direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C5⋊D12, C60⋊15D4, C20⋊8D12, C5⋊5(C4×D12), D6⋊7(C4×D5), C15⋊21(C4×D4), D30⋊25(C2×C4), C12⋊8(C5⋊D4), Dic5⋊6(C4×S3), D6⋊Dic5⋊40C2, (C4×Dic5)⋊16S3, C30.145(C2×D4), (C2×C20).341D6, C10.61(C2×D12), D30⋊4C4⋊40C2, (C12×Dic5)⋊12C2, C30.80(C4○D4), C6.36(C4○D20), (C2×C12).345D10, C30.64(C22×C4), C30.Q8⋊43C2, C10.39(C4○D12), (C2×C60).243C22, (C2×C30).135C23, (C2×Dic5).180D6, (C22×S3).71D10, (C2×Dic3).153D10, C2.6(D6.D10), (C6×Dic5).207C22, (C10×Dic3).186C22, (C2×Dic15).212C22, (C22×D15).107C22, C3⋊1(C4×C5⋊D4), (S3×C2×C20)⋊8C2, (S3×C2×C4)⋊10D5, C6.32(C2×C4×D5), C2.34(C4×S3×D5), (C2×C4×D15)⋊29C2, C10.65(S3×C2×C4), C6.15(C2×C5⋊D4), C2.1(C2×C5⋊D12), (S3×C10)⋊20(C2×C4), C22.67(C2×S3×D5), (C2×C4).246(S3×D5), (C3×Dic5)⋊16(C2×C4), (C2×C5⋊D12).11C2, (S3×C2×C10).86C22, (C2×C6).147(C22×D5), (C2×C10).147(C22×S3), SmallGroup(480,521)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C5⋊D12
G = < a,b,c,d | a4=b5=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 940 in 188 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, S3×C10, S3×C10, D30, D30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×D12, C5⋊D12, C6×Dic5, S3×C20, C10×Dic3, C4×D15, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C4×C5⋊D4, D6⋊Dic5, D30⋊4C4, C30.Q8, C12×Dic5, C2×C5⋊D12, S3×C2×C20, C2×C4×D15, C4×C5⋊D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, D12, C22×S3, C4×D4, C4×D5, C5⋊D4, C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×D12, C5⋊D12, C2×S3×D5, C4×C5⋊D4, D6.D10, C4×S3×D5, C2×C5⋊D12, C4×C5⋊D12
►On 240 points
Generators in S240
(1 238 203 216)(2 239 204 205)(3 240 193 206)(4 229 194 207)(5 230 195 208)(6 231 196 209)(7 232 197 210)(8 233 198 211)(9 234 199 212)(10 235 200 213)(11 236 201 214)(12 237 202 215)(13 166 155 120)(14 167 156 109)(15 168 145 110)(16 157 146 111)(17 158 147 112)(18 159 148 113)(19 160 149 114)(20 161 150 115)(21 162 151 116)(22 163 152 117)(23 164 153 118)(24 165 154 119)(25 178 55 140)(26 179 56 141)(27 180 57 142)(28 169 58 143)(29 170 59 144)(30 171 60 133)(31 172 49 134)(32 173 50 135)(33 174 51 136)(34 175 52 137)(35 176 53 138)(36 177 54 139)(37 92 100 125)(38 93 101 126)(39 94 102 127)(40 95 103 128)(41 96 104 129)(42 85 105 130)(43 86 106 131)(44 87 107 132)(45 88 108 121)(46 89 97 122)(47 90 98 123)(48 91 99 124)(61 82 190 217)(62 83 191 218)(63 84 192 219)(64 73 181 220)(65 74 182 221)(66 75 183 222)(67 76 184 223)(68 77 185 224)(69 78 186 225)(70 79 187 226)(71 80 188 227)(72 81 189 228)
(1 188 114 101 175)(2 176 102 115 189)(3 190 116 103 177)(4 178 104 117 191)(5 192 118 105 179)(6 180 106 119 181)(7 182 120 107 169)(8 170 108 109 183)(9 184 110 97 171)(10 172 98 111 185)(11 186 112 99 173)(12 174 100 113 187)(13 132 58 232 221)(14 222 233 59 121)(15 122 60 234 223)(16 224 235 49 123)(17 124 50 236 225)(18 226 237 51 125)(19 126 52 238 227)(20 228 239 53 127)(21 128 54 240 217)(22 218 229 55 129)(23 130 56 230 219)(24 220 231 57 131)(25 96 152 83 207)(26 208 84 153 85)(27 86 154 73 209)(28 210 74 155 87)(29 88 156 75 211)(30 212 76 145 89)(31 90 146 77 213)(32 214 78 147 91)(33 92 148 79 215)(34 216 80 149 93)(35 94 150 81 205)(36 206 82 151 95)(37 159 70 202 136)(38 137 203 71 160)(39 161 72 204 138)(40 139 193 61 162)(41 163 62 194 140)(42 141 195 63 164)(43 165 64 196 142)(44 143 197 65 166)(45 167 66 198 144)(46 133 199 67 168)(47 157 68 200 134)(48 135 201 69 158)
(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)
(1 199)(2 198)(3 197)(4 196)(5 195)(6 194)(7 193)(8 204)(9 203)(10 202)(11 201)(12 200)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 96)(25 220)(26 219)(27 218)(28 217)(29 228)(30 227)(31 226)(32 225)(33 224)(34 223)(35 222)(36 221)(37 111)(38 110)(39 109)(40 120)(41 119)(42 118)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 84)(57 83)(58 82)(59 81)(60 80)(61 169)(62 180)(63 179)(64 178)(65 177)(66 176)(67 175)(68 174)(69 173)(70 172)(71 171)(72 170)(97 160)(98 159)(99 158)(100 157)(101 168)(102 167)(103 166)(104 165)(105 164)(106 163)(107 162)(108 161)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 156)(128 155)(129 154)(130 153)(131 152)(132 151)(133 188)(134 187)(135 186)(136 185)(137 184)(138 183)(139 182)(140 181)(141 192)(142 191)(143 190)(144 189)(205 233)(206 232)(207 231)(208 230)(209 229)(210 240)(211 239)(212 238)(213 237)(214 236)(215 235)(216 234)
G:=sub<Sym(240)| (1,238,203,216)(2,239,204,205)(3,240,193,206)(4,229,194,207)(5,230,195,208)(6,231,196,209)(7,232,197,210)(8,233,198,211)(9,234,199,212)(10,235,200,213)(11,236,201,214)(12,237,202,215)(13,166,155,120)(14,167,156,109)(15,168,145,110)(16,157,146,111)(17,158,147,112)(18,159,148,113)(19,160,149,114)(20,161,150,115)(21,162,151,116)(22,163,152,117)(23,164,153,118)(24,165,154,119)(25,178,55,140)(26,179,56,141)(27,180,57,142)(28,169,58,143)(29,170,59,144)(30,171,60,133)(31,172,49,134)(32,173,50,135)(33,174,51,136)(34,175,52,137)(35,176,53,138)(36,177,54,139)(37,92,100,125)(38,93,101,126)(39,94,102,127)(40,95,103,128)(41,96,104,129)(42,85,105,130)(43,86,106,131)(44,87,107,132)(45,88,108,121)(46,89,97,122)(47,90,98,123)(48,91,99,124)(61,82,190,217)(62,83,191,218)(63,84,192,219)(64,73,181,220)(65,74,182,221)(66,75,183,222)(67,76,184,223)(68,77,185,224)(69,78,186,225)(70,79,187,226)(71,80,188,227)(72,81,189,228), (1,188,114,101,175)(2,176,102,115,189)(3,190,116,103,177)(4,178,104,117,191)(5,192,118,105,179)(6,180,106,119,181)(7,182,120,107,169)(8,170,108,109,183)(9,184,110,97,171)(10,172,98,111,185)(11,186,112,99,173)(12,174,100,113,187)(13,132,58,232,221)(14,222,233,59,121)(15,122,60,234,223)(16,224,235,49,123)(17,124,50,236,225)(18,226,237,51,125)(19,126,52,238,227)(20,228,239,53,127)(21,128,54,240,217)(22,218,229,55,129)(23,130,56,230,219)(24,220,231,57,131)(25,96,152,83,207)(26,208,84,153,85)(27,86,154,73,209)(28,210,74,155,87)(29,88,156,75,211)(30,212,76,145,89)(31,90,146,77,213)(32,214,78,147,91)(33,92,148,79,215)(34,216,80,149,93)(35,94,150,81,205)(36,206,82,151,95)(37,159,70,202,136)(38,137,203,71,160)(39,161,72,204,138)(40,139,193,61,162)(41,163,62,194,140)(42,141,195,63,164)(43,165,64,196,142)(44,143,197,65,166)(45,167,66,198,144)(46,133,199,67,168)(47,157,68,200,134)(48,135,201,69,158), (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), (1,199)(2,198)(3,197)(4,196)(5,195)(6,194)(7,193)(8,204)(9,203)(10,202)(11,201)(12,200)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,96)(25,220)(26,219)(27,218)(28,217)(29,228)(30,227)(31,226)(32,225)(33,224)(34,223)(35,222)(36,221)(37,111)(38,110)(39,109)(40,120)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,84)(57,83)(58,82)(59,81)(60,80)(61,169)(62,180)(63,179)(64,178)(65,177)(66,176)(67,175)(68,174)(69,173)(70,172)(71,171)(72,170)(97,160)(98,159)(99,158)(100,157)(101,168)(102,167)(103,166)(104,165)(105,164)(106,163)(107,162)(108,161)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,188)(134,187)(135,186)(136,185)(137,184)(138,183)(139,182)(140,181)(141,192)(142,191)(143,190)(144,189)(205,233)(206,232)(207,231)(208,230)(209,229)(210,240)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234)>;
G:=Group( (1,238,203,216)(2,239,204,205)(3,240,193,206)(4,229,194,207)(5,230,195,208)(6,231,196,209)(7,232,197,210)(8,233,198,211)(9,234,199,212)(10,235,200,213)(11,236,201,214)(12,237,202,215)(13,166,155,120)(14,167,156,109)(15,168,145,110)(16,157,146,111)(17,158,147,112)(18,159,148,113)(19,160,149,114)(20,161,150,115)(21,162,151,116)(22,163,152,117)(23,164,153,118)(24,165,154,119)(25,178,55,140)(26,179,56,141)(27,180,57,142)(28,169,58,143)(29,170,59,144)(30,171,60,133)(31,172,49,134)(32,173,50,135)(33,174,51,136)(34,175,52,137)(35,176,53,138)(36,177,54,139)(37,92,100,125)(38,93,101,126)(39,94,102,127)(40,95,103,128)(41,96,104,129)(42,85,105,130)(43,86,106,131)(44,87,107,132)(45,88,108,121)(46,89,97,122)(47,90,98,123)(48,91,99,124)(61,82,190,217)(62,83,191,218)(63,84,192,219)(64,73,181,220)(65,74,182,221)(66,75,183,222)(67,76,184,223)(68,77,185,224)(69,78,186,225)(70,79,187,226)(71,80,188,227)(72,81,189,228), (1,188,114,101,175)(2,176,102,115,189)(3,190,116,103,177)(4,178,104,117,191)(5,192,118,105,179)(6,180,106,119,181)(7,182,120,107,169)(8,170,108,109,183)(9,184,110,97,171)(10,172,98,111,185)(11,186,112,99,173)(12,174,100,113,187)(13,132,58,232,221)(14,222,233,59,121)(15,122,60,234,223)(16,224,235,49,123)(17,124,50,236,225)(18,226,237,51,125)(19,126,52,238,227)(20,228,239,53,127)(21,128,54,240,217)(22,218,229,55,129)(23,130,56,230,219)(24,220,231,57,131)(25,96,152,83,207)(26,208,84,153,85)(27,86,154,73,209)(28,210,74,155,87)(29,88,156,75,211)(30,212,76,145,89)(31,90,146,77,213)(32,214,78,147,91)(33,92,148,79,215)(34,216,80,149,93)(35,94,150,81,205)(36,206,82,151,95)(37,159,70,202,136)(38,137,203,71,160)(39,161,72,204,138)(40,139,193,61,162)(41,163,62,194,140)(42,141,195,63,164)(43,165,64,196,142)(44,143,197,65,166)(45,167,66,198,144)(46,133,199,67,168)(47,157,68,200,134)(48,135,201,69,158), (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), (1,199)(2,198)(3,197)(4,196)(5,195)(6,194)(7,193)(8,204)(9,203)(10,202)(11,201)(12,200)(13,95)(14,94)(15,93)(16,92)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,96)(25,220)(26,219)(27,218)(28,217)(29,228)(30,227)(31,226)(32,225)(33,224)(34,223)(35,222)(36,221)(37,111)(38,110)(39,109)(40,120)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,84)(57,83)(58,82)(59,81)(60,80)(61,169)(62,180)(63,179)(64,178)(65,177)(66,176)(67,175)(68,174)(69,173)(70,172)(71,171)(72,170)(97,160)(98,159)(99,158)(100,157)(101,168)(102,167)(103,166)(104,165)(105,164)(106,163)(107,162)(108,161)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,188)(134,187)(135,186)(136,185)(137,184)(138,183)(139,182)(140,181)(141,192)(142,191)(143,190)(144,189)(205,233)(206,232)(207,231)(208,230)(209,229)(210,240)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234) );
G=PermutationGroup([[(1,238,203,216),(2,239,204,205),(3,240,193,206),(4,229,194,207),(5,230,195,208),(6,231,196,209),(7,232,197,210),(8,233,198,211),(9,234,199,212),(10,235,200,213),(11,236,201,214),(12,237,202,215),(13,166,155,120),(14,167,156,109),(15,168,145,110),(16,157,146,111),(17,158,147,112),(18,159,148,113),(19,160,149,114),(20,161,150,115),(21,162,151,116),(22,163,152,117),(23,164,153,118),(24,165,154,119),(25,178,55,140),(26,179,56,141),(27,180,57,142),(28,169,58,143),(29,170,59,144),(30,171,60,133),(31,172,49,134),(32,173,50,135),(33,174,51,136),(34,175,52,137),(35,176,53,138),(36,177,54,139),(37,92,100,125),(38,93,101,126),(39,94,102,127),(40,95,103,128),(41,96,104,129),(42,85,105,130),(43,86,106,131),(44,87,107,132),(45,88,108,121),(46,89,97,122),(47,90,98,123),(48,91,99,124),(61,82,190,217),(62,83,191,218),(63,84,192,219),(64,73,181,220),(65,74,182,221),(66,75,183,222),(67,76,184,223),(68,77,185,224),(69,78,186,225),(70,79,187,226),(71,80,188,227),(72,81,189,228)], [(1,188,114,101,175),(2,176,102,115,189),(3,190,116,103,177),(4,178,104,117,191),(5,192,118,105,179),(6,180,106,119,181),(7,182,120,107,169),(8,170,108,109,183),(9,184,110,97,171),(10,172,98,111,185),(11,186,112,99,173),(12,174,100,113,187),(13,132,58,232,221),(14,222,233,59,121),(15,122,60,234,223),(16,224,235,49,123),(17,124,50,236,225),(18,226,237,51,125),(19,126,52,238,227),(20,228,239,53,127),(21,128,54,240,217),(22,218,229,55,129),(23,130,56,230,219),(24,220,231,57,131),(25,96,152,83,207),(26,208,84,153,85),(27,86,154,73,209),(28,210,74,155,87),(29,88,156,75,211),(30,212,76,145,89),(31,90,146,77,213),(32,214,78,147,91),(33,92,148,79,215),(34,216,80,149,93),(35,94,150,81,205),(36,206,82,151,95),(37,159,70,202,136),(38,137,203,71,160),(39,161,72,204,138),(40,139,193,61,162),(41,163,62,194,140),(42,141,195,63,164),(43,165,64,196,142),(44,143,197,65,166),(45,167,66,198,144),(46,133,199,67,168),(47,157,68,200,134),(48,135,201,69,158)], [(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)], [(1,199),(2,198),(3,197),(4,196),(5,195),(6,194),(7,193),(8,204),(9,203),(10,202),(11,201),(12,200),(13,95),(14,94),(15,93),(16,92),(17,91),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,96),(25,220),(26,219),(27,218),(28,217),(29,228),(30,227),(31,226),(32,225),(33,224),(34,223),(35,222),(36,221),(37,111),(38,110),(39,109),(40,120),(41,119),(42,118),(43,117),(44,116),(45,115),(46,114),(47,113),(48,112),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,84),(57,83),(58,82),(59,81),(60,80),(61,169),(62,180),(63,179),(64,178),(65,177),(66,176),(67,175),(68,174),(69,173),(70,172),(71,171),(72,170),(97,160),(98,159),(99,158),(100,157),(101,168),(102,167),(103,166),(104,165),(105,164),(106,163),(107,162),(108,161),(121,150),(122,149),(123,148),(124,147),(125,146),(126,145),(127,156),(128,155),(129,154),(130,153),(131,152),(132,151),(133,188),(134,187),(135,186),(136,185),(137,184),(138,183),(139,182),(140,181),(141,192),(142,191),(143,190),(144,189),(205,233),(206,232),(207,231),(208,230),(209,229),(210,240),(211,239),(212,238),(213,237),(214,236),(215,235),(216,234)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4×S3 | D12 | C5⋊D4 | C4×D5 | C4○D12 | C4○D20 | S3×D5 | C5⋊D12 | C2×S3×D5 | D6.D10 | C4×S3×D5 |
| kernel | C4×C5⋊D12 | D6⋊Dic5 | D30⋊4C4 | C30.Q8 | C12×Dic5 | C2×C5⋊D12 | S3×C2×C20 | C2×C4×D15 | C5⋊D12 | C4×Dic5 | C60 | S3×C2×C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | C20 | C12 | D6 | C10 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C4×C5⋊D12 ►in GL4(𝔽61) generated by
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 16 | 0 | 0 |
| 1 | 44 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 17 | 0 | 0 |
| 1 | 44 | 0 | 0 |
| 0 | 0 | 15 | 38 |
| 0 | 0 | 23 | 38 |
| 44 | 44 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [11,0,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,16,44,0,0,0,0,1,0,0,0,0,1],[17,1,0,0,17,44,0,0,0,0,15,23,0,0,38,38],[44,60,0,0,44,17,0,0,0,0,0,1,0,0,1,0] >;
C4×C5⋊D12 in GAP, Magma, Sage, TeX
C_4\times C_5\rtimes D_{12} % in TeX
G:=Group("C4xC5:D12"); // GroupNames label
G:=SmallGroup(480,521);
// by ID
G=gap.SmallGroup(480,521);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^5=c^12=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