direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D60, C60⋊9C23, D30⋊6C23, C30.56C24, C23.40D30, (C2×C6)⋊6D20, C6⋊2(C2×D20), (C2×C4)⋊9D30, (C2×C10)⋊9D12, (C2×C20)⋊33D6, C10⋊2(C2×D12), C30⋊10(C2×D4), (C2×C30)⋊24D4, C5⋊2(C22×D12), C3⋊2(C22×D20), (C2×C12)⋊33D10, C20⋊8(C22×S3), (C22×C4)⋊7D15, C4⋊2(C22×D15), C12⋊8(C22×D5), (C22×C12)⋊7D5, C15⋊11(C22×D4), (C22×C60)⋊11C2, (C22×C20)⋊11S3, (C2×C60)⋊44C22, (C23×D15)⋊3C2, C6.56(C23×D5), C2.4(C23×D15), C10.56(S3×C23), (C2×C30).320C23, (C22×C10).144D6, (C22×C6).126D10, (C22×D15)⋊17C22, C22.30(C22×D15), (C22×C30).149C22, (C2×C6).316(C22×D5), (C2×C10).315(C22×S3), SmallGroup(480,1167)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3252 in 472 conjugacy classes, 159 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, S3 [×8], C6, C6 [×6], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×8], C10, C10 [×6], C12 [×4], D6 [×32], C2×C6 [×7], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], D10 [×32], C2×C10 [×7], D12 [×16], C2×C12 [×6], C22×S3 [×20], C22×C6, D15 [×8], C30, C30 [×6], C22×D4, D20 [×16], C2×C20 [×6], C22×D5 [×20], C22×C10, C2×D12 [×12], C22×C12, S3×C23 [×2], C60 [×4], D30 [×8], D30 [×24], C2×C30 [×7], C2×D20 [×12], C22×C20, C23×D5 [×2], C22×D12, D60 [×16], C2×C60 [×6], C22×D15 [×12], C22×D15 [×8], C22×C30, C22×D20, C2×D60 [×12], C22×C60, C23×D15 [×2], C22×D60
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], D12 [×4], C22×S3 [×7], D15, C22×D4, D20 [×4], C22×D5 [×7], C2×D12 [×6], S3×C23, D30 [×7], C2×D20 [×6], C23×D5, C22×D12, D60 [×4], C22×D15 [×7], C22×D20, C2×D60 [×6], C23×D15, C22×D60
Generators and relations
G = < a,b,c,d | a2=b2=c60=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 240 points
Generators in S240
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 218)(62 219)(63 220)(64 221)(65 222)(66 223)(67 224)(68 225)(69 226)(70 227)(71 228)(72 229)(73 230)(74 231)(75 232)(76 233)(77 234)(78 235)(79 236)(80 237)(81 238)(82 239)(83 240)(84 181)(85 182)(86 183)(87 184)(88 185)(89 186)(90 187)(91 188)(92 189)(93 190)(94 191)(95 192)(96 193)(97 194)(98 195)(99 196)(100 197)(101 198)(102 199)(103 200)(104 201)(105 202)(106 203)(107 204)(108 205)(109 206)(110 207)(111 208)(112 209)(113 210)(114 211)(115 212)(116 213)(117 214)(118 215)(119 216)(120 217)
(1 189)(2 190)(3 191)(4 192)(5 193)(6 194)(7 195)(8 196)(9 197)(10 198)(11 199)(12 200)(13 201)(14 202)(15 203)(16 204)(17 205)(18 206)(19 207)(20 208)(21 209)(22 210)(23 211)(24 212)(25 213)(26 214)(27 215)(28 216)(29 217)(30 218)(31 219)(32 220)(33 221)(34 222)(35 223)(36 224)(37 225)(38 226)(39 227)(40 228)(41 229)(42 230)(43 231)(44 232)(45 233)(46 234)(47 235)(48 236)(49 237)(50 238)(51 239)(52 240)(53 181)(54 182)(55 183)(56 184)(57 185)(58 186)(59 187)(60 188)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 136)(72 137)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 145)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 161)(97 162)(98 163)(99 164)(100 165)(101 166)(102 167)(103 168)(104 169)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 177)(113 178)(114 179)(115 180)(116 121)(117 122)(118 123)(119 124)(120 125)
(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)
(1 218)(2 217)(3 216)(4 215)(5 214)(6 213)(7 212)(8 211)(9 210)(10 209)(11 208)(12 207)(13 206)(14 205)(15 204)(16 203)(17 202)(18 201)(19 200)(20 199)(21 198)(22 197)(23 196)(24 195)(25 194)(26 193)(27 192)(28 191)(29 190)(30 189)(31 188)(32 187)(33 186)(34 185)(35 184)(36 183)(37 182)(38 181)(39 240)(40 239)(41 238)(42 237)(43 236)(44 235)(45 234)(46 233)(47 232)(48 231)(49 230)(50 229)(51 228)(52 227)(53 226)(54 225)(55 224)(56 223)(57 222)(58 221)(59 220)(60 219)(61 157)(62 156)(63 155)(64 154)(65 153)(66 152)(67 151)(68 150)(69 149)(70 148)(71 147)(72 146)(73 145)(74 144)(75 143)(76 142)(77 141)(78 140)(79 139)(80 138)(81 137)(82 136)(83 135)(84 134)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)(91 127)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 180)(99 179)(100 178)(101 177)(102 176)(103 175)(104 174)(105 173)(106 172)(107 171)(108 170)(109 169)(110 168)(111 167)(112 166)(113 165)(114 164)(115 163)(116 162)(117 161)(118 160)(119 159)(120 158)
G:=sub<Sym(240)| (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,229)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201)(105,202)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,209)(113,210)(114,211)(115,212)(116,213)(117,214)(118,215)(119,216)(120,217), (1,189)(2,190)(3,191)(4,192)(5,193)(6,194)(7,195)(8,196)(9,197)(10,198)(11,199)(12,200)(13,201)(14,202)(15,203)(16,204)(17,205)(18,206)(19,207)(20,208)(21,209)(22,210)(23,211)(24,212)(25,213)(26,214)(27,215)(28,216)(29,217)(30,218)(31,219)(32,220)(33,221)(34,222)(35,223)(36,224)(37,225)(38,226)(39,227)(40,228)(41,229)(42,230)(43,231)(44,232)(45,233)(46,234)(47,235)(48,236)(49,237)(50,238)(51,239)(52,240)(53,181)(54,182)(55,183)(56,184)(57,185)(58,186)(59,187)(60,188)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,121)(117,122)(118,123)(119,124)(120,125), (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), (1,218)(2,217)(3,216)(4,215)(5,214)(6,213)(7,212)(8,211)(9,210)(10,209)(11,208)(12,207)(13,206)(14,205)(15,204)(16,203)(17,202)(18,201)(19,200)(20,199)(21,198)(22,197)(23,196)(24,195)(25,194)(26,193)(27,192)(28,191)(29,190)(30,189)(31,188)(32,187)(33,186)(34,185)(35,184)(36,183)(37,182)(38,181)(39,240)(40,239)(41,238)(42,237)(43,236)(44,235)(45,234)(46,233)(47,232)(48,231)(49,230)(50,229)(51,228)(52,227)(53,226)(54,225)(55,224)(56,223)(57,222)(58,221)(59,220)(60,219)(61,157)(62,156)(63,155)(64,154)(65,153)(66,152)(67,151)(68,150)(69,149)(70,148)(71,147)(72,146)(73,145)(74,144)(75,143)(76,142)(77,141)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,180)(99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)(120,158)>;
G:=Group( (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,229)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201)(105,202)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,209)(113,210)(114,211)(115,212)(116,213)(117,214)(118,215)(119,216)(120,217), (1,189)(2,190)(3,191)(4,192)(5,193)(6,194)(7,195)(8,196)(9,197)(10,198)(11,199)(12,200)(13,201)(14,202)(15,203)(16,204)(17,205)(18,206)(19,207)(20,208)(21,209)(22,210)(23,211)(24,212)(25,213)(26,214)(27,215)(28,216)(29,217)(30,218)(31,219)(32,220)(33,221)(34,222)(35,223)(36,224)(37,225)(38,226)(39,227)(40,228)(41,229)(42,230)(43,231)(44,232)(45,233)(46,234)(47,235)(48,236)(49,237)(50,238)(51,239)(52,240)(53,181)(54,182)(55,183)(56,184)(57,185)(58,186)(59,187)(60,188)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,121)(117,122)(118,123)(119,124)(120,125), (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), (1,218)(2,217)(3,216)(4,215)(5,214)(6,213)(7,212)(8,211)(9,210)(10,209)(11,208)(12,207)(13,206)(14,205)(15,204)(16,203)(17,202)(18,201)(19,200)(20,199)(21,198)(22,197)(23,196)(24,195)(25,194)(26,193)(27,192)(28,191)(29,190)(30,189)(31,188)(32,187)(33,186)(34,185)(35,184)(36,183)(37,182)(38,181)(39,240)(40,239)(41,238)(42,237)(43,236)(44,235)(45,234)(46,233)(47,232)(48,231)(49,230)(50,229)(51,228)(52,227)(53,226)(54,225)(55,224)(56,223)(57,222)(58,221)(59,220)(60,219)(61,157)(62,156)(63,155)(64,154)(65,153)(66,152)(67,151)(68,150)(69,149)(70,148)(71,147)(72,146)(73,145)(74,144)(75,143)(76,142)(77,141)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,180)(99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)(120,158) );
G=PermutationGroup([(1,157),(2,158),(3,159),(4,160),(5,161),(6,162),(7,163),(8,164),(9,165),(10,166),(11,167),(12,168),(13,169),(14,170),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,177),(22,178),(23,179),(24,180),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,218),(62,219),(63,220),(64,221),(65,222),(66,223),(67,224),(68,225),(69,226),(70,227),(71,228),(72,229),(73,230),(74,231),(75,232),(76,233),(77,234),(78,235),(79,236),(80,237),(81,238),(82,239),(83,240),(84,181),(85,182),(86,183),(87,184),(88,185),(89,186),(90,187),(91,188),(92,189),(93,190),(94,191),(95,192),(96,193),(97,194),(98,195),(99,196),(100,197),(101,198),(102,199),(103,200),(104,201),(105,202),(106,203),(107,204),(108,205),(109,206),(110,207),(111,208),(112,209),(113,210),(114,211),(115,212),(116,213),(117,214),(118,215),(119,216),(120,217)], [(1,189),(2,190),(3,191),(4,192),(5,193),(6,194),(7,195),(8,196),(9,197),(10,198),(11,199),(12,200),(13,201),(14,202),(15,203),(16,204),(17,205),(18,206),(19,207),(20,208),(21,209),(22,210),(23,211),(24,212),(25,213),(26,214),(27,215),(28,216),(29,217),(30,218),(31,219),(32,220),(33,221),(34,222),(35,223),(36,224),(37,225),(38,226),(39,227),(40,228),(41,229),(42,230),(43,231),(44,232),(45,233),(46,234),(47,235),(48,236),(49,237),(50,238),(51,239),(52,240),(53,181),(54,182),(55,183),(56,184),(57,185),(58,186),(59,187),(60,188),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,136),(72,137),(73,138),(74,139),(75,140),(76,141),(77,142),(78,143),(79,144),(80,145),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,161),(97,162),(98,163),(99,164),(100,165),(101,166),(102,167),(103,168),(104,169),(105,170),(106,171),(107,172),(108,173),(109,174),(110,175),(111,176),(112,177),(113,178),(114,179),(115,180),(116,121),(117,122),(118,123),(119,124),(120,125)], [(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)], [(1,218),(2,217),(3,216),(4,215),(5,214),(6,213),(7,212),(8,211),(9,210),(10,209),(11,208),(12,207),(13,206),(14,205),(15,204),(16,203),(17,202),(18,201),(19,200),(20,199),(21,198),(22,197),(23,196),(24,195),(25,194),(26,193),(27,192),(28,191),(29,190),(30,189),(31,188),(32,187),(33,186),(34,185),(35,184),(36,183),(37,182),(38,181),(39,240),(40,239),(41,238),(42,237),(43,236),(44,235),(45,234),(46,233),(47,232),(48,231),(49,230),(50,229),(51,228),(52,227),(53,226),(54,225),(55,224),(56,223),(57,222),(58,221),(59,220),(60,219),(61,157),(62,156),(63,155),(64,154),(65,153),(66,152),(67,151),(68,150),(69,149),(70,148),(71,147),(72,146),(73,145),(74,144),(75,143),(76,142),(77,141),(78,140),(79,139),(80,138),(81,137),(82,136),(83,135),(84,134),(85,133),(86,132),(87,131),(88,130),(89,129),(90,128),(91,127),(92,126),(93,125),(94,124),(95,123),(96,122),(97,121),(98,180),(99,179),(100,178),(101,177),(102,176),(103,175),(104,174),(105,173),(106,172),(107,171),(108,170),(109,169),(110,168),(111,167),(112,166),(113,165),(114,164),(115,163),(116,162),(117,161),(118,160),(119,159),(120,158)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 2 | 8 |
| 0 | 0 | 31 | 33 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 34 |
| 0 | 0 | 14 | 47 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,2,31,0,0,8,33],[1,0,0,0,0,1,0,0,0,0,14,14,0,0,34,47] >;
132 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | D15 | D20 | D30 | D30 | D60 |
| kernel | C22×D60 | C2×D60 | C22×C60 | C23×D15 | C22×C20 | C2×C30 | C22×C12 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 1 | 4 | 2 | 6 | 1 | 12 | 2 | 8 | 4 | 16 | 24 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2^2\times D_{60} % in TeX
G:=Group("C2^2xD60"); // GroupNames label
G:=SmallGroup(480,1167);
// by ID
G=gap.SmallGroup(480,1167);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^60=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀