direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4⋊2D20, C60⋊16D4, C12⋊9D20, C20⋊1(C3×D4), C4⋊2(C3×D20), (C2×D20)⋊4C6, D10⋊2(C3×D4), (C6×D5)⋊14D4, C2.9(C6×D20), (C6×D20)⋊20C2, C6.78(C2×D20), C6.179(D4×D5), C10.25(C6×D4), D10⋊C4⋊7C6, C15⋊29(C4⋊D4), C30.338(C2×D4), (C2×C12).234D10, C30.264(C4○D4), (C2×C60).268C22, (C2×C30).353C23, C6.51(Q8⋊2D5), (C6×Dic5).243C22, (C2×C4×D5)⋊1C6, (C5×C4⋊C4)⋊6C6, C4⋊C4⋊3(C3×D5), C5⋊2(C3×C4⋊D4), C2.13(C3×D4×D5), (C3×C4⋊C4)⋊12D5, (D5×C2×C12)⋊11C2, (C15×C4⋊C4)⋊15C2, (C2×C4).42(C6×D5), C22.50(D5×C2×C6), (C2×C20).20(C2×C6), C10.34(C3×C4○D4), C2.6(C3×Q8⋊2D5), (C3×D10⋊C4)⋊19C2, (C22×D5).9(C2×C6), (D5×C2×C6).130C22, (C2×C10).36(C22×C6), (C2×Dic5).32(C2×C6), (C2×C6).349(C22×D5), SmallGroup(480,688)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊2D20
G = < a,b,c,d | a3=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 768 in 188 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], D5 [×4], C10 [×3], C12 [×2], C12 [×3], C2×C6, C2×C6 [×10], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×3], C3×D4 [×6], C22×C6 [×3], C3×D5 [×4], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4 [×3], C3×Dic5, C60 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×8], C2×C30, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20 [×2], C3×C4⋊D4, D5×C12 [×2], C3×D20 [×6], C6×Dic5, C2×C60, C2×C60 [×2], D5×C2×C6, D5×C2×C6 [×2], C4⋊2D20, C3×D10⋊C4 [×2], C15×C4⋊C4, D5×C2×C12, C6×D20, C6×D20 [×2], C3×C4⋊2D20
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×4], C23, D5, C2×C6 [×7], C2×D4 [×2], C4○D4, D10 [×3], C3×D4 [×4], C22×C6, C3×D5, C4⋊D4, D20 [×2], C22×D5, C6×D4 [×2], C3×C4○D4, C6×D5 [×3], C2×D20, D4×D5, Q8⋊2D5, C3×C4⋊D4, C3×D20 [×2], D5×C2×C6, C4⋊2D20, C6×D20, C3×D4×D5, C3×Q8⋊2D5, C3×C4⋊2D20
►On 240 points
Generators in S240
(1 190 130)(2 191 131)(3 192 132)(4 193 133)(5 194 134)(6 195 135)(7 196 136)(8 197 137)(9 198 138)(10 199 139)(11 200 140)(12 181 121)(13 182 122)(14 183 123)(15 184 124)(16 185 125)(17 186 126)(18 187 127)(19 188 128)(20 189 129)(21 98 64)(22 99 65)(23 100 66)(24 81 67)(25 82 68)(26 83 69)(27 84 70)(28 85 71)(29 86 72)(30 87 73)(31 88 74)(32 89 75)(33 90 76)(34 91 77)(35 92 78)(36 93 79)(37 94 80)(38 95 61)(39 96 62)(40 97 63)(41 164 201)(42 165 202)(43 166 203)(44 167 204)(45 168 205)(46 169 206)(47 170 207)(48 171 208)(49 172 209)(50 173 210)(51 174 211)(52 175 212)(53 176 213)(54 177 214)(55 178 215)(56 179 216)(57 180 217)(58 161 218)(59 162 219)(60 163 220)(101 157 236)(102 158 237)(103 159 238)(104 160 239)(105 141 240)(106 142 221)(107 143 222)(108 144 223)(109 145 224)(110 146 225)(111 147 226)(112 148 227)(113 149 228)(114 150 229)(115 151 230)(116 152 231)(117 153 232)(118 154 233)(119 155 234)(120 156 235)
(1 229 90 167)(2 168 91 230)(3 231 92 169)(4 170 93 232)(5 233 94 171)(6 172 95 234)(7 235 96 173)(8 174 97 236)(9 237 98 175)(10 176 99 238)(11 239 100 177)(12 178 81 240)(13 221 82 179)(14 180 83 222)(15 223 84 161)(16 162 85 224)(17 225 86 163)(18 164 87 226)(19 227 88 165)(20 166 89 228)(21 52 138 158)(22 159 139 53)(23 54 140 160)(24 141 121 55)(25 56 122 142)(26 143 123 57)(27 58 124 144)(28 145 125 59)(29 60 126 146)(30 147 127 41)(31 42 128 148)(32 149 129 43)(33 44 130 150)(34 151 131 45)(35 46 132 152)(36 153 133 47)(37 48 134 154)(38 155 135 49)(39 50 136 156)(40 157 137 51)(61 119 195 209)(62 210 196 120)(63 101 197 211)(64 212 198 102)(65 103 199 213)(66 214 200 104)(67 105 181 215)(68 216 182 106)(69 107 183 217)(70 218 184 108)(71 109 185 219)(72 220 186 110)(73 111 187 201)(74 202 188 112)(75 113 189 203)(76 204 190 114)(77 115 191 205)(78 206 192 116)(79 117 193 207)(80 208 194 118)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 140)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)(161 172)(162 171)(163 170)(164 169)(165 168)(166 167)(173 180)(174 179)(175 178)(176 177)(181 198)(182 197)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(199 200)(201 206)(202 205)(203 204)(207 220)(208 219)(209 218)(210 217)(211 216)(212 215)(213 214)(221 236)(222 235)(223 234)(224 233)(225 232)(226 231)(227 230)(228 229)(237 240)(238 239)
G:=sub<Sym(240)| (1,190,130)(2,191,131)(3,192,132)(4,193,133)(5,194,134)(6,195,135)(7,196,136)(8,197,137)(9,198,138)(10,199,139)(11,200,140)(12,181,121)(13,182,122)(14,183,123)(15,184,124)(16,185,125)(17,186,126)(18,187,127)(19,188,128)(20,189,129)(21,98,64)(22,99,65)(23,100,66)(24,81,67)(25,82,68)(26,83,69)(27,84,70)(28,85,71)(29,86,72)(30,87,73)(31,88,74)(32,89,75)(33,90,76)(34,91,77)(35,92,78)(36,93,79)(37,94,80)(38,95,61)(39,96,62)(40,97,63)(41,164,201)(42,165,202)(43,166,203)(44,167,204)(45,168,205)(46,169,206)(47,170,207)(48,171,208)(49,172,209)(50,173,210)(51,174,211)(52,175,212)(53,176,213)(54,177,214)(55,178,215)(56,179,216)(57,180,217)(58,161,218)(59,162,219)(60,163,220)(101,157,236)(102,158,237)(103,159,238)(104,160,239)(105,141,240)(106,142,221)(107,143,222)(108,144,223)(109,145,224)(110,146,225)(111,147,226)(112,148,227)(113,149,228)(114,150,229)(115,151,230)(116,152,231)(117,153,232)(118,154,233)(119,155,234)(120,156,235), (1,229,90,167)(2,168,91,230)(3,231,92,169)(4,170,93,232)(5,233,94,171)(6,172,95,234)(7,235,96,173)(8,174,97,236)(9,237,98,175)(10,176,99,238)(11,239,100,177)(12,178,81,240)(13,221,82,179)(14,180,83,222)(15,223,84,161)(16,162,85,224)(17,225,86,163)(18,164,87,226)(19,227,88,165)(20,166,89,228)(21,52,138,158)(22,159,139,53)(23,54,140,160)(24,141,121,55)(25,56,122,142)(26,143,123,57)(27,58,124,144)(28,145,125,59)(29,60,126,146)(30,147,127,41)(31,42,128,148)(32,149,129,43)(33,44,130,150)(34,151,131,45)(35,46,132,152)(36,153,133,47)(37,48,134,154)(38,155,135,49)(39,50,136,156)(40,157,137,51)(61,119,195,209)(62,210,196,120)(63,101,197,211)(64,212,198,102)(65,103,199,213)(66,214,200,104)(67,105,181,215)(68,216,182,106)(69,107,183,217)(70,218,184,108)(71,109,185,219)(72,220,186,110)(73,111,187,201)(74,202,188,112)(75,113,189,203)(76,204,190,114)(77,115,191,205)(78,206,192,116)(79,117,193,207)(80,208,194,118), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,140)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(199,200)(201,206)(202,205)(203,204)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)(221,236)(222,235)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229)(237,240)(238,239)>;
G:=Group( (1,190,130)(2,191,131)(3,192,132)(4,193,133)(5,194,134)(6,195,135)(7,196,136)(8,197,137)(9,198,138)(10,199,139)(11,200,140)(12,181,121)(13,182,122)(14,183,123)(15,184,124)(16,185,125)(17,186,126)(18,187,127)(19,188,128)(20,189,129)(21,98,64)(22,99,65)(23,100,66)(24,81,67)(25,82,68)(26,83,69)(27,84,70)(28,85,71)(29,86,72)(30,87,73)(31,88,74)(32,89,75)(33,90,76)(34,91,77)(35,92,78)(36,93,79)(37,94,80)(38,95,61)(39,96,62)(40,97,63)(41,164,201)(42,165,202)(43,166,203)(44,167,204)(45,168,205)(46,169,206)(47,170,207)(48,171,208)(49,172,209)(50,173,210)(51,174,211)(52,175,212)(53,176,213)(54,177,214)(55,178,215)(56,179,216)(57,180,217)(58,161,218)(59,162,219)(60,163,220)(101,157,236)(102,158,237)(103,159,238)(104,160,239)(105,141,240)(106,142,221)(107,143,222)(108,144,223)(109,145,224)(110,146,225)(111,147,226)(112,148,227)(113,149,228)(114,150,229)(115,151,230)(116,152,231)(117,153,232)(118,154,233)(119,155,234)(120,156,235), (1,229,90,167)(2,168,91,230)(3,231,92,169)(4,170,93,232)(5,233,94,171)(6,172,95,234)(7,235,96,173)(8,174,97,236)(9,237,98,175)(10,176,99,238)(11,239,100,177)(12,178,81,240)(13,221,82,179)(14,180,83,222)(15,223,84,161)(16,162,85,224)(17,225,86,163)(18,164,87,226)(19,227,88,165)(20,166,89,228)(21,52,138,158)(22,159,139,53)(23,54,140,160)(24,141,121,55)(25,56,122,142)(26,143,123,57)(27,58,124,144)(28,145,125,59)(29,60,126,146)(30,147,127,41)(31,42,128,148)(32,149,129,43)(33,44,130,150)(34,151,131,45)(35,46,132,152)(36,153,133,47)(37,48,134,154)(38,155,135,49)(39,50,136,156)(40,157,137,51)(61,119,195,209)(62,210,196,120)(63,101,197,211)(64,212,198,102)(65,103,199,213)(66,214,200,104)(67,105,181,215)(68,216,182,106)(69,107,183,217)(70,218,184,108)(71,109,185,219)(72,220,186,110)(73,111,187,201)(74,202,188,112)(75,113,189,203)(76,204,190,114)(77,115,191,205)(78,206,192,116)(79,117,193,207)(80,208,194,118), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(99,100)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,140)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(199,200)(201,206)(202,205)(203,204)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)(221,236)(222,235)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229)(237,240)(238,239) );
G=PermutationGroup([(1,190,130),(2,191,131),(3,192,132),(4,193,133),(5,194,134),(6,195,135),(7,196,136),(8,197,137),(9,198,138),(10,199,139),(11,200,140),(12,181,121),(13,182,122),(14,183,123),(15,184,124),(16,185,125),(17,186,126),(18,187,127),(19,188,128),(20,189,129),(21,98,64),(22,99,65),(23,100,66),(24,81,67),(25,82,68),(26,83,69),(27,84,70),(28,85,71),(29,86,72),(30,87,73),(31,88,74),(32,89,75),(33,90,76),(34,91,77),(35,92,78),(36,93,79),(37,94,80),(38,95,61),(39,96,62),(40,97,63),(41,164,201),(42,165,202),(43,166,203),(44,167,204),(45,168,205),(46,169,206),(47,170,207),(48,171,208),(49,172,209),(50,173,210),(51,174,211),(52,175,212),(53,176,213),(54,177,214),(55,178,215),(56,179,216),(57,180,217),(58,161,218),(59,162,219),(60,163,220),(101,157,236),(102,158,237),(103,159,238),(104,160,239),(105,141,240),(106,142,221),(107,143,222),(108,144,223),(109,145,224),(110,146,225),(111,147,226),(112,148,227),(113,149,228),(114,150,229),(115,151,230),(116,152,231),(117,153,232),(118,154,233),(119,155,234),(120,156,235)], [(1,229,90,167),(2,168,91,230),(3,231,92,169),(4,170,93,232),(5,233,94,171),(6,172,95,234),(7,235,96,173),(8,174,97,236),(9,237,98,175),(10,176,99,238),(11,239,100,177),(12,178,81,240),(13,221,82,179),(14,180,83,222),(15,223,84,161),(16,162,85,224),(17,225,86,163),(18,164,87,226),(19,227,88,165),(20,166,89,228),(21,52,138,158),(22,159,139,53),(23,54,140,160),(24,141,121,55),(25,56,122,142),(26,143,123,57),(27,58,124,144),(28,145,125,59),(29,60,126,146),(30,147,127,41),(31,42,128,148),(32,149,129,43),(33,44,130,150),(34,151,131,45),(35,46,132,152),(36,153,133,47),(37,48,134,154),(38,155,135,49),(39,50,136,156),(40,157,137,51),(61,119,195,209),(62,210,196,120),(63,101,197,211),(64,212,198,102),(65,103,199,213),(66,214,200,104),(67,105,181,215),(68,216,182,106),(69,107,183,217),(70,218,184,108),(71,109,185,219),(72,220,186,110),(73,111,187,201),(74,202,188,112),(75,113,189,203),(76,204,190,114),(77,115,191,205),(78,206,192,116),(79,117,193,207),(80,208,194,118)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(99,100),(101,106),(102,105),(103,104),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(139,140),(141,158),(142,157),(143,156),(144,155),(145,154),(146,153),(147,152),(148,151),(149,150),(159,160),(161,172),(162,171),(163,170),(164,169),(165,168),(166,167),(173,180),(174,179),(175,178),(176,177),(181,198),(182,197),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(199,200),(201,206),(202,205),(203,204),(207,220),(208,219),(209,218),(210,217),(211,216),(212,215),(213,214),(221,236),(222,235),(223,234),(224,233),(225,232),(226,231),(227,230),(228,229),(237,240),(238,239)])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | C4○D4 | D10 | C3×D4 | C3×D4 | C3×D5 | D20 | C3×C4○D4 | C6×D5 | C3×D20 | D4×D5 | Q8⋊2D5 | C3×D4×D5 | C3×Q8⋊2D5 |
| kernel | C3×C4⋊2D20 | C3×D10⋊C4 | C15×C4⋊C4 | D5×C2×C12 | C6×D20 | C4⋊2D20 | D10⋊C4 | C5×C4⋊C4 | C2×C4×D5 | C2×D20 | C60 | C6×D5 | C3×C4⋊C4 | C30 | C2×C12 | C20 | D10 | C4⋊C4 | C12 | C10 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 2 | 2 | 6 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 4 | 8 | 4 | 12 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C4⋊2D20 ►in GL6(𝔽61)
| 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 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 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 21 | 20 | 0 | 0 | 0 | 0 |
| 45 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 21 | 20 | 0 | 0 | 0 | 0 |
| 39 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,11],[21,45,0,0,0,0,20,40,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[21,39,0,0,0,0,20,40,0,0,0,0,0,0,43,18,0,0,0,0,60,18,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C3×C4⋊2D20 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes_2D_{20} % in TeX
G:=Group("C3xC4:2D20"); // GroupNames label
G:=SmallGroup(480,688);
// by ID
G=gap.SmallGroup(480,688);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,142,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations