,53,37,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,52,60],[36,30,0,0,0,0,28,25,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C60⋊2D4 in GAP, Magma, Sage, TeX
C_{60}\rtimes_2D_4 % in TeX
G:=Group("C60:2D4"); // GroupNames label
G:=SmallGroup(480,903);
// by ID
G=gap.SmallGroup(480,903);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations