metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (S3×C20)⋊5C4, C4⋊Dic5⋊16S3, (C4×S3)⋊1Dic5, C60⋊5C4⋊23C2, C20.104(C4×S3), C60.103(C2×C4), C30.8(C4○D4), (C2×C20).290D6, D6⋊Dic5.1C2, D6.4(C2×Dic5), C4.15(S3×Dic5), C15⋊6(C42⋊C2), (Dic3×Dic5)⋊3C2, C6.44(C4○D20), (C2×C12).108D10, (C2×C30).28C23, (C2×Dic5).88D6, C12.12(C2×Dic5), C2.2(D60⋊C2), C2.1(D20⋊5S3), C6.7(C22×Dic5), (C2×C60).109C22, C30.115(C22×C4), C10.4(Q8⋊3S3), (C22×S3).63D10, C10.17(D4⋊2S3), (C2×Dic3).173D10, Dic3.10(C2×Dic5), C3⋊2(C23.21D10), (C6×Dic5).13C22, (C2×Dic15).35C22, (C10×Dic3).156C22, (S3×C2×C4).2D5, (S3×C2×C20).2C2, C5⋊6(C4⋊C4⋊7S3), C2.9(C2×S3×Dic5), C10.115(S3×C2×C4), (C3×C4⋊Dic5)⋊2C2, C22.28(C2×S3×D5), (C2×C4).100(S3×D5), (S3×C10).36(C2×C4), (S3×C2×C10).75C22, (C2×C6).40(C22×D5), (C2×C10).40(C22×S3), (C5×Dic3).44(C2×C4), SmallGroup(480,414)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (S3×C20)⋊5C4
G = < a,b,c,d | a20=b3=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=a10c >
Subgroups: 556 in 152 conjugacy classes, 68 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C42⋊C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10, C2×C30, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C23.D5, C22×C20, C4⋊C4⋊7S3, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C23.21D10, Dic3×Dic5, D6⋊Dic5, C3×C4⋊Dic5, C60⋊5C4, S3×C2×C20, (S3×C20)⋊5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, Dic5, D10, C4×S3, C22×S3, C42⋊C2, C2×Dic5, C22×D5, S3×C2×C4, D4⋊2S3, Q8⋊3S3, S3×D5, C4○D20, C22×Dic5, C4⋊C4⋊7S3, S3×Dic5, C2×S3×D5, C23.21D10, D20⋊5S3, D60⋊C2, C2×S3×Dic5, (S3×C20)⋊5C4
►On 240 points
Generators in S240
(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 59 232)(2 60 233)(3 41 234)(4 42 235)(5 43 236)(6 44 237)(7 45 238)(8 46 239)(9 47 240)(10 48 221)(11 49 222)(12 50 223)(13 51 224)(14 52 225)(15 53 226)(16 54 227)(17 55 228)(18 56 229)(19 57 230)(20 58 231)(21 179 148)(22 180 149)(23 161 150)(24 162 151)(25 163 152)(26 164 153)(27 165 154)(28 166 155)(29 167 156)(30 168 157)(31 169 158)(32 170 159)(33 171 160)(34 172 141)(35 173 142)(36 174 143)(37 175 144)(38 176 145)(39 177 146)(40 178 147)(61 97 212)(62 98 213)(63 99 214)(64 100 215)(65 81 216)(66 82 217)(67 83 218)(68 84 219)(69 85 220)(70 86 201)(71 87 202)(72 88 203)(73 89 204)(74 90 205)(75 91 206)(76 92 207)(77 93 208)(78 94 209)(79 95 210)(80 96 211)(101 183 136)(102 184 137)(103 185 138)(104 186 139)(105 187 140)(106 188 121)(107 189 122)(108 190 123)(109 191 124)(110 192 125)(111 193 126)(112 194 127)(113 195 128)(114 196 129)(115 197 130)(116 198 131)(117 199 132)(118 200 133)(119 181 134)(120 182 135)
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 121)(19 122)(20 123)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 61)(37 62)(38 63)(39 64)(40 65)(41 193)(42 194)(43 195)(44 196)(45 197)(46 198)(47 199)(48 200)(49 181)(50 182)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 224)(102 225)(103 226)(104 227)(105 228)(106 229)(107 230)(108 231)(109 232)(110 233)(111 234)(112 235)(113 236)(114 237)(115 238)(116 239)(117 240)(118 221)(119 222)(120 223)(161 219)(162 220)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 208)(171 209)(172 210)(173 211)(174 212)(175 213)(176 214)(177 215)(178 216)(179 217)(180 218)
(1 176 134 214)(2 175 135 213)(3 174 136 212)(4 173 137 211)(5 172 138 210)(6 171 139 209)(7 170 140 208)(8 169 121 207)(9 168 122 206)(10 167 123 205)(11 166 124 204)(12 165 125 203)(13 164 126 202)(14 163 127 201)(15 162 128 220)(16 161 129 219)(17 180 130 218)(18 179 131 217)(19 178 132 216)(20 177 133 215)(21 198 82 229)(22 197 83 228)(23 196 84 227)(24 195 85 226)(25 194 86 225)(26 193 87 224)(27 192 88 223)(28 191 89 222)(29 190 90 221)(30 189 91 240)(31 188 92 239)(32 187 93 238)(33 186 94 237)(34 185 95 236)(35 184 96 235)(36 183 97 234)(37 182 98 233)(38 181 99 232)(39 200 100 231)(40 199 81 230)(41 143 101 61)(42 142 102 80)(43 141 103 79)(44 160 104 78)(45 159 105 77)(46 158 106 76)(47 157 107 75)(48 156 108 74)(49 155 109 73)(50 154 110 72)(51 153 111 71)(52 152 112 70)(53 151 113 69)(54 150 114 68)(55 149 115 67)(56 148 116 66)(57 147 117 65)(58 146 118 64)(59 145 119 63)(60 144 120 62)
G:=sub<Sym(240)| (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), (1,59,232)(2,60,233)(3,41,234)(4,42,235)(5,43,236)(6,44,237)(7,45,238)(8,46,239)(9,47,240)(10,48,221)(11,49,222)(12,50,223)(13,51,224)(14,52,225)(15,53,226)(16,54,227)(17,55,228)(18,56,229)(19,57,230)(20,58,231)(21,179,148)(22,180,149)(23,161,150)(24,162,151)(25,163,152)(26,164,153)(27,165,154)(28,166,155)(29,167,156)(30,168,157)(31,169,158)(32,170,159)(33,171,160)(34,172,141)(35,173,142)(36,174,143)(37,175,144)(38,176,145)(39,177,146)(40,178,147)(61,97,212)(62,98,213)(63,99,214)(64,100,215)(65,81,216)(66,82,217)(67,83,218)(68,84,219)(69,85,220)(70,86,201)(71,87,202)(72,88,203)(73,89,204)(74,90,205)(75,91,206)(76,92,207)(77,93,208)(78,94,209)(79,95,210)(80,96,211)(101,183,136)(102,184,137)(103,185,138)(104,186,139)(105,187,140)(106,188,121)(107,189,122)(108,190,123)(109,191,124)(110,192,125)(111,193,126)(112,194,127)(113,195,128)(114,196,129)(115,197,130)(116,198,131)(117,199,132)(118,200,133)(119,181,134)(120,182,135), (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,121)(19,122)(20,123)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,61)(37,62)(38,63)(39,64)(40,65)(41,193)(42,194)(43,195)(44,196)(45,197)(46,198)(47,199)(48,200)(49,181)(50,182)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,224)(102,225)(103,226)(104,227)(105,228)(106,229)(107,230)(108,231)(109,232)(110,233)(111,234)(112,235)(113,236)(114,237)(115,238)(116,239)(117,240)(118,221)(119,222)(120,223)(161,219)(162,220)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218), (1,176,134,214)(2,175,135,213)(3,174,136,212)(4,173,137,211)(5,172,138,210)(6,171,139,209)(7,170,140,208)(8,169,121,207)(9,168,122,206)(10,167,123,205)(11,166,124,204)(12,165,125,203)(13,164,126,202)(14,163,127,201)(15,162,128,220)(16,161,129,219)(17,180,130,218)(18,179,131,217)(19,178,132,216)(20,177,133,215)(21,198,82,229)(22,197,83,228)(23,196,84,227)(24,195,85,226)(25,194,86,225)(26,193,87,224)(27,192,88,223)(28,191,89,222)(29,190,90,221)(30,189,91,240)(31,188,92,239)(32,187,93,238)(33,186,94,237)(34,185,95,236)(35,184,96,235)(36,183,97,234)(37,182,98,233)(38,181,99,232)(39,200,100,231)(40,199,81,230)(41,143,101,61)(42,142,102,80)(43,141,103,79)(44,160,104,78)(45,159,105,77)(46,158,106,76)(47,157,107,75)(48,156,108,74)(49,155,109,73)(50,154,110,72)(51,153,111,71)(52,152,112,70)(53,151,113,69)(54,150,114,68)(55,149,115,67)(56,148,116,66)(57,147,117,65)(58,146,118,64)(59,145,119,63)(60,144,120,62)>;
G:=Group( (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), (1,59,232)(2,60,233)(3,41,234)(4,42,235)(5,43,236)(6,44,237)(7,45,238)(8,46,239)(9,47,240)(10,48,221)(11,49,222)(12,50,223)(13,51,224)(14,52,225)(15,53,226)(16,54,227)(17,55,228)(18,56,229)(19,57,230)(20,58,231)(21,179,148)(22,180,149)(23,161,150)(24,162,151)(25,163,152)(26,164,153)(27,165,154)(28,166,155)(29,167,156)(30,168,157)(31,169,158)(32,170,159)(33,171,160)(34,172,141)(35,173,142)(36,174,143)(37,175,144)(38,176,145)(39,177,146)(40,178,147)(61,97,212)(62,98,213)(63,99,214)(64,100,215)(65,81,216)(66,82,217)(67,83,218)(68,84,219)(69,85,220)(70,86,201)(71,87,202)(72,88,203)(73,89,204)(74,90,205)(75,91,206)(76,92,207)(77,93,208)(78,94,209)(79,95,210)(80,96,211)(101,183,136)(102,184,137)(103,185,138)(104,186,139)(105,187,140)(106,188,121)(107,189,122)(108,190,123)(109,191,124)(110,192,125)(111,193,126)(112,194,127)(113,195,128)(114,196,129)(115,197,130)(116,198,131)(117,199,132)(118,200,133)(119,181,134)(120,182,135), (1,124)(2,125)(3,126)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,121)(19,122)(20,123)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,61)(37,62)(38,63)(39,64)(40,65)(41,193)(42,194)(43,195)(44,196)(45,197)(46,198)(47,199)(48,200)(49,181)(50,182)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,189)(58,190)(59,191)(60,192)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,224)(102,225)(103,226)(104,227)(105,228)(106,229)(107,230)(108,231)(109,232)(110,233)(111,234)(112,235)(113,236)(114,237)(115,238)(116,239)(117,240)(118,221)(119,222)(120,223)(161,219)(162,220)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218), (1,176,134,214)(2,175,135,213)(3,174,136,212)(4,173,137,211)(5,172,138,210)(6,171,139,209)(7,170,140,208)(8,169,121,207)(9,168,122,206)(10,167,123,205)(11,166,124,204)(12,165,125,203)(13,164,126,202)(14,163,127,201)(15,162,128,220)(16,161,129,219)(17,180,130,218)(18,179,131,217)(19,178,132,216)(20,177,133,215)(21,198,82,229)(22,197,83,228)(23,196,84,227)(24,195,85,226)(25,194,86,225)(26,193,87,224)(27,192,88,223)(28,191,89,222)(29,190,90,221)(30,189,91,240)(31,188,92,239)(32,187,93,238)(33,186,94,237)(34,185,95,236)(35,184,96,235)(36,183,97,234)(37,182,98,233)(38,181,99,232)(39,200,100,231)(40,199,81,230)(41,143,101,61)(42,142,102,80)(43,141,103,79)(44,160,104,78)(45,159,105,77)(46,158,106,76)(47,157,107,75)(48,156,108,74)(49,155,109,73)(50,154,110,72)(51,153,111,71)(52,152,112,70)(53,151,113,69)(54,150,114,68)(55,149,115,67)(56,148,116,66)(57,147,117,65)(58,146,118,64)(59,145,119,63)(60,144,120,62) );
G=PermutationGroup([[(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)], [(1,59,232),(2,60,233),(3,41,234),(4,42,235),(5,43,236),(6,44,237),(7,45,238),(8,46,239),(9,47,240),(10,48,221),(11,49,222),(12,50,223),(13,51,224),(14,52,225),(15,53,226),(16,54,227),(17,55,228),(18,56,229),(19,57,230),(20,58,231),(21,179,148),(22,180,149),(23,161,150),(24,162,151),(25,163,152),(26,164,153),(27,165,154),(28,166,155),(29,167,156),(30,168,157),(31,169,158),(32,170,159),(33,171,160),(34,172,141),(35,173,142),(36,174,143),(37,175,144),(38,176,145),(39,177,146),(40,178,147),(61,97,212),(62,98,213),(63,99,214),(64,100,215),(65,81,216),(66,82,217),(67,83,218),(68,84,219),(69,85,220),(70,86,201),(71,87,202),(72,88,203),(73,89,204),(74,90,205),(75,91,206),(76,92,207),(77,93,208),(78,94,209),(79,95,210),(80,96,211),(101,183,136),(102,184,137),(103,185,138),(104,186,139),(105,187,140),(106,188,121),(107,189,122),(108,190,123),(109,191,124),(110,192,125),(111,193,126),(112,194,127),(113,195,128),(114,196,129),(115,197,130),(116,198,131),(117,199,132),(118,200,133),(119,181,134),(120,182,135)], [(1,124),(2,125),(3,126),(4,127),(5,128),(6,129),(7,130),(8,131),(9,132),(10,133),(11,134),(12,135),(13,136),(14,137),(15,138),(16,139),(17,140),(18,121),(19,122),(20,123),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,61),(37,62),(38,63),(39,64),(40,65),(41,193),(42,194),(43,195),(44,196),(45,197),(46,198),(47,199),(48,200),(49,181),(50,182),(51,183),(52,184),(53,185),(54,186),(55,187),(56,188),(57,189),(58,190),(59,191),(60,192),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,224),(102,225),(103,226),(104,227),(105,228),(106,229),(107,230),(108,231),(109,232),(110,233),(111,234),(112,235),(113,236),(114,237),(115,238),(116,239),(117,240),(118,221),(119,222),(120,223),(161,219),(162,220),(163,201),(164,202),(165,203),(166,204),(167,205),(168,206),(169,207),(170,208),(171,209),(172,210),(173,211),(174,212),(175,213),(176,214),(177,215),(178,216),(179,217),(180,218)], [(1,176,134,214),(2,175,135,213),(3,174,136,212),(4,173,137,211),(5,172,138,210),(6,171,139,209),(7,170,140,208),(8,169,121,207),(9,168,122,206),(10,167,123,205),(11,166,124,204),(12,165,125,203),(13,164,126,202),(14,163,127,201),(15,162,128,220),(16,161,129,219),(17,180,130,218),(18,179,131,217),(19,178,132,216),(20,177,133,215),(21,198,82,229),(22,197,83,228),(23,196,84,227),(24,195,85,226),(25,194,86,225),(26,193,87,224),(27,192,88,223),(28,191,89,222),(29,190,90,221),(30,189,91,240),(31,188,92,239),(32,187,93,238),(33,186,94,237),(34,185,95,236),(35,184,96,235),(36,183,97,234),(37,182,98,233),(38,181,99,232),(39,200,100,231),(40,199,81,230),(41,143,101,61),(42,142,102,80),(43,141,103,79),(44,160,104,78),(45,159,105,77),(46,158,106,76),(47,157,107,75),(48,156,108,74),(49,155,109,73),(50,154,110,72),(51,153,111,71),(52,152,112,70),(53,151,113,69),(54,150,114,68),(55,149,115,67),(56,148,116,66),(57,147,117,65),(58,146,118,64),(59,145,119,63),(60,144,120,62)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | C4○D4 | Dic5 | D10 | D10 | D10 | C4×S3 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | S3×Dic5 | C2×S3×D5 | D20⋊5S3 | D60⋊C2 |
| kernel | (S3×C20)⋊5C4 | Dic3×Dic5 | D6⋊Dic5 | C3×C4⋊Dic5 | C60⋊5C4 | S3×C2×C20 | S3×C20 | C4⋊Dic5 | S3×C2×C4 | C2×Dic5 | C2×C20 | C30 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C20 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 8 | 2 | 2 | 2 | 4 | 16 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of (S3×C20)⋊5C4 ►in GL6(𝔽61)
| 38 | 0 | 0 | 0 | 0 | 0 |
| 36 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 6 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 17 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 7 | 8 | 0 | 0 | 0 | 0 |
| 55 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 21 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [38,36,0,0,0,0,0,53,0,0,0,0,0,0,34,6,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,17,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[7,55,0,0,0,0,8,54,0,0,0,0,0,0,51,21,0,0,0,0,1,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
(S3×C20)⋊5C4 in GAP, Magma, Sage, TeX
(S_3\times C_{20})\rtimes_5C_4 % in TeX
G:=Group("(S3xC20):5C4"); // GroupNames label
G:=SmallGroup(480,414);
// by ID
G=gap.SmallGroup(480,414);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^3=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^10*c>;
// generators/relations