direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4⋊D20, C60⋊16D4, C12⋊9D20, C20⋊1(C3×D4), C4⋊2(C3×D20), D10⋊2(C3×D4), (C6×D5)⋊14D4, (C2×D20)⋊4C6, C2.9(C6×D20), (C6×D20)⋊20C2, C6.78(C2×D20), C10.25(C6×D4), C6.179(D4×D5), D10⋊C4⋊7C6, C15⋊29(C4⋊D4), C30.338(C2×D4), (C2×C12).234D10, C30.264(C4○D4), (C2×C30).353C23, (C2×C60).268C22, 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⋊D20
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, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, C3×C4⋊D4, D5×C12, C3×D20, C6×Dic5, C2×C60, C2×C60, D5×C2×C6, D5×C2×C6, C4⋊D20, C3×D10⋊C4, C15×C4⋊C4, D5×C2×C12, C6×D20, C6×D20, C3×C4⋊D20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C4⋊D4, D20, C22×D5, C6×D4, C3×C4○D4, C6×D5, C2×D20, D4×D5, Q8⋊2D5, C3×C4⋊D4, C3×D20, D5×C2×C6, C4⋊D20, C6×D20, C3×D4×D5, C3×Q8⋊2D5, C3×C4⋊D20
►On 240 points
Generators in S240
(1 46 150)(2 47 151)(3 48 152)(4 49 153)(5 50 154)(6 51 155)(7 52 156)(8 53 157)(9 54 158)(10 55 159)(11 56 160)(12 57 141)(13 58 142)(14 59 143)(15 60 144)(16 41 145)(17 42 146)(18 43 147)(19 44 148)(20 45 149)(21 120 87)(22 101 88)(23 102 89)(24 103 90)(25 104 91)(26 105 92)(27 106 93)(28 107 94)(29 108 95)(30 109 96)(31 110 97)(32 111 98)(33 112 99)(34 113 100)(35 114 81)(36 115 82)(37 116 83)(38 117 84)(39 118 85)(40 119 86)(61 132 207)(62 133 208)(63 134 209)(64 135 210)(65 136 211)(66 137 212)(67 138 213)(68 139 214)(69 140 215)(70 121 216)(71 122 217)(72 123 218)(73 124 219)(74 125 220)(75 126 201)(76 127 202)(77 128 203)(78 129 204)(79 130 205)(80 131 206)(161 230 188)(162 231 189)(163 232 190)(164 233 191)(165 234 192)(166 235 193)(167 236 194)(168 237 195)(169 238 196)(170 239 197)(171 240 198)(172 221 199)(173 222 200)(174 223 181)(175 224 182)(176 225 183)(177 226 184)(178 227 185)(179 228 186)(180 229 187)
(1 67 104 230)(2 231 105 68)(3 69 106 232)(4 233 107 70)(5 71 108 234)(6 235 109 72)(7 73 110 236)(8 237 111 74)(9 75 112 238)(10 239 113 76)(11 77 114 240)(12 221 115 78)(13 79 116 222)(14 223 117 80)(15 61 118 224)(16 225 119 62)(17 63 120 226)(18 227 101 64)(19 65 102 228)(20 229 103 66)(21 177 146 209)(22 210 147 178)(23 179 148 211)(24 212 149 180)(25 161 150 213)(26 214 151 162)(27 163 152 215)(28 216 153 164)(29 165 154 217)(30 218 155 166)(31 167 156 219)(32 220 157 168)(33 169 158 201)(34 202 159 170)(35 171 160 203)(36 204 141 172)(37 173 142 205)(38 206 143 174)(39 175 144 207)(40 208 145 176)(41 183 86 133)(42 134 87 184)(43 185 88 135)(44 136 89 186)(45 187 90 137)(46 138 91 188)(47 189 92 139)(48 140 93 190)(49 191 94 121)(50 122 95 192)(51 193 96 123)(52 124 97 194)(53 195 98 125)(54 126 99 196)(55 197 100 127)(56 128 81 198)(57 199 82 129)(58 130 83 200)(59 181 84 131)(60 132 85 182)
(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 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 140)(136 139)(137 138)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)(161 180)(162 179)(163 178)(164 177)(165 176)(166 175)(167 174)(168 173)(169 172)(170 171)(181 194)(182 193)(183 192)(184 191)(185 190)(186 189)(187 188)(195 200)(196 199)(197 198)(201 204)(202 203)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)(221 238)(222 237)(223 236)(224 235)(225 234)(226 233)(227 232)(228 231)(229 230)(239 240)
G:=sub<Sym(240)| (1,46,150)(2,47,151)(3,48,152)(4,49,153)(5,50,154)(6,51,155)(7,52,156)(8,53,157)(9,54,158)(10,55,159)(11,56,160)(12,57,141)(13,58,142)(14,59,143)(15,60,144)(16,41,145)(17,42,146)(18,43,147)(19,44,148)(20,45,149)(21,120,87)(22,101,88)(23,102,89)(24,103,90)(25,104,91)(26,105,92)(27,106,93)(28,107,94)(29,108,95)(30,109,96)(31,110,97)(32,111,98)(33,112,99)(34,113,100)(35,114,81)(36,115,82)(37,116,83)(38,117,84)(39,118,85)(40,119,86)(61,132,207)(62,133,208)(63,134,209)(64,135,210)(65,136,211)(66,137,212)(67,138,213)(68,139,214)(69,140,215)(70,121,216)(71,122,217)(72,123,218)(73,124,219)(74,125,220)(75,126,201)(76,127,202)(77,128,203)(78,129,204)(79,130,205)(80,131,206)(161,230,188)(162,231,189)(163,232,190)(164,233,191)(165,234,192)(166,235,193)(167,236,194)(168,237,195)(169,238,196)(170,239,197)(171,240,198)(172,221,199)(173,222,200)(174,223,181)(175,224,182)(176,225,183)(177,226,184)(178,227,185)(179,228,186)(180,229,187), (1,67,104,230)(2,231,105,68)(3,69,106,232)(4,233,107,70)(5,71,108,234)(6,235,109,72)(7,73,110,236)(8,237,111,74)(9,75,112,238)(10,239,113,76)(11,77,114,240)(12,221,115,78)(13,79,116,222)(14,223,117,80)(15,61,118,224)(16,225,119,62)(17,63,120,226)(18,227,101,64)(19,65,102,228)(20,229,103,66)(21,177,146,209)(22,210,147,178)(23,179,148,211)(24,212,149,180)(25,161,150,213)(26,214,151,162)(27,163,152,215)(28,216,153,164)(29,165,154,217)(30,218,155,166)(31,167,156,219)(32,220,157,168)(33,169,158,201)(34,202,159,170)(35,171,160,203)(36,204,141,172)(37,173,142,205)(38,206,143,174)(39,175,144,207)(40,208,145,176)(41,183,86,133)(42,134,87,184)(43,185,88,135)(44,136,89,186)(45,187,90,137)(46,138,91,188)(47,189,92,139)(48,140,93,190)(49,191,94,121)(50,122,95,192)(51,193,96,123)(52,124,97,194)(53,195,98,125)(54,126,99,196)(55,197,100,127)(56,128,81,198)(57,199,82,129)(58,130,83,200)(59,181,84,131)(60,132,85,182), (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)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,180)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)(168,173)(169,172)(170,171)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,200)(196,199)(197,198)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,238)(222,237)(223,236)(224,235)(225,234)(226,233)(227,232)(228,231)(229,230)(239,240)>;
G:=Group( (1,46,150)(2,47,151)(3,48,152)(4,49,153)(5,50,154)(6,51,155)(7,52,156)(8,53,157)(9,54,158)(10,55,159)(11,56,160)(12,57,141)(13,58,142)(14,59,143)(15,60,144)(16,41,145)(17,42,146)(18,43,147)(19,44,148)(20,45,149)(21,120,87)(22,101,88)(23,102,89)(24,103,90)(25,104,91)(26,105,92)(27,106,93)(28,107,94)(29,108,95)(30,109,96)(31,110,97)(32,111,98)(33,112,99)(34,113,100)(35,114,81)(36,115,82)(37,116,83)(38,117,84)(39,118,85)(40,119,86)(61,132,207)(62,133,208)(63,134,209)(64,135,210)(65,136,211)(66,137,212)(67,138,213)(68,139,214)(69,140,215)(70,121,216)(71,122,217)(72,123,218)(73,124,219)(74,125,220)(75,126,201)(76,127,202)(77,128,203)(78,129,204)(79,130,205)(80,131,206)(161,230,188)(162,231,189)(163,232,190)(164,233,191)(165,234,192)(166,235,193)(167,236,194)(168,237,195)(169,238,196)(170,239,197)(171,240,198)(172,221,199)(173,222,200)(174,223,181)(175,224,182)(176,225,183)(177,226,184)(178,227,185)(179,228,186)(180,229,187), (1,67,104,230)(2,231,105,68)(3,69,106,232)(4,233,107,70)(5,71,108,234)(6,235,109,72)(7,73,110,236)(8,237,111,74)(9,75,112,238)(10,239,113,76)(11,77,114,240)(12,221,115,78)(13,79,116,222)(14,223,117,80)(15,61,118,224)(16,225,119,62)(17,63,120,226)(18,227,101,64)(19,65,102,228)(20,229,103,66)(21,177,146,209)(22,210,147,178)(23,179,148,211)(24,212,149,180)(25,161,150,213)(26,214,151,162)(27,163,152,215)(28,216,153,164)(29,165,154,217)(30,218,155,166)(31,167,156,219)(32,220,157,168)(33,169,158,201)(34,202,159,170)(35,171,160,203)(36,204,141,172)(37,173,142,205)(38,206,143,174)(39,175,144,207)(40,208,145,176)(41,183,86,133)(42,134,87,184)(43,185,88,135)(44,136,89,186)(45,187,90,137)(46,138,91,188)(47,189,92,139)(48,140,93,190)(49,191,94,121)(50,122,95,192)(51,193,96,123)(52,124,97,194)(53,195,98,125)(54,126,99,196)(55,197,100,127)(56,128,81,198)(57,199,82,129)(58,130,83,200)(59,181,84,131)(60,132,85,182), (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)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,140)(136,139)(137,138)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,180)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)(168,173)(169,172)(170,171)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,200)(196,199)(197,198)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,238)(222,237)(223,236)(224,235)(225,234)(226,233)(227,232)(228,231)(229,230)(239,240) );
G=PermutationGroup([[(1,46,150),(2,47,151),(3,48,152),(4,49,153),(5,50,154),(6,51,155),(7,52,156),(8,53,157),(9,54,158),(10,55,159),(11,56,160),(12,57,141),(13,58,142),(14,59,143),(15,60,144),(16,41,145),(17,42,146),(18,43,147),(19,44,148),(20,45,149),(21,120,87),(22,101,88),(23,102,89),(24,103,90),(25,104,91),(26,105,92),(27,106,93),(28,107,94),(29,108,95),(30,109,96),(31,110,97),(32,111,98),(33,112,99),(34,113,100),(35,114,81),(36,115,82),(37,116,83),(38,117,84),(39,118,85),(40,119,86),(61,132,207),(62,133,208),(63,134,209),(64,135,210),(65,136,211),(66,137,212),(67,138,213),(68,139,214),(69,140,215),(70,121,216),(71,122,217),(72,123,218),(73,124,219),(74,125,220),(75,126,201),(76,127,202),(77,128,203),(78,129,204),(79,130,205),(80,131,206),(161,230,188),(162,231,189),(163,232,190),(164,233,191),(165,234,192),(166,235,193),(167,236,194),(168,237,195),(169,238,196),(170,239,197),(171,240,198),(172,221,199),(173,222,200),(174,223,181),(175,224,182),(176,225,183),(177,226,184),(178,227,185),(179,228,186),(180,229,187)], [(1,67,104,230),(2,231,105,68),(3,69,106,232),(4,233,107,70),(5,71,108,234),(6,235,109,72),(7,73,110,236),(8,237,111,74),(9,75,112,238),(10,239,113,76),(11,77,114,240),(12,221,115,78),(13,79,116,222),(14,223,117,80),(15,61,118,224),(16,225,119,62),(17,63,120,226),(18,227,101,64),(19,65,102,228),(20,229,103,66),(21,177,146,209),(22,210,147,178),(23,179,148,211),(24,212,149,180),(25,161,150,213),(26,214,151,162),(27,163,152,215),(28,216,153,164),(29,165,154,217),(30,218,155,166),(31,167,156,219),(32,220,157,168),(33,169,158,201),(34,202,159,170),(35,171,160,203),(36,204,141,172),(37,173,142,205),(38,206,143,174),(39,175,144,207),(40,208,145,176),(41,183,86,133),(42,134,87,184),(43,185,88,135),(44,136,89,186),(45,187,90,137),(46,138,91,188),(47,189,92,139),(48,140,93,190),(49,191,94,121),(50,122,95,192),(51,193,96,123),(52,124,97,194),(53,195,98,125),(54,126,99,196),(55,197,100,127),(56,128,81,198),(57,199,82,129),(58,130,83,200),(59,181,84,131),(60,132,85,182)], [(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),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,80),(74,79),(75,78),(76,77),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(101,106),(102,105),(103,104),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(135,140),(136,139),(137,138),(141,158),(142,157),(143,156),(144,155),(145,154),(146,153),(147,152),(148,151),(149,150),(159,160),(161,180),(162,179),(163,178),(164,177),(165,176),(166,175),(167,174),(168,173),(169,172),(170,171),(181,194),(182,193),(183,192),(184,191),(185,190),(186,189),(187,188),(195,200),(196,199),(197,198),(201,204),(202,203),(205,220),(206,219),(207,218),(208,217),(209,216),(210,215),(211,214),(212,213),(221,238),(222,237),(223,236),(224,235),(225,234),(226,233),(227,232),(228,231),(229,230),(239,240)]])
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⋊D20 | C3×D10⋊C4 | C15×C4⋊C4 | D5×C2×C12 | C6×D20 | C4⋊D20 | 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⋊D20 ►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⋊D20 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes D_{20} % in TeX
G:=Group("C3xC4:D20"); // 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