Copied to
clipboard

## G = C22×D60order 480 = 25·3·5

### Direct product of C22 and D60

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C22×D60
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C23×D15 — C22×D60
 Lower central C15 — C30 — C22×D60
 Upper central C1 — C23 — C22×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, 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

Smallest permutation representation of 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

׿
×
𝔽