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