direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C4×C20, (C4×C60)⋊22C2, (C4×C12)⋊8C10, C60⋊37(C2×C4), C12⋊5(C2×C20), C15⋊11(C2×C42), D6.6(C2×C20), Dic3⋊5(C2×C20), (C2×C20).450D6, C6.2(C22×C20), (C4×Dic3)⋊17C10, (Dic3×C20)⋊35C2, (C2×C60).562C22, C30.193(C22×C4), (C2×C30).391C23, (C10×Dic3).237C22, C3⋊1(C2×C4×C20), C2.1(S3×C2×C20), (S3×C2×C20).24C2, (S3×C2×C4).11C10, C10.129(S3×C2×C4), C22.9(S3×C2×C10), (C2×C4).97(S3×C10), (S3×C10).42(C2×C4), (C5×Dic3)⋊25(C2×C4), (C2×C12).114(C2×C10), (S3×C2×C10).124C22, (C2×C6).12(C22×C10), (C22×S3).33(C2×C10), (C2×C10).325(C22×S3), (C2×Dic3).46(C2×C10), SmallGroup(480,750)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C4×C20 |
Generators and relations for S3×C4×C20
G = < a,b,c,d | a4=b20=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 372 in 216 conjugacy classes, 138 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C42, C22×C4, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C2×C42, C2×C20, C2×C20, C22×C10, C4×Dic3, C4×C12, S3×C2×C4, C5×Dic3, C60, S3×C10, C2×C30, C4×C20, C4×C20, C22×C20, S3×C42, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C2×C4×C20, Dic3×C20, C4×C60, S3×C2×C20, S3×C4×C20
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, C42, C22×C4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C2×C42, C2×C20, C22×C10, S3×C2×C4, S3×C10, C4×C20, C22×C20, S3×C42, S3×C20, S3×C2×C10, C2×C4×C20, S3×C2×C20, S3×C4×C20
►On 240 points
Generators in S240
(1 174 85 193)(2 175 86 194)(3 176 87 195)(4 177 88 196)(5 178 89 197)(6 179 90 198)(7 180 91 199)(8 161 92 200)(9 162 93 181)(10 163 94 182)(11 164 95 183)(12 165 96 184)(13 166 97 185)(14 167 98 186)(15 168 99 187)(16 169 100 188)(17 170 81 189)(18 171 82 190)(19 172 83 191)(20 173 84 192)(21 71 240 101)(22 72 221 102)(23 73 222 103)(24 74 223 104)(25 75 224 105)(26 76 225 106)(27 77 226 107)(28 78 227 108)(29 79 228 109)(30 80 229 110)(31 61 230 111)(32 62 231 112)(33 63 232 113)(34 64 233 114)(35 65 234 115)(36 66 235 116)(37 67 236 117)(38 68 237 118)(39 69 238 119)(40 70 239 120)(41 203 153 129)(42 204 154 130)(43 205 155 131)(44 206 156 132)(45 207 157 133)(46 208 158 134)(47 209 159 135)(48 210 160 136)(49 211 141 137)(50 212 142 138)(51 213 143 139)(52 214 144 140)(53 215 145 121)(54 216 146 122)(55 217 147 123)(56 218 148 124)(57 219 149 125)(58 220 150 126)(59 201 151 127)(60 202 152 128)
(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)
(1 132 235)(2 133 236)(3 134 237)(4 135 238)(5 136 239)(6 137 240)(7 138 221)(8 139 222)(9 140 223)(10 121 224)(11 122 225)(12 123 226)(13 124 227)(14 125 228)(15 126 229)(16 127 230)(17 128 231)(18 129 232)(19 130 233)(20 131 234)(21 90 211)(22 91 212)(23 92 213)(24 93 214)(25 94 215)(26 95 216)(27 96 217)(28 97 218)(29 98 219)(30 99 220)(31 100 201)(32 81 202)(33 82 203)(34 83 204)(35 84 205)(36 85 206)(37 86 207)(38 87 208)(39 88 209)(40 89 210)(41 113 171)(42 114 172)(43 115 173)(44 116 174)(45 117 175)(46 118 176)(47 119 177)(48 120 178)(49 101 179)(50 102 180)(51 103 161)(52 104 162)(53 105 163)(54 106 164)(55 107 165)(56 108 166)(57 109 167)(58 110 168)(59 111 169)(60 112 170)(61 188 151)(62 189 152)(63 190 153)(64 191 154)(65 192 155)(66 193 156)(67 194 157)(68 195 158)(69 196 159)(70 197 160)(71 198 141)(72 199 142)(73 200 143)(74 181 144)(75 182 145)(76 183 146)(77 184 147)(78 185 148)(79 186 149)(80 187 150)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 201)(22 202)(23 203)(24 204)(25 205)(26 206)(27 207)(28 208)(29 209)(30 210)(31 211)(32 212)(33 213)(34 214)(35 215)(36 216)(37 217)(38 218)(39 219)(40 220)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 101)(60 102)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(121 234)(122 235)(123 236)(124 237)(125 238)(126 239)(127 240)(128 221)(129 222)(130 223)(131 224)(132 225)(133 226)(134 227)(135 228)(136 229)(137 230)(138 231)(139 232)(140 233)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)
G:=sub<Sym(240)| (1,174,85,193)(2,175,86,194)(3,176,87,195)(4,177,88,196)(5,178,89,197)(6,179,90,198)(7,180,91,199)(8,161,92,200)(9,162,93,181)(10,163,94,182)(11,164,95,183)(12,165,96,184)(13,166,97,185)(14,167,98,186)(15,168,99,187)(16,169,100,188)(17,170,81,189)(18,171,82,190)(19,172,83,191)(20,173,84,192)(21,71,240,101)(22,72,221,102)(23,73,222,103)(24,74,223,104)(25,75,224,105)(26,76,225,106)(27,77,226,107)(28,78,227,108)(29,79,228,109)(30,80,229,110)(31,61,230,111)(32,62,231,112)(33,63,232,113)(34,64,233,114)(35,65,234,115)(36,66,235,116)(37,67,236,117)(38,68,237,118)(39,69,238,119)(40,70,239,120)(41,203,153,129)(42,204,154,130)(43,205,155,131)(44,206,156,132)(45,207,157,133)(46,208,158,134)(47,209,159,135)(48,210,160,136)(49,211,141,137)(50,212,142,138)(51,213,143,139)(52,214,144,140)(53,215,145,121)(54,216,146,122)(55,217,147,123)(56,218,148,124)(57,219,149,125)(58,220,150,126)(59,201,151,127)(60,202,152,128), (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), (1,132,235)(2,133,236)(3,134,237)(4,135,238)(5,136,239)(6,137,240)(7,138,221)(8,139,222)(9,140,223)(10,121,224)(11,122,225)(12,123,226)(13,124,227)(14,125,228)(15,126,229)(16,127,230)(17,128,231)(18,129,232)(19,130,233)(20,131,234)(21,90,211)(22,91,212)(23,92,213)(24,93,214)(25,94,215)(26,95,216)(27,96,217)(28,97,218)(29,98,219)(30,99,220)(31,100,201)(32,81,202)(33,82,203)(34,83,204)(35,84,205)(36,85,206)(37,86,207)(38,87,208)(39,88,209)(40,89,210)(41,113,171)(42,114,172)(43,115,173)(44,116,174)(45,117,175)(46,118,176)(47,119,177)(48,120,178)(49,101,179)(50,102,180)(51,103,161)(52,104,162)(53,105,163)(54,106,164)(55,107,165)(56,108,166)(57,109,167)(58,110,168)(59,111,169)(60,112,170)(61,188,151)(62,189,152)(63,190,153)(64,191,154)(65,192,155)(66,193,156)(67,194,157)(68,195,158)(69,196,159)(70,197,160)(71,198,141)(72,199,142)(73,200,143)(74,181,144)(75,182,145)(76,183,146)(77,184,147)(78,185,148)(79,186,149)(80,187,150), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,210)(31,211)(32,212)(33,213)(34,214)(35,215)(36,216)(37,217)(38,218)(39,219)(40,220)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,221)(129,222)(130,223)(131,224)(132,225)(133,226)(134,227)(135,228)(136,229)(137,230)(138,231)(139,232)(140,233)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)>;
G:=Group( (1,174,85,193)(2,175,86,194)(3,176,87,195)(4,177,88,196)(5,178,89,197)(6,179,90,198)(7,180,91,199)(8,161,92,200)(9,162,93,181)(10,163,94,182)(11,164,95,183)(12,165,96,184)(13,166,97,185)(14,167,98,186)(15,168,99,187)(16,169,100,188)(17,170,81,189)(18,171,82,190)(19,172,83,191)(20,173,84,192)(21,71,240,101)(22,72,221,102)(23,73,222,103)(24,74,223,104)(25,75,224,105)(26,76,225,106)(27,77,226,107)(28,78,227,108)(29,79,228,109)(30,80,229,110)(31,61,230,111)(32,62,231,112)(33,63,232,113)(34,64,233,114)(35,65,234,115)(36,66,235,116)(37,67,236,117)(38,68,237,118)(39,69,238,119)(40,70,239,120)(41,203,153,129)(42,204,154,130)(43,205,155,131)(44,206,156,132)(45,207,157,133)(46,208,158,134)(47,209,159,135)(48,210,160,136)(49,211,141,137)(50,212,142,138)(51,213,143,139)(52,214,144,140)(53,215,145,121)(54,216,146,122)(55,217,147,123)(56,218,148,124)(57,219,149,125)(58,220,150,126)(59,201,151,127)(60,202,152,128), (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), (1,132,235)(2,133,236)(3,134,237)(4,135,238)(5,136,239)(6,137,240)(7,138,221)(8,139,222)(9,140,223)(10,121,224)(11,122,225)(12,123,226)(13,124,227)(14,125,228)(15,126,229)(16,127,230)(17,128,231)(18,129,232)(19,130,233)(20,131,234)(21,90,211)(22,91,212)(23,92,213)(24,93,214)(25,94,215)(26,95,216)(27,96,217)(28,97,218)(29,98,219)(30,99,220)(31,100,201)(32,81,202)(33,82,203)(34,83,204)(35,84,205)(36,85,206)(37,86,207)(38,87,208)(39,88,209)(40,89,210)(41,113,171)(42,114,172)(43,115,173)(44,116,174)(45,117,175)(46,118,176)(47,119,177)(48,120,178)(49,101,179)(50,102,180)(51,103,161)(52,104,162)(53,105,163)(54,106,164)(55,107,165)(56,108,166)(57,109,167)(58,110,168)(59,111,169)(60,112,170)(61,188,151)(62,189,152)(63,190,153)(64,191,154)(65,192,155)(66,193,156)(67,194,157)(68,195,158)(69,196,159)(70,197,160)(71,198,141)(72,199,142)(73,200,143)(74,181,144)(75,182,145)(76,183,146)(77,184,147)(78,185,148)(79,186,149)(80,187,150), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,210)(31,211)(32,212)(33,213)(34,214)(35,215)(36,216)(37,217)(38,218)(39,219)(40,220)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,221)(129,222)(130,223)(131,224)(132,225)(133,226)(134,227)(135,228)(136,229)(137,230)(138,231)(139,232)(140,233)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200) );
G=PermutationGroup([[(1,174,85,193),(2,175,86,194),(3,176,87,195),(4,177,88,196),(5,178,89,197),(6,179,90,198),(7,180,91,199),(8,161,92,200),(9,162,93,181),(10,163,94,182),(11,164,95,183),(12,165,96,184),(13,166,97,185),(14,167,98,186),(15,168,99,187),(16,169,100,188),(17,170,81,189),(18,171,82,190),(19,172,83,191),(20,173,84,192),(21,71,240,101),(22,72,221,102),(23,73,222,103),(24,74,223,104),(25,75,224,105),(26,76,225,106),(27,77,226,107),(28,78,227,108),(29,79,228,109),(30,80,229,110),(31,61,230,111),(32,62,231,112),(33,63,232,113),(34,64,233,114),(35,65,234,115),(36,66,235,116),(37,67,236,117),(38,68,237,118),(39,69,238,119),(40,70,239,120),(41,203,153,129),(42,204,154,130),(43,205,155,131),(44,206,156,132),(45,207,157,133),(46,208,158,134),(47,209,159,135),(48,210,160,136),(49,211,141,137),(50,212,142,138),(51,213,143,139),(52,214,144,140),(53,215,145,121),(54,216,146,122),(55,217,147,123),(56,218,148,124),(57,219,149,125),(58,220,150,126),(59,201,151,127),(60,202,152,128)], [(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)], [(1,132,235),(2,133,236),(3,134,237),(4,135,238),(5,136,239),(6,137,240),(7,138,221),(8,139,222),(9,140,223),(10,121,224),(11,122,225),(12,123,226),(13,124,227),(14,125,228),(15,126,229),(16,127,230),(17,128,231),(18,129,232),(19,130,233),(20,131,234),(21,90,211),(22,91,212),(23,92,213),(24,93,214),(25,94,215),(26,95,216),(27,96,217),(28,97,218),(29,98,219),(30,99,220),(31,100,201),(32,81,202),(33,82,203),(34,83,204),(35,84,205),(36,85,206),(37,86,207),(38,87,208),(39,88,209),(40,89,210),(41,113,171),(42,114,172),(43,115,173),(44,116,174),(45,117,175),(46,118,176),(47,119,177),(48,120,178),(49,101,179),(50,102,180),(51,103,161),(52,104,162),(53,105,163),(54,106,164),(55,107,165),(56,108,166),(57,109,167),(58,110,168),(59,111,169),(60,112,170),(61,188,151),(62,189,152),(63,190,153),(64,191,154),(65,192,155),(66,193,156),(67,194,157),(68,195,158),(69,196,159),(70,197,160),(71,198,141),(72,199,142),(73,200,143),(74,181,144),(75,182,145),(76,183,146),(77,184,147),(78,185,148),(79,186,149),(80,187,150)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,201),(22,202),(23,203),(24,204),(25,205),(26,206),(27,207),(28,208),(29,209),(30,210),(31,211),(32,212),(33,213),(34,214),(35,215),(36,216),(37,217),(38,218),(39,219),(40,220),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,101),(60,102),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(121,234),(122,235),(123,236),(124,237),(125,238),(126,239),(127,240),(128,221),(129,222),(130,223),(131,224),(132,225),(133,226),(134,227),(135,228),(136,229),(137,230),(138,231),(139,232),(140,233),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178),(169,179),(170,180),(181,191),(182,192),(183,193),(184,194),(185,195),(186,196),(187,197),(188,198),(189,199),(190,200)]])
240 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20AV | 20AW | ··· | 20CR | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
240 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D6 | C4×S3 | C5×S3 | S3×C10 | S3×C20 |
| kernel | S3×C4×C20 | Dic3×C20 | C4×C60 | S3×C2×C20 | S3×C20 | S3×C42 | C4×Dic3 | C4×C12 | S3×C2×C4 | C4×S3 | C4×C20 | C2×C20 | C20 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 24 | 4 | 12 | 4 | 12 | 96 | 1 | 3 | 12 | 4 | 12 | 48 |
Matrix representation of S3×C4×C20 ►in GL3(𝔽61) generated by
| 50 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 60 | 0 | 0 |
| 0 | 53 | 0 |
| 0 | 0 | 53 |
| 1 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 1 | 60 |
| 60 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 60 | 0 |
G:=sub<GL(3,GF(61))| [50,0,0,0,1,0,0,0,1],[60,0,0,0,53,0,0,0,53],[1,0,0,0,0,1,0,60,60],[60,0,0,0,0,60,0,60,0] >;
S3×C4×C20 in GAP, Magma, Sage, TeX
S_3\times C_4\times C_{20} % in TeX
G:=Group("S3xC4xC20"); // GroupNames label
G:=SmallGroup(480,750);
// by ID
G=gap.SmallGroup(480,750);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,226,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^20=c^3=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