direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C42×D15, (C4×C12)⋊8D5, C5⋊3(S3×C42), C20⋊15(C4×S3), C3⋊2(D5×C42), C60⋊33(C2×C4), (C4×C60)⋊10C2, (C4×C20)⋊10S3, C12⋊11(C4×D5), C15⋊9(C2×C42), (C2×C4).96D30, D30.42(C2×C4), (C2×C20).411D6, (C4×Dic15)⋊35C2, Dic15⋊25(C2×C4), (C2×C12).414D10, (C2×C60).496C22, C30.153(C22×C4), (C2×C30).268C23, C22.9(C22×D15), (C2×Dic15).236C22, (C22×D15).124C22, C6.58(C2×C4×D5), C2.1(C2×C4×D15), C10.90(S3×C2×C4), (C2×C4×D15).24C2, (C2×C6).264(C22×D5), (C2×C10).263(C22×S3), SmallGroup(480,836)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C42×D15 |
Generators and relations for C42×D15
G = < a,b,c,d | a4=b4=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 996 in 216 conjugacy classes, 99 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C42, C42, C22×C4, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C2×C42, C4×D5, C2×Dic5, C2×C20, C22×D5, C4×Dic3, C4×C12, S3×C2×C4, Dic15, C60, D30, C2×C30, C4×Dic5, C4×C20, C2×C4×D5, S3×C42, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×C42, C4×Dic15, C4×C60, C2×C4×D15, C42×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C42, C22×C4, D10, C4×S3, C22×S3, D15, C2×C42, C4×D5, C22×D5, S3×C2×C4, D30, C2×C4×D5, S3×C42, C4×D15, C22×D15, D5×C42, C2×C4×D15, C42×D15
►On 240 points
Generators in S240
(1 167 24 152)(2 168 25 153)(3 169 26 154)(4 170 27 155)(5 171 28 156)(6 172 29 157)(7 173 30 158)(8 174 16 159)(9 175 17 160)(10 176 18 161)(11 177 19 162)(12 178 20 163)(13 179 21 164)(14 180 22 165)(15 166 23 151)(31 148 54 132)(32 149 55 133)(33 150 56 134)(34 136 57 135)(35 137 58 121)(36 138 59 122)(37 139 60 123)(38 140 46 124)(39 141 47 125)(40 142 48 126)(41 143 49 127)(42 144 50 128)(43 145 51 129)(44 146 52 130)(45 147 53 131)(61 233 87 222)(62 234 88 223)(63 235 89 224)(64 236 90 225)(65 237 76 211)(66 238 77 212)(67 239 78 213)(68 240 79 214)(69 226 80 215)(70 227 81 216)(71 228 82 217)(72 229 83 218)(73 230 84 219)(74 231 85 220)(75 232 86 221)(91 203 109 184)(92 204 110 185)(93 205 111 186)(94 206 112 187)(95 207 113 188)(96 208 114 189)(97 209 115 190)(98 210 116 191)(99 196 117 192)(100 197 118 193)(101 198 119 194)(102 199 120 195)(103 200 106 181)(104 201 107 182)(105 202 108 183)
(1 103 35 73)(2 104 36 74)(3 105 37 75)(4 91 38 61)(5 92 39 62)(6 93 40 63)(7 94 41 64)(8 95 42 65)(9 96 43 66)(10 97 44 67)(11 98 45 68)(12 99 31 69)(13 100 32 70)(14 101 33 71)(15 102 34 72)(16 113 50 76)(17 114 51 77)(18 115 52 78)(19 116 53 79)(20 117 54 80)(21 118 55 81)(22 119 56 82)(23 120 57 83)(24 106 58 84)(25 107 59 85)(26 108 60 86)(27 109 46 87)(28 110 47 88)(29 111 48 89)(30 112 49 90)(121 219 152 181)(122 220 153 182)(123 221 154 183)(124 222 155 184)(125 223 156 185)(126 224 157 186)(127 225 158 187)(128 211 159 188)(129 212 160 189)(130 213 161 190)(131 214 162 191)(132 215 163 192)(133 216 164 193)(134 217 165 194)(135 218 151 195)(136 229 166 199)(137 230 167 200)(138 231 168 201)(139 232 169 202)(140 233 170 203)(141 234 171 204)(142 235 172 205)(143 236 173 206)(144 237 174 207)(145 238 175 208)(146 239 176 209)(147 240 177 210)(148 226 178 196)(149 227 179 197)(150 228 180 198)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 120)(104 119)(105 118)(121 136)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)(151 167)(152 166)(153 180)(154 179)(155 178)(156 177)(157 176)(158 175)(159 174)(160 173)(161 172)(162 171)(163 170)(164 169)(165 168)(181 199)(182 198)(183 197)(184 196)(185 210)(186 209)(187 208)(188 207)(189 206)(190 205)(191 204)(192 203)(193 202)(194 201)(195 200)(211 237)(212 236)(213 235)(214 234)(215 233)(216 232)(217 231)(218 230)(219 229)(220 228)(221 227)(222 226)(223 240)(224 239)(225 238)
G:=sub<Sym(240)| (1,167,24,152)(2,168,25,153)(3,169,26,154)(4,170,27,155)(5,171,28,156)(6,172,29,157)(7,173,30,158)(8,174,16,159)(9,175,17,160)(10,176,18,161)(11,177,19,162)(12,178,20,163)(13,179,21,164)(14,180,22,165)(15,166,23,151)(31,148,54,132)(32,149,55,133)(33,150,56,134)(34,136,57,135)(35,137,58,121)(36,138,59,122)(37,139,60,123)(38,140,46,124)(39,141,47,125)(40,142,48,126)(41,143,49,127)(42,144,50,128)(43,145,51,129)(44,146,52,130)(45,147,53,131)(61,233,87,222)(62,234,88,223)(63,235,89,224)(64,236,90,225)(65,237,76,211)(66,238,77,212)(67,239,78,213)(68,240,79,214)(69,226,80,215)(70,227,81,216)(71,228,82,217)(72,229,83,218)(73,230,84,219)(74,231,85,220)(75,232,86,221)(91,203,109,184)(92,204,110,185)(93,205,111,186)(94,206,112,187)(95,207,113,188)(96,208,114,189)(97,209,115,190)(98,210,116,191)(99,196,117,192)(100,197,118,193)(101,198,119,194)(102,199,120,195)(103,200,106,181)(104,201,107,182)(105,202,108,183), (1,103,35,73)(2,104,36,74)(3,105,37,75)(4,91,38,61)(5,92,39,62)(6,93,40,63)(7,94,41,64)(8,95,42,65)(9,96,43,66)(10,97,44,67)(11,98,45,68)(12,99,31,69)(13,100,32,70)(14,101,33,71)(15,102,34,72)(16,113,50,76)(17,114,51,77)(18,115,52,78)(19,116,53,79)(20,117,54,80)(21,118,55,81)(22,119,56,82)(23,120,57,83)(24,106,58,84)(25,107,59,85)(26,108,60,86)(27,109,46,87)(28,110,47,88)(29,111,48,89)(30,112,49,90)(121,219,152,181)(122,220,153,182)(123,221,154,183)(124,222,155,184)(125,223,156,185)(126,224,157,186)(127,225,158,187)(128,211,159,188)(129,212,160,189)(130,213,161,190)(131,214,162,191)(132,215,163,192)(133,216,164,193)(134,217,165,194)(135,218,151,195)(136,229,166,199)(137,230,167,200)(138,231,168,201)(139,232,169,202)(140,233,170,203)(141,234,171,204)(142,235,172,205)(143,236,173,206)(144,237,174,207)(145,238,175,208)(146,239,176,209)(147,240,177,210)(148,226,178,196)(149,227,179,197)(150,228,180,198), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,120)(104,119)(105,118)(121,136)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(151,167)(152,166)(153,180)(154,179)(155,178)(156,177)(157,176)(158,175)(159,174)(160,173)(161,172)(162,171)(163,170)(164,169)(165,168)(181,199)(182,198)(183,197)(184,196)(185,210)(186,209)(187,208)(188,207)(189,206)(190,205)(191,204)(192,203)(193,202)(194,201)(195,200)(211,237)(212,236)(213,235)(214,234)(215,233)(216,232)(217,231)(218,230)(219,229)(220,228)(221,227)(222,226)(223,240)(224,239)(225,238)>;
G:=Group( (1,167,24,152)(2,168,25,153)(3,169,26,154)(4,170,27,155)(5,171,28,156)(6,172,29,157)(7,173,30,158)(8,174,16,159)(9,175,17,160)(10,176,18,161)(11,177,19,162)(12,178,20,163)(13,179,21,164)(14,180,22,165)(15,166,23,151)(31,148,54,132)(32,149,55,133)(33,150,56,134)(34,136,57,135)(35,137,58,121)(36,138,59,122)(37,139,60,123)(38,140,46,124)(39,141,47,125)(40,142,48,126)(41,143,49,127)(42,144,50,128)(43,145,51,129)(44,146,52,130)(45,147,53,131)(61,233,87,222)(62,234,88,223)(63,235,89,224)(64,236,90,225)(65,237,76,211)(66,238,77,212)(67,239,78,213)(68,240,79,214)(69,226,80,215)(70,227,81,216)(71,228,82,217)(72,229,83,218)(73,230,84,219)(74,231,85,220)(75,232,86,221)(91,203,109,184)(92,204,110,185)(93,205,111,186)(94,206,112,187)(95,207,113,188)(96,208,114,189)(97,209,115,190)(98,210,116,191)(99,196,117,192)(100,197,118,193)(101,198,119,194)(102,199,120,195)(103,200,106,181)(104,201,107,182)(105,202,108,183), (1,103,35,73)(2,104,36,74)(3,105,37,75)(4,91,38,61)(5,92,39,62)(6,93,40,63)(7,94,41,64)(8,95,42,65)(9,96,43,66)(10,97,44,67)(11,98,45,68)(12,99,31,69)(13,100,32,70)(14,101,33,71)(15,102,34,72)(16,113,50,76)(17,114,51,77)(18,115,52,78)(19,116,53,79)(20,117,54,80)(21,118,55,81)(22,119,56,82)(23,120,57,83)(24,106,58,84)(25,107,59,85)(26,108,60,86)(27,109,46,87)(28,110,47,88)(29,111,48,89)(30,112,49,90)(121,219,152,181)(122,220,153,182)(123,221,154,183)(124,222,155,184)(125,223,156,185)(126,224,157,186)(127,225,158,187)(128,211,159,188)(129,212,160,189)(130,213,161,190)(131,214,162,191)(132,215,163,192)(133,216,164,193)(134,217,165,194)(135,218,151,195)(136,229,166,199)(137,230,167,200)(138,231,168,201)(139,232,169,202)(140,233,170,203)(141,234,171,204)(142,235,172,205)(143,236,173,206)(144,237,174,207)(145,238,175,208)(146,239,176,209)(147,240,177,210)(148,226,178,196)(149,227,179,197)(150,228,180,198), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(31,46)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,80)(62,79)(63,78)(64,77)(65,76)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,120)(104,119)(105,118)(121,136)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(151,167)(152,166)(153,180)(154,179)(155,178)(156,177)(157,176)(158,175)(159,174)(160,173)(161,172)(162,171)(163,170)(164,169)(165,168)(181,199)(182,198)(183,197)(184,196)(185,210)(186,209)(187,208)(188,207)(189,206)(190,205)(191,204)(192,203)(193,202)(194,201)(195,200)(211,237)(212,236)(213,235)(214,234)(215,233)(216,232)(217,231)(218,230)(219,229)(220,228)(221,227)(222,226)(223,240)(224,239)(225,238) );
G=PermutationGroup([[(1,167,24,152),(2,168,25,153),(3,169,26,154),(4,170,27,155),(5,171,28,156),(6,172,29,157),(7,173,30,158),(8,174,16,159),(9,175,17,160),(10,176,18,161),(11,177,19,162),(12,178,20,163),(13,179,21,164),(14,180,22,165),(15,166,23,151),(31,148,54,132),(32,149,55,133),(33,150,56,134),(34,136,57,135),(35,137,58,121),(36,138,59,122),(37,139,60,123),(38,140,46,124),(39,141,47,125),(40,142,48,126),(41,143,49,127),(42,144,50,128),(43,145,51,129),(44,146,52,130),(45,147,53,131),(61,233,87,222),(62,234,88,223),(63,235,89,224),(64,236,90,225),(65,237,76,211),(66,238,77,212),(67,239,78,213),(68,240,79,214),(69,226,80,215),(70,227,81,216),(71,228,82,217),(72,229,83,218),(73,230,84,219),(74,231,85,220),(75,232,86,221),(91,203,109,184),(92,204,110,185),(93,205,111,186),(94,206,112,187),(95,207,113,188),(96,208,114,189),(97,209,115,190),(98,210,116,191),(99,196,117,192),(100,197,118,193),(101,198,119,194),(102,199,120,195),(103,200,106,181),(104,201,107,182),(105,202,108,183)], [(1,103,35,73),(2,104,36,74),(3,105,37,75),(4,91,38,61),(5,92,39,62),(6,93,40,63),(7,94,41,64),(8,95,42,65),(9,96,43,66),(10,97,44,67),(11,98,45,68),(12,99,31,69),(13,100,32,70),(14,101,33,71),(15,102,34,72),(16,113,50,76),(17,114,51,77),(18,115,52,78),(19,116,53,79),(20,117,54,80),(21,118,55,81),(22,119,56,82),(23,120,57,83),(24,106,58,84),(25,107,59,85),(26,108,60,86),(27,109,46,87),(28,110,47,88),(29,111,48,89),(30,112,49,90),(121,219,152,181),(122,220,153,182),(123,221,154,183),(124,222,155,184),(125,223,156,185),(126,224,157,186),(127,225,158,187),(128,211,159,188),(129,212,160,189),(130,213,161,190),(131,214,162,191),(132,215,163,192),(133,216,164,193),(134,217,165,194),(135,218,151,195),(136,229,166,199),(137,230,167,200),(138,231,168,201),(139,232,169,202),(140,233,170,203),(141,234,171,204),(142,235,172,205),(143,236,173,206),(144,237,174,207),(145,238,175,208),(146,239,176,209),(147,240,177,210),(148,226,178,196),(149,227,179,197),(150,228,180,198)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(31,46),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,80),(62,79),(63,78),(64,77),(65,76),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,120),(104,119),(105,118),(121,136),(122,150),(123,149),(124,148),(125,147),(126,146),(127,145),(128,144),(129,143),(130,142),(131,141),(132,140),(133,139),(134,138),(135,137),(151,167),(152,166),(153,180),(154,179),(155,178),(156,177),(157,176),(158,175),(159,174),(160,173),(161,172),(162,171),(163,170),(164,169),(165,168),(181,199),(182,198),(183,197),(184,196),(185,210),(186,209),(187,208),(188,207),(189,206),(190,205),(191,204),(192,203),(193,202),(194,201),(195,200),(211,237),(212,236),(213,235),(214,234),(215,233),(216,232),(217,231),(218,230),(219,229),(220,228),(221,227),(222,226),(223,240),(224,239),(225,238)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 1 | ··· | 1 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 |
| kernel | C42×D15 | C4×Dic15 | C4×C60 | C2×C4×D15 | C4×D15 | C4×C20 | C4×C12 | C2×C20 | C2×C12 | C20 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 24 | 1 | 2 | 3 | 6 | 12 | 4 | 24 | 12 | 48 |
Matrix representation of C42×D15 ►in GL4(𝔽61) generated by
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 17 | 43 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 25 | 30 |
| 0 | 0 | 31 | 47 |
| 36 | 9 | 0 | 0 |
| 12 | 25 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [11,0,0,0,0,11,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,11],[17,17,0,0,43,0,0,0,0,0,25,31,0,0,30,47],[36,12,0,0,9,25,0,0,0,0,0,1,0,0,1,0] >;
C42×D15 in GAP, Magma, Sage, TeX
C_4^2\times D_{15} % in TeX
G:=Group("C4^2xD15"); // GroupNames label
G:=SmallGroup(480,836);
// by ID
G=gap.SmallGroup(480,836);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations