metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.67D4, D10⋊6Dic6, (C6×D5)⋊6Q8, C6.27(Q8×D5), (C2×Dic6)⋊2D5, C60⋊5C4⋊30C2, C15⋊3(C22⋊Q8), C30.21(C2×Q8), (C10×Dic6)⋊5C2, C3⋊4(D10⋊3Q8), (C2×C20).111D6, C30.108(C2×D4), Dic15⋊5C4⋊5C2, C2.11(D5×Dic6), C10.9(C2×Dic6), C30.26(C4○D4), C10.5(C4○D12), (C2×C12).296D10, C20.33(C3⋊D4), C12.84(C5⋊D4), C4.18(C15⋊D4), C5⋊4(C12.48D4), (C2×C30).49C23, C6.7(Q8⋊2D5), (C22×D5).83D6, (C2×C60).140C22, (C2×Dic5).160D6, (C2×Dic3).10D10, D10⋊Dic3.3C2, C2.10(C12.28D10), (C6×Dic5).182C22, (C2×Dic15).51C22, (C10×Dic3).29C22, (C2×C4×D5).3S3, (D5×C2×C12).3C2, C6.80(C2×C5⋊D4), (C2×C4).153(S3×D5), C2.14(C2×C15⋊D4), C10.81(C2×C3⋊D4), (D5×C2×C6).96C22, C22.136(C2×S3×D5), (C2×C6).61(C22×D5), (C2×C10).61(C22×S3), SmallGroup(480,435)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.67D4
G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a49, cbc=a30b-1 >
Subgroups: 636 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊Q8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C2×C4×D5, Q8×C10, C12.48D4, D5×C12, C6×Dic5, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D10⋊3Q8, D10⋊Dic3, Dic15⋊5C4, C60⋊5C4, D5×C2×C12, C10×Dic6, C60.67D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C5⋊D4, C22×D5, C2×Dic6, C4○D12, C2×C3⋊D4, S3×D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, C12.48D4, C15⋊D4, C2×S3×D5, D10⋊3Q8, D5×Dic6, C12.28D10, C2×C15⋊D4, C60.67D4
►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)
(1 125 89 240)(2 124 90 239)(3 123 91 238)(4 122 92 237)(5 121 93 236)(6 180 94 235)(7 179 95 234)(8 178 96 233)(9 177 97 232)(10 176 98 231)(11 175 99 230)(12 174 100 229)(13 173 101 228)(14 172 102 227)(15 171 103 226)(16 170 104 225)(17 169 105 224)(18 168 106 223)(19 167 107 222)(20 166 108 221)(21 165 109 220)(22 164 110 219)(23 163 111 218)(24 162 112 217)(25 161 113 216)(26 160 114 215)(27 159 115 214)(28 158 116 213)(29 157 117 212)(30 156 118 211)(31 155 119 210)(32 154 120 209)(33 153 61 208)(34 152 62 207)(35 151 63 206)(36 150 64 205)(37 149 65 204)(38 148 66 203)(39 147 67 202)(40 146 68 201)(41 145 69 200)(42 144 70 199)(43 143 71 198)(44 142 72 197)(45 141 73 196)(46 140 74 195)(47 139 75 194)(48 138 76 193)(49 137 77 192)(50 136 78 191)(51 135 79 190)(52 134 80 189)(53 133 81 188)(54 132 82 187)(55 131 83 186)(56 130 84 185)(57 129 85 184)(58 128 86 183)(59 127 87 182)(60 126 88 181)
(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 97)(62 86)(63 75)(65 113)(66 102)(67 91)(68 80)(70 118)(71 107)(72 96)(73 85)(76 112)(77 101)(78 90)(81 117)(82 106)(83 95)(87 111)(88 100)(92 116)(93 105)(98 110)(103 115)(108 120)(121 194)(122 183)(123 232)(124 221)(125 210)(126 199)(127 188)(128 237)(129 226)(130 215)(131 204)(132 193)(133 182)(134 231)(135 220)(136 209)(137 198)(138 187)(139 236)(140 225)(141 214)(142 203)(143 192)(144 181)(145 230)(146 219)(147 208)(148 197)(149 186)(150 235)(151 224)(152 213)(153 202)(154 191)(155 240)(156 229)(157 218)(158 207)(159 196)(160 185)(161 234)(162 223)(163 212)(164 201)(165 190)(166 239)(167 228)(168 217)(169 206)(170 195)(171 184)(172 233)(173 222)(174 211)(175 200)(176 189)(177 238)(178 227)(179 216)(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), (1,125,89,240)(2,124,90,239)(3,123,91,238)(4,122,92,237)(5,121,93,236)(6,180,94,235)(7,179,95,234)(8,178,96,233)(9,177,97,232)(10,176,98,231)(11,175,99,230)(12,174,100,229)(13,173,101,228)(14,172,102,227)(15,171,103,226)(16,170,104,225)(17,169,105,224)(18,168,106,223)(19,167,107,222)(20,166,108,221)(21,165,109,220)(22,164,110,219)(23,163,111,218)(24,162,112,217)(25,161,113,216)(26,160,114,215)(27,159,115,214)(28,158,116,213)(29,157,117,212)(30,156,118,211)(31,155,119,210)(32,154,120,209)(33,153,61,208)(34,152,62,207)(35,151,63,206)(36,150,64,205)(37,149,65,204)(38,148,66,203)(39,147,67,202)(40,146,68,201)(41,145,69,200)(42,144,70,199)(43,143,71,198)(44,142,72,197)(45,141,73,196)(46,140,74,195)(47,139,75,194)(48,138,76,193)(49,137,77,192)(50,136,78,191)(51,135,79,190)(52,134,80,189)(53,133,81,188)(54,132,82,187)(55,131,83,186)(56,130,84,185)(57,129,85,184)(58,128,86,183)(59,127,87,182)(60,126,88,181), (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)(61,97)(62,86)(63,75)(65,113)(66,102)(67,91)(68,80)(70,118)(71,107)(72,96)(73,85)(76,112)(77,101)(78,90)(81,117)(82,106)(83,95)(87,111)(88,100)(92,116)(93,105)(98,110)(103,115)(108,120)(121,194)(122,183)(123,232)(124,221)(125,210)(126,199)(127,188)(128,237)(129,226)(130,215)(131,204)(132,193)(133,182)(134,231)(135,220)(136,209)(137,198)(138,187)(139,236)(140,225)(141,214)(142,203)(143,192)(144,181)(145,230)(146,219)(147,208)(148,197)(149,186)(150,235)(151,224)(152,213)(153,202)(154,191)(155,240)(156,229)(157,218)(158,207)(159,196)(160,185)(161,234)(162,223)(163,212)(164,201)(165,190)(166,239)(167,228)(168,217)(169,206)(170,195)(171,184)(172,233)(173,222)(174,211)(175,200)(176,189)(177,238)(178,227)(179,216)(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), (1,125,89,240)(2,124,90,239)(3,123,91,238)(4,122,92,237)(5,121,93,236)(6,180,94,235)(7,179,95,234)(8,178,96,233)(9,177,97,232)(10,176,98,231)(11,175,99,230)(12,174,100,229)(13,173,101,228)(14,172,102,227)(15,171,103,226)(16,170,104,225)(17,169,105,224)(18,168,106,223)(19,167,107,222)(20,166,108,221)(21,165,109,220)(22,164,110,219)(23,163,111,218)(24,162,112,217)(25,161,113,216)(26,160,114,215)(27,159,115,214)(28,158,116,213)(29,157,117,212)(30,156,118,211)(31,155,119,210)(32,154,120,209)(33,153,61,208)(34,152,62,207)(35,151,63,206)(36,150,64,205)(37,149,65,204)(38,148,66,203)(39,147,67,202)(40,146,68,201)(41,145,69,200)(42,144,70,199)(43,143,71,198)(44,142,72,197)(45,141,73,196)(46,140,74,195)(47,139,75,194)(48,138,76,193)(49,137,77,192)(50,136,78,191)(51,135,79,190)(52,134,80,189)(53,133,81,188)(54,132,82,187)(55,131,83,186)(56,130,84,185)(57,129,85,184)(58,128,86,183)(59,127,87,182)(60,126,88,181), (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)(61,97)(62,86)(63,75)(65,113)(66,102)(67,91)(68,80)(70,118)(71,107)(72,96)(73,85)(76,112)(77,101)(78,90)(81,117)(82,106)(83,95)(87,111)(88,100)(92,116)(93,105)(98,110)(103,115)(108,120)(121,194)(122,183)(123,232)(124,221)(125,210)(126,199)(127,188)(128,237)(129,226)(130,215)(131,204)(132,193)(133,182)(134,231)(135,220)(136,209)(137,198)(138,187)(139,236)(140,225)(141,214)(142,203)(143,192)(144,181)(145,230)(146,219)(147,208)(148,197)(149,186)(150,235)(151,224)(152,213)(153,202)(154,191)(155,240)(156,229)(157,218)(158,207)(159,196)(160,185)(161,234)(162,223)(163,212)(164,201)(165,190)(166,239)(167,228)(168,217)(169,206)(170,195)(171,184)(172,233)(173,222)(174,211)(175,200)(176,189)(177,238)(178,227)(179,216)(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)], [(1,125,89,240),(2,124,90,239),(3,123,91,238),(4,122,92,237),(5,121,93,236),(6,180,94,235),(7,179,95,234),(8,178,96,233),(9,177,97,232),(10,176,98,231),(11,175,99,230),(12,174,100,229),(13,173,101,228),(14,172,102,227),(15,171,103,226),(16,170,104,225),(17,169,105,224),(18,168,106,223),(19,167,107,222),(20,166,108,221),(21,165,109,220),(22,164,110,219),(23,163,111,218),(24,162,112,217),(25,161,113,216),(26,160,114,215),(27,159,115,214),(28,158,116,213),(29,157,117,212),(30,156,118,211),(31,155,119,210),(32,154,120,209),(33,153,61,208),(34,152,62,207),(35,151,63,206),(36,150,64,205),(37,149,65,204),(38,148,66,203),(39,147,67,202),(40,146,68,201),(41,145,69,200),(42,144,70,199),(43,143,71,198),(44,142,72,197),(45,141,73,196),(46,140,74,195),(47,139,75,194),(48,138,76,193),(49,137,77,192),(50,136,78,191),(51,135,79,190),(52,134,80,189),(53,133,81,188),(54,132,82,187),(55,131,83,186),(56,130,84,185),(57,129,85,184),(58,128,86,183),(59,127,87,182),(60,126,88,181)], [(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),(61,97),(62,86),(63,75),(65,113),(66,102),(67,91),(68,80),(70,118),(71,107),(72,96),(73,85),(76,112),(77,101),(78,90),(81,117),(82,106),(83,95),(87,111),(88,100),(92,116),(93,105),(98,110),(103,115),(108,120),(121,194),(122,183),(123,232),(124,221),(125,210),(126,199),(127,188),(128,237),(129,226),(130,215),(131,204),(132,193),(133,182),(134,231),(135,220),(136,209),(137,198),(138,187),(139,236),(140,225),(141,214),(142,203),(143,192),(144,181),(145,230),(146,219),(147,208),(148,197),(149,186),(150,235),(151,224),(152,213),(153,202),(154,191),(155,240),(156,229),(157,218),(158,207),(159,196),(160,185),(161,234),(162,223),(163,212),(164,201),(165,190),(166,239),(167,228),(168,217),(169,206),(170,195),(171,184),(172,233),(173,222),(174,211),(175,200),(176,189),(177,238),(178,227),(179,216),(180,205)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 10 | 10 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 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 | Q8 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | Dic6 | C5⋊D4 | C4○D12 | S3×D5 | Q8×D5 | Q8⋊2D5 | C15⋊D4 | C2×S3×D5 | D5×Dic6 | C12.28D10 |
| kernel | C60.67D4 | D10⋊Dic3 | Dic15⋊5C4 | C60⋊5C4 | D5×C2×C12 | C10×Dic6 | C2×C4×D5 | C60 | C6×D5 | C2×Dic6 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C20 | D10 | C12 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60.67D4 ►in GL4(𝔽61) generated by
| 18 | 18 | 0 | 0 |
| 43 | 60 | 0 | 0 |
| 0 | 0 | 29 | 0 |
| 0 | 0 | 25 | 40 |
| 31 | 17 | 0 | 0 |
| 8 | 30 | 0 | 0 |
| 0 | 0 | 33 | 56 |
| 0 | 0 | 59 | 28 |
| 1 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 60 |
G:=sub<GL(4,GF(61))| [18,43,0,0,18,60,0,0,0,0,29,25,0,0,0,40],[31,8,0,0,17,30,0,0,0,0,33,59,0,0,56,28],[1,43,0,0,0,60,0,0,0,0,1,1,0,0,0,60] >;
C60.67D4 in GAP, Magma, Sage, TeX
C_{60}._{67}D_4 % in TeX
G:=Group("C60.67D4"); // GroupNames label
G:=SmallGroup(480,435);
// by ID
G=gap.SmallGroup(480,435);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,219,100,1356,18822]);
// 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=a^30*b^-1>;
// generators/relations