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 [×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
►On 240 points
Generators in S240
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 217)(137 218)(138 219)(139 220)(140 221)(141 222)(142 223)(143 224)(144 225)(145 226)(146 227)(147 228)(148 229)(149 230)(150 231)(151 232)(152 233)(153 234)(154 235)(155 236)(156 237)(157 238)(158 239)(159 240)(160 181)(161 182)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(169 190)(170 191)(171 192)(172 193)(173 194)(174 195)(175 196)(176 197)(177 198)(178 199)(179 200)(180 201)
(1 188)(2 189)(3 190)(4 191)(5 192)(6 193)(7 194)(8 195)(9 196)(10 197)(11 198)(12 199)(13 200)(14 201)(15 202)(16 203)(17 204)(18 205)(19 206)(20 207)(21 208)(22 209)(23 210)(24 211)(25 212)(26 213)(27 214)(28 215)(29 216)(30 217)(31 218)(32 219)(33 220)(34 221)(35 222)(36 223)(37 224)(38 225)(39 226)(40 227)(41 228)(42 229)(43 230)(44 231)(45 232)(46 233)(47 234)(48 235)(49 236)(50 237)(51 238)(52 239)(53 240)(54 181)(55 182)(56 183)(57 184)(58 185)(59 186)(60 187)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(91 169)(92 170)(93 171)(94 172)(95 173)(96 174)(97 175)(98 176)(99 177)(100 178)(101 179)(102 180)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)
(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)
(1 217)(2 216)(3 215)(4 214)(5 213)(6 212)(7 211)(8 210)(9 209)(10 208)(11 207)(12 206)(13 205)(14 204)(15 203)(16 202)(17 201)(18 200)(19 199)(20 198)(21 197)(22 196)(23 195)(24 194)(25 193)(26 192)(27 191)(28 190)(29 189)(30 188)(31 187)(32 186)(33 185)(34 184)(35 183)(36 182)(37 181)(38 240)(39 239)(40 238)(41 237)(42 236)(43 235)(44 234)(45 233)(46 232)(47 231)(48 230)(49 229)(50 228)(51 227)(52 226)(53 225)(54 224)(55 223)(56 222)(57 221)(58 220)(59 219)(60 218)(61 164)(62 163)(63 162)(64 161)(65 160)(66 159)(67 158)(68 157)(69 156)(70 155)(71 154)(72 153)(73 152)(74 151)(75 150)(76 149)(77 148)(78 147)(79 146)(80 145)(81 144)(82 143)(83 142)(84 141)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 126)(100 125)(101 124)(102 123)(103 122)(104 121)(105 180)(106 179)(107 178)(108 177)(109 176)(110 175)(111 174)(112 173)(113 172)(114 171)(115 170)(116 169)(117 168)(118 167)(119 166)(120 165)
G:=sub<Sym(240)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,181)(161,182)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,190)(170,191)(171,192)(172,193)(173,194)(174,195)(175,196)(176,197)(177,198)(178,199)(179,200)(180,201), (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,197)(11,198)(12,199)(13,200)(14,201)(15,202)(16,203)(17,204)(18,205)(19,206)(20,207)(21,208)(22,209)(23,210)(24,211)(25,212)(26,213)(27,214)(28,215)(29,216)(30,217)(31,218)(32,219)(33,220)(34,221)(35,222)(36,223)(37,224)(38,225)(39,226)(40,227)(41,228)(42,229)(43,230)(44,231)(45,232)(46,233)(47,234)(48,235)(49,236)(50,237)(51,238)(52,239)(53,240)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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), (1,217)(2,216)(3,215)(4,214)(5,213)(6,212)(7,211)(8,210)(9,209)(10,208)(11,207)(12,206)(13,205)(14,204)(15,203)(16,202)(17,201)(18,200)(19,199)(20,198)(21,197)(22,196)(23,195)(24,194)(25,193)(26,192)(27,191)(28,190)(29,189)(30,188)(31,187)(32,186)(33,185)(34,184)(35,183)(36,182)(37,181)(38,240)(39,239)(40,238)(41,237)(42,236)(43,235)(44,234)(45,233)(46,232)(47,231)(48,230)(49,229)(50,228)(51,227)(52,226)(53,225)(54,224)(55,223)(56,222)(57,221)(58,220)(59,219)(60,218)(61,164)(62,163)(63,162)(64,161)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,144)(82,143)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,123)(103,122)(104,121)(105,180)(106,179)(107,178)(108,177)(109,176)(110,175)(111,174)(112,173)(113,172)(114,171)(115,170)(116,169)(117,168)(118,167)(119,166)(120,165)>;
G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,181)(161,182)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,190)(170,191)(171,192)(172,193)(173,194)(174,195)(175,196)(176,197)(177,198)(178,199)(179,200)(180,201), (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,197)(11,198)(12,199)(13,200)(14,201)(15,202)(16,203)(17,204)(18,205)(19,206)(20,207)(21,208)(22,209)(23,210)(24,211)(25,212)(26,213)(27,214)(28,215)(29,216)(30,217)(31,218)(32,219)(33,220)(34,221)(35,222)(36,223)(37,224)(38,225)(39,226)(40,227)(41,228)(42,229)(43,230)(44,231)(45,232)(46,233)(47,234)(48,235)(49,236)(50,237)(51,238)(52,239)(53,240)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,179)(102,180)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138), (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), (1,217)(2,216)(3,215)(4,214)(5,213)(6,212)(7,211)(8,210)(9,209)(10,208)(11,207)(12,206)(13,205)(14,204)(15,203)(16,202)(17,201)(18,200)(19,199)(20,198)(21,197)(22,196)(23,195)(24,194)(25,193)(26,192)(27,191)(28,190)(29,189)(30,188)(31,187)(32,186)(33,185)(34,184)(35,183)(36,182)(37,181)(38,240)(39,239)(40,238)(41,237)(42,236)(43,235)(44,234)(45,233)(46,232)(47,231)(48,230)(49,229)(50,228)(51,227)(52,226)(53,225)(54,224)(55,223)(56,222)(57,221)(58,220)(59,219)(60,218)(61,164)(62,163)(63,162)(64,161)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,144)(82,143)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,123)(103,122)(104,121)(105,180)(106,179)(107,178)(108,177)(109,176)(110,175)(111,174)(112,173)(113,172)(114,171)(115,170)(116,169)(117,168)(118,167)(119,166)(120,165) );
G=PermutationGroup([(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(57,85),(58,86),(59,87),(60,88),(121,202),(122,203),(123,204),(124,205),(125,206),(126,207),(127,208),(128,209),(129,210),(130,211),(131,212),(132,213),(133,214),(134,215),(135,216),(136,217),(137,218),(138,219),(139,220),(140,221),(141,222),(142,223),(143,224),(144,225),(145,226),(146,227),(147,228),(148,229),(149,230),(150,231),(151,232),(152,233),(153,234),(154,235),(155,236),(156,237),(157,238),(158,239),(159,240),(160,181),(161,182),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189),(169,190),(170,191),(171,192),(172,193),(173,194),(174,195),(175,196),(176,197),(177,198),(178,199),(179,200),(180,201)], [(1,188),(2,189),(3,190),(4,191),(5,192),(6,193),(7,194),(8,195),(9,196),(10,197),(11,198),(12,199),(13,200),(14,201),(15,202),(16,203),(17,204),(18,205),(19,206),(20,207),(21,208),(22,209),(23,210),(24,211),(25,212),(26,213),(27,214),(28,215),(29,216),(30,217),(31,218),(32,219),(33,220),(34,221),(35,222),(36,223),(37,224),(38,225),(39,226),(40,227),(41,228),(42,229),(43,230),(44,231),(45,232),(46,233),(47,234),(48,235),(49,236),(50,237),(51,238),(52,239),(53,240),(54,181),(55,182),(56,183),(57,184),(58,185),(59,186),(60,187),(61,139),(62,140),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,155),(78,156),(79,157),(80,158),(81,159),(82,160),(83,161),(84,162),(85,163),(86,164),(87,165),(88,166),(89,167),(90,168),(91,169),(92,170),(93,171),(94,172),(95,173),(96,174),(97,175),(98,176),(99,177),(100,178),(101,179),(102,180),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138)], [(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)], [(1,217),(2,216),(3,215),(4,214),(5,213),(6,212),(7,211),(8,210),(9,209),(10,208),(11,207),(12,206),(13,205),(14,204),(15,203),(16,202),(17,201),(18,200),(19,199),(20,198),(21,197),(22,196),(23,195),(24,194),(25,193),(26,192),(27,191),(28,190),(29,189),(30,188),(31,187),(32,186),(33,185),(34,184),(35,183),(36,182),(37,181),(38,240),(39,239),(40,238),(41,237),(42,236),(43,235),(44,234),(45,233),(46,232),(47,231),(48,230),(49,229),(50,228),(51,227),(52,226),(53,225),(54,224),(55,223),(56,222),(57,221),(58,220),(59,219),(60,218),(61,164),(62,163),(63,162),(64,161),(65,160),(66,159),(67,158),(68,157),(69,156),(70,155),(71,154),(72,153),(73,152),(74,151),(75,150),(76,149),(77,148),(78,147),(79,146),(80,145),(81,144),(82,143),(83,142),(84,141),(85,140),(86,139),(87,138),(88,137),(89,136),(90,135),(91,134),(92,133),(93,132),(94,131),(95,130),(96,129),(97,128),(98,127),(99,126),(100,125),(101,124),(102,123),(103,122),(104,121),(105,180),(106,179),(107,178),(108,177),(109,176),(110,175),(111,174),(112,173),(113,172),(114,171),(115,170),(116,169),(117,168),(118,167),(119,166),(120,165)])
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