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G = C22×D60order 480 = 25·3·5

Direct product of C22 and D60

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×D60, C609C23, D306C23, C30.56C24, C23.40D30, (C2×C4)⋊9D30, C62(C2×D20), (C2×C6)⋊6D20, (C2×C30)⋊24D4, C3010(C2×D4), C102(C2×D12), (C2×C20)⋊33D6, (C2×C10)⋊9D12, C32(C22×D20), C52(C22×D12), (C2×C12)⋊33D10, C208(C22×S3), C128(C22×D5), (C22×C12)⋊7D5, C42(C22×D15), (C22×C4)⋊7D15, C1511(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

C1C30 — C22×D60
C1C5C15C30D30C22×D15C23×D15 — C22×D60
C15C30 — C22×D60
C1C23C22×C4

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

Smallest permutation representation of 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···2G2H···2O 3 4A4B4C4D5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12···22···234444556···610···1012···121515151520···2030···3060···60
size11···130···3022222222···22···22···222222···22···22···2

132 irreducible representations

dim11112222222222222
type+++++++++++++++++
imageC1C2C2C2S3D4D5D6D6D10D10D12D15D20D30D30D60
kernelC22×D60C2×D60C22×C60C23×D15C22×C20C2×C30C22×C12C2×C20C22×C10C2×C12C22×C6C2×C10C22×C4C2×C6C2×C4C23C22
# reps1121214261122841624432

Matrix representation of C22×D60 in GL4(𝔽61) generated by

1000
06000
0010
0001
,
60000
0100
00600
00060
,
1000
06000
0028
003133
,
1000
0100
001434
001447
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

׿
×
𝔽