direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C6.D20, C30⋊3SD16, C60.40D4, D20.30D6, C12.14D20, C60.103C23, Dic30⋊26C22, C3⋊C8⋊26D10, C15⋊6(C2×SD16), C6⋊3(C40⋊C2), (C6×D20).3C2, (C2×D20).3S3, C6.48(C2×D20), (C2×C30).53D4, C30.85(C2×D4), (C2×C6).39D20, C10⋊1(D4.S3), (C2×C20).286D6, (C2×C12).96D10, C4.6(C3⋊D20), (C2×Dic30)⋊21C2, C20.54(C3⋊D4), C12.94(C22×D5), (C2×C60).105C22, C20.153(C22×S3), (C3×D20).35C22, C22.20(C3⋊D20), (C2×C3⋊C8)⋊6D5, (C10×C3⋊C8)⋊7C2, C3⋊4(C2×C40⋊C2), C5⋊1(C2×D4.S3), C4.102(C2×S3×D5), (C5×C3⋊C8)⋊30C22, (C2×C4).95(S3×D5), C10.3(C2×C3⋊D4), C2.7(C2×C3⋊D20), (C2×C10).31(C3⋊D4), SmallGroup(480,386)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6.D20
G = < a,b,c,d | a2=b6=1, c20=d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c19 >
Subgroups: 764 in 136 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C2×SD16, C40, Dic10, D20, D20, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, D4.S3, C2×Dic6, C6×D4, Dic15, C60, C6×D5, C2×C30, C40⋊C2, C2×C40, C2×Dic10, C2×D20, C2×D4.S3, C5×C3⋊C8, C3×D20, C3×D20, Dic30, Dic30, C2×Dic15, C2×C60, D5×C2×C6, C2×C40⋊C2, C6.D20, C10×C3⋊C8, C6×D20, C2×Dic30, C2×C6.D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C3⋊D4, C22×S3, C2×SD16, D20, C22×D5, D4.S3, C2×C3⋊D4, S3×D5, C40⋊C2, C2×D20, C2×D4.S3, C3⋊D20, C2×S3×D5, C2×C40⋊C2, C6.D20, C2×C3⋊D20, C2×C6.D20
►On 240 points
Generators in S240
(1 198)(2 199)(3 200)(4 161)(5 162)(6 163)(7 164)(8 165)(9 166)(10 167)(11 168)(12 169)(13 170)(14 171)(15 172)(16 173)(17 174)(18 175)(19 176)(20 177)(21 178)(22 179)(23 180)(24 181)(25 182)(26 183)(27 184)(28 185)(29 186)(30 187)(31 188)(32 189)(33 190)(34 191)(35 192)(36 193)(37 194)(38 195)(39 196)(40 197)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 233)(82 234)(83 235)(84 236)(85 237)(86 238)(87 239)(88 240)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)(113 225)(114 226)(115 227)(116 228)(117 229)(118 230)(119 231)(120 232)
(1 145 216 21 125 236)(2 237 126 22 217 146)(3 147 218 23 127 238)(4 239 128 24 219 148)(5 149 220 25 129 240)(6 201 130 26 221 150)(7 151 222 27 131 202)(8 203 132 28 223 152)(9 153 224 29 133 204)(10 205 134 30 225 154)(11 155 226 31 135 206)(12 207 136 32 227 156)(13 157 228 33 137 208)(14 209 138 34 229 158)(15 159 230 35 139 210)(16 211 140 36 231 160)(17 121 232 37 141 212)(18 213 142 38 233 122)(19 123 234 39 143 214)(20 215 144 40 235 124)(41 106 180 61 86 200)(42 161 87 62 181 107)(43 108 182 63 88 162)(44 163 89 64 183 109)(45 110 184 65 90 164)(46 165 91 66 185 111)(47 112 186 67 92 166)(48 167 93 68 187 113)(49 114 188 69 94 168)(50 169 95 70 189 115)(51 116 190 71 96 170)(52 171 97 72 191 117)(53 118 192 73 98 172)(54 173 99 74 193 119)(55 120 194 75 100 174)(56 175 101 76 195 81)(57 82 196 77 102 176)(58 177 103 78 197 83)(59 84 198 79 104 178)(60 179 105 80 199 85)
(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)
(1 177 21 197)(2 196 22 176)(3 175 23 195)(4 194 24 174)(5 173 25 193)(6 192 26 172)(7 171 27 191)(8 190 28 170)(9 169 29 189)(10 188 30 168)(11 167 31 187)(12 186 32 166)(13 165 33 185)(14 184 34 164)(15 163 35 183)(16 182 36 162)(17 161 37 181)(18 180 38 200)(19 199 39 179)(20 178 40 198)(41 122 61 142)(42 141 62 121)(43 160 63 140)(44 139 64 159)(45 158 65 138)(46 137 66 157)(47 156 67 136)(48 135 68 155)(49 154 69 134)(50 133 70 153)(51 152 71 132)(52 131 72 151)(53 150 73 130)(54 129 74 149)(55 148 75 128)(56 127 76 147)(57 146 77 126)(58 125 78 145)(59 144 79 124)(60 123 80 143)(81 238 101 218)(82 217 102 237)(83 236 103 216)(84 215 104 235)(85 234 105 214)(86 213 106 233)(87 232 107 212)(88 211 108 231)(89 230 109 210)(90 209 110 229)(91 228 111 208)(92 207 112 227)(93 226 113 206)(94 205 114 225)(95 224 115 204)(96 203 116 223)(97 222 117 202)(98 201 118 221)(99 220 119 240)(100 239 120 219)
G:=sub<Sym(240)| (1,198)(2,199)(3,200)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,182)(26,183)(27,184)(28,185)(29,186)(30,187)(31,188)(32,189)(33,190)(34,191)(35,192)(36,193)(37,194)(38,195)(39,196)(40,197)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,233)(82,234)(83,235)(84,236)(85,237)(86,238)(87,239)(88,240)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224)(113,225)(114,226)(115,227)(116,228)(117,229)(118,230)(119,231)(120,232), (1,145,216,21,125,236)(2,237,126,22,217,146)(3,147,218,23,127,238)(4,239,128,24,219,148)(5,149,220,25,129,240)(6,201,130,26,221,150)(7,151,222,27,131,202)(8,203,132,28,223,152)(9,153,224,29,133,204)(10,205,134,30,225,154)(11,155,226,31,135,206)(12,207,136,32,227,156)(13,157,228,33,137,208)(14,209,138,34,229,158)(15,159,230,35,139,210)(16,211,140,36,231,160)(17,121,232,37,141,212)(18,213,142,38,233,122)(19,123,234,39,143,214)(20,215,144,40,235,124)(41,106,180,61,86,200)(42,161,87,62,181,107)(43,108,182,63,88,162)(44,163,89,64,183,109)(45,110,184,65,90,164)(46,165,91,66,185,111)(47,112,186,67,92,166)(48,167,93,68,187,113)(49,114,188,69,94,168)(50,169,95,70,189,115)(51,116,190,71,96,170)(52,171,97,72,191,117)(53,118,192,73,98,172)(54,173,99,74,193,119)(55,120,194,75,100,174)(56,175,101,76,195,81)(57,82,196,77,102,176)(58,177,103,78,197,83)(59,84,198,79,104,178)(60,179,105,80,199,85), (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), (1,177,21,197)(2,196,22,176)(3,175,23,195)(4,194,24,174)(5,173,25,193)(6,192,26,172)(7,171,27,191)(8,190,28,170)(9,169,29,189)(10,188,30,168)(11,167,31,187)(12,186,32,166)(13,165,33,185)(14,184,34,164)(15,163,35,183)(16,182,36,162)(17,161,37,181)(18,180,38,200)(19,199,39,179)(20,178,40,198)(41,122,61,142)(42,141,62,121)(43,160,63,140)(44,139,64,159)(45,158,65,138)(46,137,66,157)(47,156,67,136)(48,135,68,155)(49,154,69,134)(50,133,70,153)(51,152,71,132)(52,131,72,151)(53,150,73,130)(54,129,74,149)(55,148,75,128)(56,127,76,147)(57,146,77,126)(58,125,78,145)(59,144,79,124)(60,123,80,143)(81,238,101,218)(82,217,102,237)(83,236,103,216)(84,215,104,235)(85,234,105,214)(86,213,106,233)(87,232,107,212)(88,211,108,231)(89,230,109,210)(90,209,110,229)(91,228,111,208)(92,207,112,227)(93,226,113,206)(94,205,114,225)(95,224,115,204)(96,203,116,223)(97,222,117,202)(98,201,118,221)(99,220,119,240)(100,239,120,219)>;
G:=Group( (1,198)(2,199)(3,200)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,182)(26,183)(27,184)(28,185)(29,186)(30,187)(31,188)(32,189)(33,190)(34,191)(35,192)(36,193)(37,194)(38,195)(39,196)(40,197)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,233)(82,234)(83,235)(84,236)(85,237)(86,238)(87,239)(88,240)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224)(113,225)(114,226)(115,227)(116,228)(117,229)(118,230)(119,231)(120,232), (1,145,216,21,125,236)(2,237,126,22,217,146)(3,147,218,23,127,238)(4,239,128,24,219,148)(5,149,220,25,129,240)(6,201,130,26,221,150)(7,151,222,27,131,202)(8,203,132,28,223,152)(9,153,224,29,133,204)(10,205,134,30,225,154)(11,155,226,31,135,206)(12,207,136,32,227,156)(13,157,228,33,137,208)(14,209,138,34,229,158)(15,159,230,35,139,210)(16,211,140,36,231,160)(17,121,232,37,141,212)(18,213,142,38,233,122)(19,123,234,39,143,214)(20,215,144,40,235,124)(41,106,180,61,86,200)(42,161,87,62,181,107)(43,108,182,63,88,162)(44,163,89,64,183,109)(45,110,184,65,90,164)(46,165,91,66,185,111)(47,112,186,67,92,166)(48,167,93,68,187,113)(49,114,188,69,94,168)(50,169,95,70,189,115)(51,116,190,71,96,170)(52,171,97,72,191,117)(53,118,192,73,98,172)(54,173,99,74,193,119)(55,120,194,75,100,174)(56,175,101,76,195,81)(57,82,196,77,102,176)(58,177,103,78,197,83)(59,84,198,79,104,178)(60,179,105,80,199,85), (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), (1,177,21,197)(2,196,22,176)(3,175,23,195)(4,194,24,174)(5,173,25,193)(6,192,26,172)(7,171,27,191)(8,190,28,170)(9,169,29,189)(10,188,30,168)(11,167,31,187)(12,186,32,166)(13,165,33,185)(14,184,34,164)(15,163,35,183)(16,182,36,162)(17,161,37,181)(18,180,38,200)(19,199,39,179)(20,178,40,198)(41,122,61,142)(42,141,62,121)(43,160,63,140)(44,139,64,159)(45,158,65,138)(46,137,66,157)(47,156,67,136)(48,135,68,155)(49,154,69,134)(50,133,70,153)(51,152,71,132)(52,131,72,151)(53,150,73,130)(54,129,74,149)(55,148,75,128)(56,127,76,147)(57,146,77,126)(58,125,78,145)(59,144,79,124)(60,123,80,143)(81,238,101,218)(82,217,102,237)(83,236,103,216)(84,215,104,235)(85,234,105,214)(86,213,106,233)(87,232,107,212)(88,211,108,231)(89,230,109,210)(90,209,110,229)(91,228,111,208)(92,207,112,227)(93,226,113,206)(94,205,114,225)(95,224,115,204)(96,203,116,223)(97,222,117,202)(98,201,118,221)(99,220,119,240)(100,239,120,219) );
G=PermutationGroup([[(1,198),(2,199),(3,200),(4,161),(5,162),(6,163),(7,164),(8,165),(9,166),(10,167),(11,168),(12,169),(13,170),(14,171),(15,172),(16,173),(17,174),(18,175),(19,176),(20,177),(21,178),(22,179),(23,180),(24,181),(25,182),(26,183),(27,184),(28,185),(29,186),(30,187),(31,188),(32,189),(33,190),(34,191),(35,192),(36,193),(37,194),(38,195),(39,196),(40,197),(41,147),(42,148),(43,149),(44,150),(45,151),(46,152),(47,153),(48,154),(49,155),(50,156),(51,157),(52,158),(53,159),(54,160),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,233),(82,234),(83,235),(84,236),(85,237),(86,238),(87,239),(88,240),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224),(113,225),(114,226),(115,227),(116,228),(117,229),(118,230),(119,231),(120,232)], [(1,145,216,21,125,236),(2,237,126,22,217,146),(3,147,218,23,127,238),(4,239,128,24,219,148),(5,149,220,25,129,240),(6,201,130,26,221,150),(7,151,222,27,131,202),(8,203,132,28,223,152),(9,153,224,29,133,204),(10,205,134,30,225,154),(11,155,226,31,135,206),(12,207,136,32,227,156),(13,157,228,33,137,208),(14,209,138,34,229,158),(15,159,230,35,139,210),(16,211,140,36,231,160),(17,121,232,37,141,212),(18,213,142,38,233,122),(19,123,234,39,143,214),(20,215,144,40,235,124),(41,106,180,61,86,200),(42,161,87,62,181,107),(43,108,182,63,88,162),(44,163,89,64,183,109),(45,110,184,65,90,164),(46,165,91,66,185,111),(47,112,186,67,92,166),(48,167,93,68,187,113),(49,114,188,69,94,168),(50,169,95,70,189,115),(51,116,190,71,96,170),(52,171,97,72,191,117),(53,118,192,73,98,172),(54,173,99,74,193,119),(55,120,194,75,100,174),(56,175,101,76,195,81),(57,82,196,77,102,176),(58,177,103,78,197,83),(59,84,198,79,104,178),(60,179,105,80,199,85)], [(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)], [(1,177,21,197),(2,196,22,176),(3,175,23,195),(4,194,24,174),(5,173,25,193),(6,192,26,172),(7,171,27,191),(8,190,28,170),(9,169,29,189),(10,188,30,168),(11,167,31,187),(12,186,32,166),(13,165,33,185),(14,184,34,164),(15,163,35,183),(16,182,36,162),(17,161,37,181),(18,180,38,200),(19,199,39,179),(20,178,40,198),(41,122,61,142),(42,141,62,121),(43,160,63,140),(44,139,64,159),(45,158,65,138),(46,137,66,157),(47,156,67,136),(48,135,68,155),(49,154,69,134),(50,133,70,153),(51,152,71,132),(52,131,72,151),(53,150,73,130),(54,129,74,149),(55,148,75,128),(56,127,76,147),(57,146,77,126),(58,125,78,145),(59,144,79,124),(60,123,80,143),(81,238,101,218),(82,217,102,237),(83,236,103,216),(84,215,104,235),(85,234,105,214),(86,213,106,233),(87,232,107,212),(88,211,108,231),(89,230,109,210),(90,209,110,229),(91,228,111,208),(92,207,112,227),(93,226,113,206),(94,205,114,225),(95,224,115,204),(96,203,116,223),(97,222,117,202),(98,201,118,221),(99,220,119,240),(100,239,120,219)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | SD16 | D10 | D10 | C3⋊D4 | C3⋊D4 | D20 | D20 | C40⋊C2 | D4.S3 | S3×D5 | C3⋊D20 | C2×S3×D5 | C3⋊D20 | C6.D20 |
| kernel | C2×C6.D20 | C6.D20 | C10×C3⋊C8 | C6×D20 | C2×Dic30 | C2×D20 | C60 | C2×C30 | C2×C3⋊C8 | D20 | C2×C20 | C30 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C6 | C10 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×C6.D20 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 15 |
| 94 | 166 | 0 | 0 |
| 106 | 200 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 |
| 57 | 173 | 0 | 0 |
| 225 | 184 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[240,0,0,0,0,240,0,0,0,0,225,0,0,0,0,15],[94,106,0,0,166,200,0,0,0,0,0,240,0,0,1,0],[57,225,0,0,173,184,0,0,0,0,0,240,0,0,240,0] >;
C2×C6.D20 in GAP, Magma, Sage, TeX
C_2\times C_6.D_{20} % in TeX
G:=Group("C2xC6.D20"); // GroupNames label
G:=SmallGroup(480,386);
// by ID
G=gap.SmallGroup(480,386);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=1,c^20=d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^19>;
// generators/relations