direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C6×D20, C60⋊11C23, C30.71C24, C10⋊1(C6×D4), C30⋊11(C2×D4), (C2×C30)⋊26D4, (C2×C12)⋊36D10, C20⋊2(C22×C6), (C23×D5)⋊6C6, C15⋊12(C22×D4), (C2×C60)⋊47C22, (C22×C20)⋊11C6, (C22×C60)⋊16C2, (C6×D5)⋊10C23, D10⋊1(C22×C6), C12⋊10(C22×D5), (C22×C12)⋊12D5, C10.3(C23×C6), C6.71(C23×D5), C23.40(C6×D5), (C2×C30).381C23, (C22×C6).137D10, (C22×C30).166C22, C5⋊1(D4×C2×C6), C4⋊2(D5×C2×C6), (C2×C4)⋊9(C6×D5), (C2×C10)⋊9(C3×D4), (D5×C22×C6)⋊9C2, (C2×C20)⋊12(C2×C6), C2.4(D5×C22×C6), (D5×C2×C6)⋊21C22, (C22×C4)⋊7(C3×D5), C22.30(D5×C2×C6), (C22×D5)⋊6(C2×C6), (C2×C10).64(C22×C6), (C22×C10).53(C2×C6), (C2×C6).377(C22×D5), SmallGroup(480,1137)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1680 in 472 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], C5, C6, C6 [×6], C6 [×8], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×8], C10, C10 [×6], C12 [×4], C2×C6 [×7], C2×C6 [×32], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×4], D10 [×8], D10 [×24], C2×C10 [×7], C2×C12 [×6], C3×D4 [×16], C22×C6, C22×C6 [×20], C3×D5 [×8], C30, C30 [×6], C22×D4, D20 [×16], C2×C20 [×6], C22×D5 [×12], C22×D5 [×8], C22×C10, C22×C12, C6×D4 [×12], C23×C6 [×2], C60 [×4], C6×D5 [×8], C6×D5 [×24], C2×C30 [×7], C2×D20 [×12], C22×C20, C23×D5 [×2], D4×C2×C6, C3×D20 [×16], C2×C60 [×6], D5×C2×C6 [×12], D5×C2×C6 [×8], C22×C30, C22×D20, C6×D20 [×12], C22×C60, D5×C22×C6 [×2], C2×C6×D20
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], D5, C2×C6 [×35], C2×D4 [×6], C24, D10 [×7], C3×D4 [×4], C22×C6 [×15], C3×D5, C22×D4, D20 [×4], C22×D5 [×7], C6×D4 [×6], C23×C6, C6×D5 [×7], C2×D20 [×6], C23×D5, D4×C2×C6, C3×D20 [×4], D5×C2×C6 [×7], C22×D20, C6×D20 [×6], D5×C22×C6, C2×C6×D20
Generators and relations
G = < a,b,c,d | a2=b6=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 240 points
Generators in S240
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 101)(56 102)(57 103)(58 104)(59 105)(60 106)(61 189)(62 190)(63 191)(64 192)(65 193)(66 194)(67 195)(68 196)(69 197)(70 198)(71 199)(72 200)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 213)(82 214)(83 215)(84 216)(85 217)(86 218)(87 219)(88 220)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(161 235)(162 236)(163 237)(164 238)(165 239)(166 240)(167 221)(168 222)(169 223)(170 224)(171 225)(172 226)(173 227)(174 228)(175 229)(176 230)(177 231)(178 232)(179 233)(180 234)
(1 193 111 177 144 217)(2 194 112 178 145 218)(3 195 113 179 146 219)(4 196 114 180 147 220)(5 197 115 161 148 201)(6 198 116 162 149 202)(7 199 117 163 150 203)(8 200 118 164 151 204)(9 181 119 165 152 205)(10 182 120 166 153 206)(11 183 101 167 154 207)(12 184 102 168 155 208)(13 185 103 169 156 209)(14 186 104 170 157 210)(15 187 105 171 158 211)(16 188 106 172 159 212)(17 189 107 173 160 213)(18 190 108 174 141 214)(19 191 109 175 142 215)(20 192 110 176 143 216)(21 94 130 74 54 240)(22 95 131 75 55 221)(23 96 132 76 56 222)(24 97 133 77 57 223)(25 98 134 78 58 224)(26 99 135 79 59 225)(27 100 136 80 60 226)(28 81 137 61 41 227)(29 82 138 62 42 228)(30 83 139 63 43 229)(31 84 140 64 44 230)(32 85 121 65 45 231)(33 86 122 66 46 232)(34 87 123 67 47 233)(35 88 124 68 48 234)(36 89 125 69 49 235)(37 90 126 70 50 236)(38 91 127 71 51 237)(39 92 128 72 52 238)(40 93 129 73 53 239)
(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)
(1 171)(2 170)(3 169)(4 168)(5 167)(6 166)(7 165)(8 164)(9 163)(10 162)(11 161)(12 180)(13 179)(14 178)(15 177)(16 176)(17 175)(18 174)(19 173)(20 172)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 83)(42 82)(43 81)(44 100)(45 99)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 90)(55 89)(56 88)(57 87)(58 86)(59 85)(60 84)(101 201)(102 220)(103 219)(104 218)(105 217)(106 216)(107 215)(108 214)(109 213)(110 212)(111 211)(112 210)(113 209)(114 208)(115 207)(116 206)(117 205)(118 204)(119 203)(120 202)(121 225)(122 224)(123 223)(124 222)(125 221)(126 240)(127 239)(128 238)(129 237)(130 236)(131 235)(132 234)(133 233)(134 232)(135 231)(136 230)(137 229)(138 228)(139 227)(140 226)(141 190)(142 189)(143 188)(144 187)(145 186)(146 185)(147 184)(148 183)(149 182)(150 181)(151 200)(152 199)(153 198)(154 197)(155 196)(156 195)(157 194)(158 193)(159 192)(160 191)
G:=sub<Sym(240)| (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(61,189)(62,190)(63,191)(64,192)(65,193)(66,194)(67,195)(68,196)(69,197)(70,198)(71,199)(72,200)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,213)(82,214)(83,215)(84,216)(85,217)(86,218)(87,219)(88,220)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(161,235)(162,236)(163,237)(164,238)(165,239)(166,240)(167,221)(168,222)(169,223)(170,224)(171,225)(172,226)(173,227)(174,228)(175,229)(176,230)(177,231)(178,232)(179,233)(180,234), (1,193,111,177,144,217)(2,194,112,178,145,218)(3,195,113,179,146,219)(4,196,114,180,147,220)(5,197,115,161,148,201)(6,198,116,162,149,202)(7,199,117,163,150,203)(8,200,118,164,151,204)(9,181,119,165,152,205)(10,182,120,166,153,206)(11,183,101,167,154,207)(12,184,102,168,155,208)(13,185,103,169,156,209)(14,186,104,170,157,210)(15,187,105,171,158,211)(16,188,106,172,159,212)(17,189,107,173,160,213)(18,190,108,174,141,214)(19,191,109,175,142,215)(20,192,110,176,143,216)(21,94,130,74,54,240)(22,95,131,75,55,221)(23,96,132,76,56,222)(24,97,133,77,57,223)(25,98,134,78,58,224)(26,99,135,79,59,225)(27,100,136,80,60,226)(28,81,137,61,41,227)(29,82,138,62,42,228)(30,83,139,63,43,229)(31,84,140,64,44,230)(32,85,121,65,45,231)(33,86,122,66,46,232)(34,87,123,67,47,233)(35,88,124,68,48,234)(36,89,125,69,49,235)(37,90,126,70,50,236)(38,91,127,71,51,237)(39,92,128,72,52,238)(40,93,129,73,53,239), (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), (1,171)(2,170)(3,169)(4,168)(5,167)(6,166)(7,165)(8,164)(9,163)(10,162)(11,161)(12,180)(13,179)(14,178)(15,177)(16,176)(17,175)(18,174)(19,173)(20,172)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,83)(42,82)(43,81)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(101,201)(102,220)(103,219)(104,218)(105,217)(106,216)(107,215)(108,214)(109,213)(110,212)(111,211)(112,210)(113,209)(114,208)(115,207)(116,206)(117,205)(118,204)(119,203)(120,202)(121,225)(122,224)(123,223)(124,222)(125,221)(126,240)(127,239)(128,238)(129,237)(130,236)(131,235)(132,234)(133,233)(134,232)(135,231)(136,230)(137,229)(138,228)(139,227)(140,226)(141,190)(142,189)(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)(149,182)(150,181)(151,200)(152,199)(153,198)(154,197)(155,196)(156,195)(157,194)(158,193)(159,192)(160,191)>;
G:=Group( (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(61,189)(62,190)(63,191)(64,192)(65,193)(66,194)(67,195)(68,196)(69,197)(70,198)(71,199)(72,200)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,213)(82,214)(83,215)(84,216)(85,217)(86,218)(87,219)(88,220)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(161,235)(162,236)(163,237)(164,238)(165,239)(166,240)(167,221)(168,222)(169,223)(170,224)(171,225)(172,226)(173,227)(174,228)(175,229)(176,230)(177,231)(178,232)(179,233)(180,234), (1,193,111,177,144,217)(2,194,112,178,145,218)(3,195,113,179,146,219)(4,196,114,180,147,220)(5,197,115,161,148,201)(6,198,116,162,149,202)(7,199,117,163,150,203)(8,200,118,164,151,204)(9,181,119,165,152,205)(10,182,120,166,153,206)(11,183,101,167,154,207)(12,184,102,168,155,208)(13,185,103,169,156,209)(14,186,104,170,157,210)(15,187,105,171,158,211)(16,188,106,172,159,212)(17,189,107,173,160,213)(18,190,108,174,141,214)(19,191,109,175,142,215)(20,192,110,176,143,216)(21,94,130,74,54,240)(22,95,131,75,55,221)(23,96,132,76,56,222)(24,97,133,77,57,223)(25,98,134,78,58,224)(26,99,135,79,59,225)(27,100,136,80,60,226)(28,81,137,61,41,227)(29,82,138,62,42,228)(30,83,139,63,43,229)(31,84,140,64,44,230)(32,85,121,65,45,231)(33,86,122,66,46,232)(34,87,123,67,47,233)(35,88,124,68,48,234)(36,89,125,69,49,235)(37,90,126,70,50,236)(38,91,127,71,51,237)(39,92,128,72,52,238)(40,93,129,73,53,239), (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), (1,171)(2,170)(3,169)(4,168)(5,167)(6,166)(7,165)(8,164)(9,163)(10,162)(11,161)(12,180)(13,179)(14,178)(15,177)(16,176)(17,175)(18,174)(19,173)(20,172)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,83)(42,82)(43,81)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(101,201)(102,220)(103,219)(104,218)(105,217)(106,216)(107,215)(108,214)(109,213)(110,212)(111,211)(112,210)(113,209)(114,208)(115,207)(116,206)(117,205)(118,204)(119,203)(120,202)(121,225)(122,224)(123,223)(124,222)(125,221)(126,240)(127,239)(128,238)(129,237)(130,236)(131,235)(132,234)(133,233)(134,232)(135,231)(136,230)(137,229)(138,228)(139,227)(140,226)(141,190)(142,189)(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)(149,182)(150,181)(151,200)(152,199)(153,198)(154,197)(155,196)(156,195)(157,194)(158,193)(159,192)(160,191) );
G=PermutationGroup([(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,153),(22,154),(23,155),(24,156),(25,157),(26,158),(27,159),(28,160),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(41,107),(42,108),(43,109),(44,110),(45,111),(46,112),(47,113),(48,114),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,101),(56,102),(57,103),(58,104),(59,105),(60,106),(61,189),(62,190),(63,191),(64,192),(65,193),(66,194),(67,195),(68,196),(69,197),(70,198),(71,199),(72,200),(73,181),(74,182),(75,183),(76,184),(77,185),(78,186),(79,187),(80,188),(81,213),(82,214),(83,215),(84,216),(85,217),(86,218),(87,219),(88,220),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(161,235),(162,236),(163,237),(164,238),(165,239),(166,240),(167,221),(168,222),(169,223),(170,224),(171,225),(172,226),(173,227),(174,228),(175,229),(176,230),(177,231),(178,232),(179,233),(180,234)], [(1,193,111,177,144,217),(2,194,112,178,145,218),(3,195,113,179,146,219),(4,196,114,180,147,220),(5,197,115,161,148,201),(6,198,116,162,149,202),(7,199,117,163,150,203),(8,200,118,164,151,204),(9,181,119,165,152,205),(10,182,120,166,153,206),(11,183,101,167,154,207),(12,184,102,168,155,208),(13,185,103,169,156,209),(14,186,104,170,157,210),(15,187,105,171,158,211),(16,188,106,172,159,212),(17,189,107,173,160,213),(18,190,108,174,141,214),(19,191,109,175,142,215),(20,192,110,176,143,216),(21,94,130,74,54,240),(22,95,131,75,55,221),(23,96,132,76,56,222),(24,97,133,77,57,223),(25,98,134,78,58,224),(26,99,135,79,59,225),(27,100,136,80,60,226),(28,81,137,61,41,227),(29,82,138,62,42,228),(30,83,139,63,43,229),(31,84,140,64,44,230),(32,85,121,65,45,231),(33,86,122,66,46,232),(34,87,123,67,47,233),(35,88,124,68,48,234),(36,89,125,69,49,235),(37,90,126,70,50,236),(38,91,127,71,51,237),(39,92,128,72,52,238),(40,93,129,73,53,239)], [(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)], [(1,171),(2,170),(3,169),(4,168),(5,167),(6,166),(7,165),(8,164),(9,163),(10,162),(11,161),(12,180),(13,179),(14,178),(15,177),(16,176),(17,175),(18,174),(19,173),(20,172),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,83),(42,82),(43,81),(44,100),(45,99),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,90),(55,89),(56,88),(57,87),(58,86),(59,85),(60,84),(101,201),(102,220),(103,219),(104,218),(105,217),(106,216),(107,215),(108,214),(109,213),(110,212),(111,211),(112,210),(113,209),(114,208),(115,207),(116,206),(117,205),(118,204),(119,203),(120,202),(121,225),(122,224),(123,223),(124,222),(125,221),(126,240),(127,239),(128,238),(129,237),(130,236),(131,235),(132,234),(133,233),(134,232),(135,231),(136,230),(137,229),(138,228),(139,227),(140,226),(141,190),(142,189),(143,188),(144,187),(145,186),(146,185),(147,184),(148,183),(149,182),(150,181),(151,200),(152,199),(153,198),(154,197),(155,196),(156,195),(157,194),(158,193),(159,192),(160,191)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 59 | 25 |
| 0 | 0 | 34 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,48,0,0,0,0,47,0,0,0,0,47],[1,0,0,0,0,1,0,0,0,0,59,34,0,0,25,32],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,60] >;
156 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6N | 6O | ··· | 6AD | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 10 | ··· | 10 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D5 | D10 | D10 | C3×D4 | C3×D5 | D20 | C6×D5 | C6×D5 | C3×D20 |
| kernel | C2×C6×D20 | C6×D20 | C22×C60 | D5×C22×C6 | C22×D20 | C2×D20 | C22×C20 | C23×D5 | C2×C30 | C22×C12 | C2×C12 | C22×C6 | C2×C10 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 2 | 24 | 2 | 4 | 4 | 2 | 12 | 2 | 8 | 4 | 16 | 24 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2\times C_6\times D_{20} % in TeX
G:=Group("C2xC6xD20"); // GroupNames label
G:=SmallGroup(480,1137);
// by ID
G=gap.SmallGroup(480,1137);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,1571,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^20=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
×
⋊
ℤ
𝔽
○
≀