direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4.D20, C60.173D4, C12.43D20, (C4×C20)⋊11C6, (C4×C60)⋊13C2, (C4×C12)⋊13D5, C2.6(C6×D20), C10.4(C6×D4), C4.5(C3×D20), C42⋊9(C3×D5), (C2×D20).3C6, C20.28(C3×D4), C6.75(C2×D20), D10⋊C4⋊1C6, (C2×Dic10)⋊1C6, (C6×D20).14C2, C30.277(C2×D4), (C6×Dic10)⋊17C2, (C2×C12).375D10, C15⋊17(C4.4D4), C30.184(C4○D4), C6.112(C4○D20), (C2×C30).333C23, (C2×C60).445C22, (C6×Dic5).152C22, C5⋊1(C3×C4.4D4), C10.5(C3×C4○D4), C2.7(C3×C4○D20), (C2×C4).78(C6×D5), C22.37(D5×C2×C6), (C2×C20).76(C2×C6), (C3×D10⋊C4)⋊1C2, (D5×C2×C6).75C22, (C2×Dic5).3(C2×C6), (C22×D5).3(C2×C6), (C2×C10).16(C22×C6), (C2×C6).329(C22×D5), SmallGroup(480,668)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.D20
G = < a,b,c,d | a3=b4=c20=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 576 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C3×Dic5, C60, C60, C6×D5, C2×C30, D10⋊C4, C4×C20, C2×Dic10, C2×D20, C3×C4.4D4, C3×Dic10, C3×D20, C6×Dic5, C2×C60, C2×C60, D5×C2×C6, C4.D20, C3×D10⋊C4, C4×C60, C6×Dic10, 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, C4.4D4, D20, C22×D5, C6×D4, C3×C4○D4, C6×D5, C2×D20, C4○D20, C3×C4.4D4, C3×D20, D5×C2×C6, C4.D20, C6×D20, C3×C4○D20, C3×C4.D20
►On 240 points
Generators in S240
(1 56 149)(2 57 150)(3 58 151)(4 59 152)(5 60 153)(6 41 154)(7 42 155)(8 43 156)(9 44 157)(10 45 158)(11 46 159)(12 47 160)(13 48 141)(14 49 142)(15 50 143)(16 51 144)(17 52 145)(18 53 146)(19 54 147)(20 55 148)(21 163 230)(22 164 231)(23 165 232)(24 166 233)(25 167 234)(26 168 235)(27 169 236)(28 170 237)(29 171 238)(30 172 239)(31 173 240)(32 174 221)(33 175 222)(34 176 223)(35 177 224)(36 178 225)(37 179 226)(38 180 227)(39 161 228)(40 162 229)(61 190 140)(62 191 121)(63 192 122)(64 193 123)(65 194 124)(66 195 125)(67 196 126)(68 197 127)(69 198 128)(70 199 129)(71 200 130)(72 181 131)(73 182 132)(74 183 133)(75 184 134)(76 185 135)(77 186 136)(78 187 137)(79 188 138)(80 189 139)(81 212 109)(82 213 110)(83 214 111)(84 215 112)(85 216 113)(86 217 114)(87 218 115)(88 219 116)(89 220 117)(90 201 118)(91 202 119)(92 203 120)(93 204 101)(94 205 102)(95 206 103)(96 207 104)(97 208 105)(98 209 106)(99 210 107)(100 211 108)
(1 81 135 28)(2 82 136 29)(3 83 137 30)(4 84 138 31)(5 85 139 32)(6 86 140 33)(7 87 121 34)(8 88 122 35)(9 89 123 36)(10 90 124 37)(11 91 125 38)(12 92 126 39)(13 93 127 40)(14 94 128 21)(15 95 129 22)(16 96 130 23)(17 97 131 24)(18 98 132 25)(19 99 133 26)(20 100 134 27)(41 217 61 175)(42 218 62 176)(43 219 63 177)(44 220 64 178)(45 201 65 179)(46 202 66 180)(47 203 67 161)(48 204 68 162)(49 205 69 163)(50 206 70 164)(51 207 71 165)(52 208 72 166)(53 209 73 167)(54 210 74 168)(55 211 75 169)(56 212 76 170)(57 213 77 171)(58 214 78 172)(59 215 79 173)(60 216 80 174)(101 197 229 141)(102 198 230 142)(103 199 231 143)(104 200 232 144)(105 181 233 145)(106 182 234 146)(107 183 235 147)(108 184 236 148)(109 185 237 149)(110 186 238 150)(111 187 239 151)(112 188 240 152)(113 189 221 153)(114 190 222 154)(115 191 223 155)(116 192 224 156)(117 193 225 157)(118 194 226 158)(119 195 227 159)(120 196 228 160)
(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 37 135 90)(2 89 136 36)(3 35 137 88)(4 87 138 34)(5 33 139 86)(6 85 140 32)(7 31 121 84)(8 83 122 30)(9 29 123 82)(10 81 124 28)(11 27 125 100)(12 99 126 26)(13 25 127 98)(14 97 128 24)(15 23 129 96)(16 95 130 22)(17 21 131 94)(18 93 132 40)(19 39 133 92)(20 91 134 38)(41 216 61 174)(42 173 62 215)(43 214 63 172)(44 171 64 213)(45 212 65 170)(46 169 66 211)(47 210 67 168)(48 167 68 209)(49 208 69 166)(50 165 70 207)(51 206 71 164)(52 163 72 205)(53 204 73 162)(54 161 74 203)(55 202 75 180)(56 179 76 201)(57 220 77 178)(58 177 78 219)(59 218 79 176)(60 175 80 217)(101 182 229 146)(102 145 230 181)(103 200 231 144)(104 143 232 199)(105 198 233 142)(106 141 234 197)(107 196 235 160)(108 159 236 195)(109 194 237 158)(110 157 238 193)(111 192 239 156)(112 155 240 191)(113 190 221 154)(114 153 222 189)(115 188 223 152)(116 151 224 187)(117 186 225 150)(118 149 226 185)(119 184 227 148)(120 147 228 183)
G:=sub<Sym(240)| (1,56,149)(2,57,150)(3,58,151)(4,59,152)(5,60,153)(6,41,154)(7,42,155)(8,43,156)(9,44,157)(10,45,158)(11,46,159)(12,47,160)(13,48,141)(14,49,142)(15,50,143)(16,51,144)(17,52,145)(18,53,146)(19,54,147)(20,55,148)(21,163,230)(22,164,231)(23,165,232)(24,166,233)(25,167,234)(26,168,235)(27,169,236)(28,170,237)(29,171,238)(30,172,239)(31,173,240)(32,174,221)(33,175,222)(34,176,223)(35,177,224)(36,178,225)(37,179,226)(38,180,227)(39,161,228)(40,162,229)(61,190,140)(62,191,121)(63,192,122)(64,193,123)(65,194,124)(66,195,125)(67,196,126)(68,197,127)(69,198,128)(70,199,129)(71,200,130)(72,181,131)(73,182,132)(74,183,133)(75,184,134)(76,185,135)(77,186,136)(78,187,137)(79,188,138)(80,189,139)(81,212,109)(82,213,110)(83,214,111)(84,215,112)(85,216,113)(86,217,114)(87,218,115)(88,219,116)(89,220,117)(90,201,118)(91,202,119)(92,203,120)(93,204,101)(94,205,102)(95,206,103)(96,207,104)(97,208,105)(98,209,106)(99,210,107)(100,211,108), (1,81,135,28)(2,82,136,29)(3,83,137,30)(4,84,138,31)(5,85,139,32)(6,86,140,33)(7,87,121,34)(8,88,122,35)(9,89,123,36)(10,90,124,37)(11,91,125,38)(12,92,126,39)(13,93,127,40)(14,94,128,21)(15,95,129,22)(16,96,130,23)(17,97,131,24)(18,98,132,25)(19,99,133,26)(20,100,134,27)(41,217,61,175)(42,218,62,176)(43,219,63,177)(44,220,64,178)(45,201,65,179)(46,202,66,180)(47,203,67,161)(48,204,68,162)(49,205,69,163)(50,206,70,164)(51,207,71,165)(52,208,72,166)(53,209,73,167)(54,210,74,168)(55,211,75,169)(56,212,76,170)(57,213,77,171)(58,214,78,172)(59,215,79,173)(60,216,80,174)(101,197,229,141)(102,198,230,142)(103,199,231,143)(104,200,232,144)(105,181,233,145)(106,182,234,146)(107,183,235,147)(108,184,236,148)(109,185,237,149)(110,186,238,150)(111,187,239,151)(112,188,240,152)(113,189,221,153)(114,190,222,154)(115,191,223,155)(116,192,224,156)(117,193,225,157)(118,194,226,158)(119,195,227,159)(120,196,228,160), (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), (1,37,135,90)(2,89,136,36)(3,35,137,88)(4,87,138,34)(5,33,139,86)(6,85,140,32)(7,31,121,84)(8,83,122,30)(9,29,123,82)(10,81,124,28)(11,27,125,100)(12,99,126,26)(13,25,127,98)(14,97,128,24)(15,23,129,96)(16,95,130,22)(17,21,131,94)(18,93,132,40)(19,39,133,92)(20,91,134,38)(41,216,61,174)(42,173,62,215)(43,214,63,172)(44,171,64,213)(45,212,65,170)(46,169,66,211)(47,210,67,168)(48,167,68,209)(49,208,69,166)(50,165,70,207)(51,206,71,164)(52,163,72,205)(53,204,73,162)(54,161,74,203)(55,202,75,180)(56,179,76,201)(57,220,77,178)(58,177,78,219)(59,218,79,176)(60,175,80,217)(101,182,229,146)(102,145,230,181)(103,200,231,144)(104,143,232,199)(105,198,233,142)(106,141,234,197)(107,196,235,160)(108,159,236,195)(109,194,237,158)(110,157,238,193)(111,192,239,156)(112,155,240,191)(113,190,221,154)(114,153,222,189)(115,188,223,152)(116,151,224,187)(117,186,225,150)(118,149,226,185)(119,184,227,148)(120,147,228,183)>;
G:=Group( (1,56,149)(2,57,150)(3,58,151)(4,59,152)(5,60,153)(6,41,154)(7,42,155)(8,43,156)(9,44,157)(10,45,158)(11,46,159)(12,47,160)(13,48,141)(14,49,142)(15,50,143)(16,51,144)(17,52,145)(18,53,146)(19,54,147)(20,55,148)(21,163,230)(22,164,231)(23,165,232)(24,166,233)(25,167,234)(26,168,235)(27,169,236)(28,170,237)(29,171,238)(30,172,239)(31,173,240)(32,174,221)(33,175,222)(34,176,223)(35,177,224)(36,178,225)(37,179,226)(38,180,227)(39,161,228)(40,162,229)(61,190,140)(62,191,121)(63,192,122)(64,193,123)(65,194,124)(66,195,125)(67,196,126)(68,197,127)(69,198,128)(70,199,129)(71,200,130)(72,181,131)(73,182,132)(74,183,133)(75,184,134)(76,185,135)(77,186,136)(78,187,137)(79,188,138)(80,189,139)(81,212,109)(82,213,110)(83,214,111)(84,215,112)(85,216,113)(86,217,114)(87,218,115)(88,219,116)(89,220,117)(90,201,118)(91,202,119)(92,203,120)(93,204,101)(94,205,102)(95,206,103)(96,207,104)(97,208,105)(98,209,106)(99,210,107)(100,211,108), (1,81,135,28)(2,82,136,29)(3,83,137,30)(4,84,138,31)(5,85,139,32)(6,86,140,33)(7,87,121,34)(8,88,122,35)(9,89,123,36)(10,90,124,37)(11,91,125,38)(12,92,126,39)(13,93,127,40)(14,94,128,21)(15,95,129,22)(16,96,130,23)(17,97,131,24)(18,98,132,25)(19,99,133,26)(20,100,134,27)(41,217,61,175)(42,218,62,176)(43,219,63,177)(44,220,64,178)(45,201,65,179)(46,202,66,180)(47,203,67,161)(48,204,68,162)(49,205,69,163)(50,206,70,164)(51,207,71,165)(52,208,72,166)(53,209,73,167)(54,210,74,168)(55,211,75,169)(56,212,76,170)(57,213,77,171)(58,214,78,172)(59,215,79,173)(60,216,80,174)(101,197,229,141)(102,198,230,142)(103,199,231,143)(104,200,232,144)(105,181,233,145)(106,182,234,146)(107,183,235,147)(108,184,236,148)(109,185,237,149)(110,186,238,150)(111,187,239,151)(112,188,240,152)(113,189,221,153)(114,190,222,154)(115,191,223,155)(116,192,224,156)(117,193,225,157)(118,194,226,158)(119,195,227,159)(120,196,228,160), (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), (1,37,135,90)(2,89,136,36)(3,35,137,88)(4,87,138,34)(5,33,139,86)(6,85,140,32)(7,31,121,84)(8,83,122,30)(9,29,123,82)(10,81,124,28)(11,27,125,100)(12,99,126,26)(13,25,127,98)(14,97,128,24)(15,23,129,96)(16,95,130,22)(17,21,131,94)(18,93,132,40)(19,39,133,92)(20,91,134,38)(41,216,61,174)(42,173,62,215)(43,214,63,172)(44,171,64,213)(45,212,65,170)(46,169,66,211)(47,210,67,168)(48,167,68,209)(49,208,69,166)(50,165,70,207)(51,206,71,164)(52,163,72,205)(53,204,73,162)(54,161,74,203)(55,202,75,180)(56,179,76,201)(57,220,77,178)(58,177,78,219)(59,218,79,176)(60,175,80,217)(101,182,229,146)(102,145,230,181)(103,200,231,144)(104,143,232,199)(105,198,233,142)(106,141,234,197)(107,196,235,160)(108,159,236,195)(109,194,237,158)(110,157,238,193)(111,192,239,156)(112,155,240,191)(113,190,221,154)(114,153,222,189)(115,188,223,152)(116,151,224,187)(117,186,225,150)(118,149,226,185)(119,184,227,148)(120,147,228,183) );
G=PermutationGroup([[(1,56,149),(2,57,150),(3,58,151),(4,59,152),(5,60,153),(6,41,154),(7,42,155),(8,43,156),(9,44,157),(10,45,158),(11,46,159),(12,47,160),(13,48,141),(14,49,142),(15,50,143),(16,51,144),(17,52,145),(18,53,146),(19,54,147),(20,55,148),(21,163,230),(22,164,231),(23,165,232),(24,166,233),(25,167,234),(26,168,235),(27,169,236),(28,170,237),(29,171,238),(30,172,239),(31,173,240),(32,174,221),(33,175,222),(34,176,223),(35,177,224),(36,178,225),(37,179,226),(38,180,227),(39,161,228),(40,162,229),(61,190,140),(62,191,121),(63,192,122),(64,193,123),(65,194,124),(66,195,125),(67,196,126),(68,197,127),(69,198,128),(70,199,129),(71,200,130),(72,181,131),(73,182,132),(74,183,133),(75,184,134),(76,185,135),(77,186,136),(78,187,137),(79,188,138),(80,189,139),(81,212,109),(82,213,110),(83,214,111),(84,215,112),(85,216,113),(86,217,114),(87,218,115),(88,219,116),(89,220,117),(90,201,118),(91,202,119),(92,203,120),(93,204,101),(94,205,102),(95,206,103),(96,207,104),(97,208,105),(98,209,106),(99,210,107),(100,211,108)], [(1,81,135,28),(2,82,136,29),(3,83,137,30),(4,84,138,31),(5,85,139,32),(6,86,140,33),(7,87,121,34),(8,88,122,35),(9,89,123,36),(10,90,124,37),(11,91,125,38),(12,92,126,39),(13,93,127,40),(14,94,128,21),(15,95,129,22),(16,96,130,23),(17,97,131,24),(18,98,132,25),(19,99,133,26),(20,100,134,27),(41,217,61,175),(42,218,62,176),(43,219,63,177),(44,220,64,178),(45,201,65,179),(46,202,66,180),(47,203,67,161),(48,204,68,162),(49,205,69,163),(50,206,70,164),(51,207,71,165),(52,208,72,166),(53,209,73,167),(54,210,74,168),(55,211,75,169),(56,212,76,170),(57,213,77,171),(58,214,78,172),(59,215,79,173),(60,216,80,174),(101,197,229,141),(102,198,230,142),(103,199,231,143),(104,200,232,144),(105,181,233,145),(106,182,234,146),(107,183,235,147),(108,184,236,148),(109,185,237,149),(110,186,238,150),(111,187,239,151),(112,188,240,152),(113,189,221,153),(114,190,222,154),(115,191,223,155),(116,192,224,156),(117,193,225,157),(118,194,226,158),(119,195,227,159),(120,196,228,160)], [(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)], [(1,37,135,90),(2,89,136,36),(3,35,137,88),(4,87,138,34),(5,33,139,86),(6,85,140,32),(7,31,121,84),(8,83,122,30),(9,29,123,82),(10,81,124,28),(11,27,125,100),(12,99,126,26),(13,25,127,98),(14,97,128,24),(15,23,129,96),(16,95,130,22),(17,21,131,94),(18,93,132,40),(19,39,133,92),(20,91,134,38),(41,216,61,174),(42,173,62,215),(43,214,63,172),(44,171,64,213),(45,212,65,170),(46,169,66,211),(47,210,67,168),(48,167,68,209),(49,208,69,166),(50,165,70,207),(51,206,71,164),(52,163,72,205),(53,204,73,162),(54,161,74,203),(55,202,75,180),(56,179,76,201),(57,220,77,178),(58,177,78,219),(59,218,79,176),(60,175,80,217),(101,182,229,146),(102,145,230,181),(103,200,231,144),(104,143,232,199),(105,198,233,142),(106,141,234,197),(107,196,235,160),(108,159,236,195),(109,194,237,158),(110,157,238,193),(111,192,239,156),(112,155,240,191),(113,190,221,154),(114,153,222,189),(115,188,223,152),(116,151,224,187),(117,186,225,150),(118,149,226,185),(119,184,227,148),(120,147,228,183)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10F | 12A | ··· | 12L | 12M | 12N | 12O | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 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 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | C3×D4 | C3×D5 | D20 | C3×C4○D4 | C6×D5 | C4○D20 | C3×D20 | C3×C4○D20 |
| kernel | C3×C4.D20 | C3×D10⋊C4 | C4×C60 | C6×Dic10 | C6×D20 | C4.D20 | D10⋊C4 | C4×C20 | C2×Dic10 | C2×D20 | C60 | C4×C12 | C30 | C2×C12 | C20 | C42 | C12 | C10 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 4 | 4 | 8 | 8 | 12 | 16 | 16 | 32 |
Matrix representation of C3×C4.D20 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 29 | 54 | 0 | 0 |
| 7 | 32 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 4 | 50 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 59 | 32 |
| 0 | 0 | 29 | 54 |
| 39 | 47 | 0 | 0 |
| 39 | 22 | 0 | 0 |
| 0 | 0 | 59 | 32 |
| 0 | 0 | 59 | 2 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[29,7,0,0,54,32,0,0,0,0,60,0,0,0,0,60],[4,11,0,0,50,0,0,0,0,0,59,29,0,0,32,54],[39,39,0,0,47,22,0,0,0,0,59,59,0,0,32,2] >;
C3×C4.D20 in GAP, Magma, Sage, TeX
C_3\times C_4.D_{20} % in TeX
G:=Group("C3xC4.D20"); // GroupNames label
G:=SmallGroup(480,668);
// by ID
G=gap.SmallGroup(480,668);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,176,590,268,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^20=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations