direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D6⋊C8, D6⋊C40, C20.71D12, C60.237D4, C30.33M4(2), (C2×C40)⋊1S3, (S3×C10)⋊5C8, (C2×C24)⋊1C10, (C2×C120)⋊3C2, C2.5(S3×C40), C6.5(C2×C40), C10.29(S3×C8), C30.57(C2×C8), C4.19(C5×D12), C12.52(C5×D4), C15⋊12(C22⋊C8), (C2×C20).448D6, C6.3(C5×M4(2)), C10.53(D6⋊C4), (C22×S3).2C20, C22.11(S3×C20), (C2×Dic3).4C20, C10.17(C8⋊S3), C20.120(C3⋊D4), C30.95(C22⋊C4), (C2×C60).560C22, (C10×Dic3).21C4, (C2×C3⋊C8)⋊9C10, (C2×C8)⋊1(C5×S3), C3⋊1(C5×C22⋊C8), (C10×C3⋊C8)⋊23C2, (S3×C2×C4).7C10, C2.1(C5×D6⋊C4), C2.3(C5×C8⋊S3), (S3×C2×C20).18C2, (S3×C2×C10).12C4, C4.27(C5×C3⋊D4), C6.6(C5×C22⋊C4), (C2×C4).95(S3×C10), (C2×C6).12(C2×C20), (C2×C10).83(C4×S3), (C2×C30).157(C2×C4), (C2×C12).112(C2×C10), SmallGroup(480,139)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D6⋊C8
G = < a,b,c,d | a5=b6=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 228 in 100 conjugacy classes, 50 normal (46 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C22⋊C8, C40, C2×C20, C2×C20, C22×C10, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C40, C2×C40, C22×C20, D6⋊C8, C5×C3⋊C8, C120, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C5×C22⋊C8, C10×C3⋊C8, C2×C120, S3×C2×C20, C5×D6⋊C8
Quotients: C1, C2, C4, C22, C5, S3, C8, C2×C4, D4, C10, D6, C22⋊C4, C2×C8, M4(2), C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C22⋊C8, C40, C2×C20, C5×D4, S3×C8, C8⋊S3, D6⋊C4, S3×C10, C5×C22⋊C4, C2×C40, C5×M4(2), D6⋊C8, S3×C20, C5×D12, C5×C3⋊D4, C5×C22⋊C8, S3×C40, C5×C8⋊S3, C5×D6⋊C4, C5×D6⋊C8
►On 240 points
Generators in S240
(1 12 119 224 95)(2 13 120 217 96)(3 14 113 218 89)(4 15 114 219 90)(5 16 115 220 91)(6 9 116 221 92)(7 10 117 222 93)(8 11 118 223 94)(17 30 45 154 148)(18 31 46 155 149)(19 32 47 156 150)(20 25 48 157 151)(21 26 41 158 152)(22 27 42 159 145)(23 28 43 160 146)(24 29 44 153 147)(33 144 192 57 168)(34 137 185 58 161)(35 138 186 59 162)(36 139 187 60 163)(37 140 188 61 164)(38 141 189 62 165)(39 142 190 63 166)(40 143 191 64 167)(49 70 175 193 177)(50 71 176 194 178)(51 72 169 195 179)(52 65 170 196 180)(53 66 171 197 181)(54 67 172 198 182)(55 68 173 199 183)(56 69 174 200 184)(73 121 225 97 201)(74 122 226 98 202)(75 123 227 99 203)(76 124 228 100 204)(77 125 229 101 205)(78 126 230 102 206)(79 127 231 103 207)(80 128 232 104 208)(81 136 240 105 216)(82 129 233 106 209)(83 130 234 107 210)(84 131 235 108 211)(85 132 236 109 212)(86 133 237 110 213)(87 134 238 111 214)(88 135 239 112 215)
(1 88 45 52 139 74)(2 81 46 53 140 75)(3 82 47 54 141 76)(4 83 48 55 142 77)(5 84 41 56 143 78)(6 85 42 49 144 79)(7 86 43 50 137 80)(8 87 44 51 138 73)(9 132 159 70 192 127)(10 133 160 71 185 128)(11 134 153 72 186 121)(12 135 154 65 187 122)(13 136 155 66 188 123)(14 129 156 67 189 124)(15 130 157 68 190 125)(16 131 158 69 191 126)(17 196 163 98 224 112)(18 197 164 99 217 105)(19 198 165 100 218 106)(20 199 166 101 219 107)(21 200 167 102 220 108)(22 193 168 103 221 109)(23 194 161 104 222 110)(24 195 162 97 223 111)(25 183 39 205 90 210)(26 184 40 206 91 211)(27 177 33 207 92 212)(28 178 34 208 93 213)(29 179 35 201 94 214)(30 180 36 202 95 215)(31 181 37 203 96 216)(32 182 38 204 89 209)(57 231 116 236 145 175)(58 232 117 237 146 176)(59 225 118 238 147 169)(60 226 119 239 148 170)(61 227 120 240 149 171)(62 228 113 233 150 172)(63 229 114 234 151 173)(64 230 115 235 152 174)
(1 74)(2 46)(3 76)(4 48)(5 78)(6 42)(7 80)(8 44)(9 159)(10 128)(11 153)(12 122)(13 155)(14 124)(15 157)(16 126)(17 196)(18 217)(19 198)(20 219)(21 200)(22 221)(23 194)(24 223)(25 90)(26 184)(27 92)(28 178)(29 94)(30 180)(31 96)(32 182)(34 213)(36 215)(38 209)(40 211)(41 56)(43 50)(45 52)(47 54)(49 79)(51 73)(53 75)(55 77)(58 237)(60 239)(62 233)(64 235)(65 154)(66 123)(67 156)(68 125)(69 158)(70 127)(71 160)(72 121)(82 141)(84 143)(86 137)(88 139)(89 204)(91 206)(93 208)(95 202)(97 195)(98 224)(99 197)(100 218)(101 199)(102 220)(103 193)(104 222)(106 165)(108 167)(110 161)(112 163)(113 228)(114 151)(115 230)(116 145)(117 232)(118 147)(119 226)(120 149)(129 189)(131 191)(133 185)(135 187)(146 176)(148 170)(150 172)(152 174)(169 225)(171 227)(173 229)(175 231)(177 207)(179 201)(181 203)(183 205)
(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)
G:=sub<Sym(240)| (1,12,119,224,95)(2,13,120,217,96)(3,14,113,218,89)(4,15,114,219,90)(5,16,115,220,91)(6,9,116,221,92)(7,10,117,222,93)(8,11,118,223,94)(17,30,45,154,148)(18,31,46,155,149)(19,32,47,156,150)(20,25,48,157,151)(21,26,41,158,152)(22,27,42,159,145)(23,28,43,160,146)(24,29,44,153,147)(33,144,192,57,168)(34,137,185,58,161)(35,138,186,59,162)(36,139,187,60,163)(37,140,188,61,164)(38,141,189,62,165)(39,142,190,63,166)(40,143,191,64,167)(49,70,175,193,177)(50,71,176,194,178)(51,72,169,195,179)(52,65,170,196,180)(53,66,171,197,181)(54,67,172,198,182)(55,68,173,199,183)(56,69,174,200,184)(73,121,225,97,201)(74,122,226,98,202)(75,123,227,99,203)(76,124,228,100,204)(77,125,229,101,205)(78,126,230,102,206)(79,127,231,103,207)(80,128,232,104,208)(81,136,240,105,216)(82,129,233,106,209)(83,130,234,107,210)(84,131,235,108,211)(85,132,236,109,212)(86,133,237,110,213)(87,134,238,111,214)(88,135,239,112,215), (1,88,45,52,139,74)(2,81,46,53,140,75)(3,82,47,54,141,76)(4,83,48,55,142,77)(5,84,41,56,143,78)(6,85,42,49,144,79)(7,86,43,50,137,80)(8,87,44,51,138,73)(9,132,159,70,192,127)(10,133,160,71,185,128)(11,134,153,72,186,121)(12,135,154,65,187,122)(13,136,155,66,188,123)(14,129,156,67,189,124)(15,130,157,68,190,125)(16,131,158,69,191,126)(17,196,163,98,224,112)(18,197,164,99,217,105)(19,198,165,100,218,106)(20,199,166,101,219,107)(21,200,167,102,220,108)(22,193,168,103,221,109)(23,194,161,104,222,110)(24,195,162,97,223,111)(25,183,39,205,90,210)(26,184,40,206,91,211)(27,177,33,207,92,212)(28,178,34,208,93,213)(29,179,35,201,94,214)(30,180,36,202,95,215)(31,181,37,203,96,216)(32,182,38,204,89,209)(57,231,116,236,145,175)(58,232,117,237,146,176)(59,225,118,238,147,169)(60,226,119,239,148,170)(61,227,120,240,149,171)(62,228,113,233,150,172)(63,229,114,234,151,173)(64,230,115,235,152,174), (1,74)(2,46)(3,76)(4,48)(5,78)(6,42)(7,80)(8,44)(9,159)(10,128)(11,153)(12,122)(13,155)(14,124)(15,157)(16,126)(17,196)(18,217)(19,198)(20,219)(21,200)(22,221)(23,194)(24,223)(25,90)(26,184)(27,92)(28,178)(29,94)(30,180)(31,96)(32,182)(34,213)(36,215)(38,209)(40,211)(41,56)(43,50)(45,52)(47,54)(49,79)(51,73)(53,75)(55,77)(58,237)(60,239)(62,233)(64,235)(65,154)(66,123)(67,156)(68,125)(69,158)(70,127)(71,160)(72,121)(82,141)(84,143)(86,137)(88,139)(89,204)(91,206)(93,208)(95,202)(97,195)(98,224)(99,197)(100,218)(101,199)(102,220)(103,193)(104,222)(106,165)(108,167)(110,161)(112,163)(113,228)(114,151)(115,230)(116,145)(117,232)(118,147)(119,226)(120,149)(129,189)(131,191)(133,185)(135,187)(146,176)(148,170)(150,172)(152,174)(169,225)(171,227)(173,229)(175,231)(177,207)(179,201)(181,203)(183,205), (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)>;
G:=Group( (1,12,119,224,95)(2,13,120,217,96)(3,14,113,218,89)(4,15,114,219,90)(5,16,115,220,91)(6,9,116,221,92)(7,10,117,222,93)(8,11,118,223,94)(17,30,45,154,148)(18,31,46,155,149)(19,32,47,156,150)(20,25,48,157,151)(21,26,41,158,152)(22,27,42,159,145)(23,28,43,160,146)(24,29,44,153,147)(33,144,192,57,168)(34,137,185,58,161)(35,138,186,59,162)(36,139,187,60,163)(37,140,188,61,164)(38,141,189,62,165)(39,142,190,63,166)(40,143,191,64,167)(49,70,175,193,177)(50,71,176,194,178)(51,72,169,195,179)(52,65,170,196,180)(53,66,171,197,181)(54,67,172,198,182)(55,68,173,199,183)(56,69,174,200,184)(73,121,225,97,201)(74,122,226,98,202)(75,123,227,99,203)(76,124,228,100,204)(77,125,229,101,205)(78,126,230,102,206)(79,127,231,103,207)(80,128,232,104,208)(81,136,240,105,216)(82,129,233,106,209)(83,130,234,107,210)(84,131,235,108,211)(85,132,236,109,212)(86,133,237,110,213)(87,134,238,111,214)(88,135,239,112,215), (1,88,45,52,139,74)(2,81,46,53,140,75)(3,82,47,54,141,76)(4,83,48,55,142,77)(5,84,41,56,143,78)(6,85,42,49,144,79)(7,86,43,50,137,80)(8,87,44,51,138,73)(9,132,159,70,192,127)(10,133,160,71,185,128)(11,134,153,72,186,121)(12,135,154,65,187,122)(13,136,155,66,188,123)(14,129,156,67,189,124)(15,130,157,68,190,125)(16,131,158,69,191,126)(17,196,163,98,224,112)(18,197,164,99,217,105)(19,198,165,100,218,106)(20,199,166,101,219,107)(21,200,167,102,220,108)(22,193,168,103,221,109)(23,194,161,104,222,110)(24,195,162,97,223,111)(25,183,39,205,90,210)(26,184,40,206,91,211)(27,177,33,207,92,212)(28,178,34,208,93,213)(29,179,35,201,94,214)(30,180,36,202,95,215)(31,181,37,203,96,216)(32,182,38,204,89,209)(57,231,116,236,145,175)(58,232,117,237,146,176)(59,225,118,238,147,169)(60,226,119,239,148,170)(61,227,120,240,149,171)(62,228,113,233,150,172)(63,229,114,234,151,173)(64,230,115,235,152,174), (1,74)(2,46)(3,76)(4,48)(5,78)(6,42)(7,80)(8,44)(9,159)(10,128)(11,153)(12,122)(13,155)(14,124)(15,157)(16,126)(17,196)(18,217)(19,198)(20,219)(21,200)(22,221)(23,194)(24,223)(25,90)(26,184)(27,92)(28,178)(29,94)(30,180)(31,96)(32,182)(34,213)(36,215)(38,209)(40,211)(41,56)(43,50)(45,52)(47,54)(49,79)(51,73)(53,75)(55,77)(58,237)(60,239)(62,233)(64,235)(65,154)(66,123)(67,156)(68,125)(69,158)(70,127)(71,160)(72,121)(82,141)(84,143)(86,137)(88,139)(89,204)(91,206)(93,208)(95,202)(97,195)(98,224)(99,197)(100,218)(101,199)(102,220)(103,193)(104,222)(106,165)(108,167)(110,161)(112,163)(113,228)(114,151)(115,230)(116,145)(117,232)(118,147)(119,226)(120,149)(129,189)(131,191)(133,185)(135,187)(146,176)(148,170)(150,172)(152,174)(169,225)(171,227)(173,229)(175,231)(177,207)(179,201)(181,203)(183,205), (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) );
G=PermutationGroup([[(1,12,119,224,95),(2,13,120,217,96),(3,14,113,218,89),(4,15,114,219,90),(5,16,115,220,91),(6,9,116,221,92),(7,10,117,222,93),(8,11,118,223,94),(17,30,45,154,148),(18,31,46,155,149),(19,32,47,156,150),(20,25,48,157,151),(21,26,41,158,152),(22,27,42,159,145),(23,28,43,160,146),(24,29,44,153,147),(33,144,192,57,168),(34,137,185,58,161),(35,138,186,59,162),(36,139,187,60,163),(37,140,188,61,164),(38,141,189,62,165),(39,142,190,63,166),(40,143,191,64,167),(49,70,175,193,177),(50,71,176,194,178),(51,72,169,195,179),(52,65,170,196,180),(53,66,171,197,181),(54,67,172,198,182),(55,68,173,199,183),(56,69,174,200,184),(73,121,225,97,201),(74,122,226,98,202),(75,123,227,99,203),(76,124,228,100,204),(77,125,229,101,205),(78,126,230,102,206),(79,127,231,103,207),(80,128,232,104,208),(81,136,240,105,216),(82,129,233,106,209),(83,130,234,107,210),(84,131,235,108,211),(85,132,236,109,212),(86,133,237,110,213),(87,134,238,111,214),(88,135,239,112,215)], [(1,88,45,52,139,74),(2,81,46,53,140,75),(3,82,47,54,141,76),(4,83,48,55,142,77),(5,84,41,56,143,78),(6,85,42,49,144,79),(7,86,43,50,137,80),(8,87,44,51,138,73),(9,132,159,70,192,127),(10,133,160,71,185,128),(11,134,153,72,186,121),(12,135,154,65,187,122),(13,136,155,66,188,123),(14,129,156,67,189,124),(15,130,157,68,190,125),(16,131,158,69,191,126),(17,196,163,98,224,112),(18,197,164,99,217,105),(19,198,165,100,218,106),(20,199,166,101,219,107),(21,200,167,102,220,108),(22,193,168,103,221,109),(23,194,161,104,222,110),(24,195,162,97,223,111),(25,183,39,205,90,210),(26,184,40,206,91,211),(27,177,33,207,92,212),(28,178,34,208,93,213),(29,179,35,201,94,214),(30,180,36,202,95,215),(31,181,37,203,96,216),(32,182,38,204,89,209),(57,231,116,236,145,175),(58,232,117,237,146,176),(59,225,118,238,147,169),(60,226,119,239,148,170),(61,227,120,240,149,171),(62,228,113,233,150,172),(63,229,114,234,151,173),(64,230,115,235,152,174)], [(1,74),(2,46),(3,76),(4,48),(5,78),(6,42),(7,80),(8,44),(9,159),(10,128),(11,153),(12,122),(13,155),(14,124),(15,157),(16,126),(17,196),(18,217),(19,198),(20,219),(21,200),(22,221),(23,194),(24,223),(25,90),(26,184),(27,92),(28,178),(29,94),(30,180),(31,96),(32,182),(34,213),(36,215),(38,209),(40,211),(41,56),(43,50),(45,52),(47,54),(49,79),(51,73),(53,75),(55,77),(58,237),(60,239),(62,233),(64,235),(65,154),(66,123),(67,156),(68,125),(69,158),(70,127),(71,160),(72,121),(82,141),(84,143),(86,137),(88,139),(89,204),(91,206),(93,208),(95,202),(97,195),(98,224),(99,197),(100,218),(101,199),(102,220),(103,193),(104,222),(106,165),(108,167),(110,161),(112,163),(113,228),(114,151),(115,230),(116,145),(117,232),(118,147),(119,226),(120,149),(129,189),(131,191),(133,185),(135,187),(146,176),(148,170),(150,172),(152,174),(169,225),(171,227),(173,229),(175,231),(177,207),(179,201),(181,203),(183,205)], [(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)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 40Q | ··· | 40AF | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | C5×S3 | C5×D4 | S3×C8 | C8⋊S3 | S3×C10 | C5×M4(2) | C5×D12 | C5×C3⋊D4 | S3×C20 | S3×C40 | C5×C8⋊S3 |
| kernel | C5×D6⋊C8 | C10×C3⋊C8 | C2×C120 | S3×C2×C20 | C10×Dic3 | S3×C2×C10 | D6⋊C8 | S3×C10 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | C2×Dic3 | C22×S3 | D6 | C2×C40 | C60 | C2×C20 | C30 | C20 | C20 | C2×C10 | C2×C8 | C12 | C10 | C10 | C2×C4 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 32 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C5×D6⋊C8 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 0 | 1 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 172 | 103 |
| 0 | 0 | 138 | 69 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,91,0,0,0,0,91],[0,240,0,0,1,240,0,0,0,0,0,1,0,0,240,1],[0,1,0,0,1,0,0,0,0,0,0,240,0,0,240,0],[30,0,0,0,0,30,0,0,0,0,172,138,0,0,103,69] >;
C5×D6⋊C8 in GAP, Magma, Sage, TeX
C_5\times D_6\rtimes C_8
% in TeX
G:=Group("C5xD6:C8"); // GroupNames label
G:=SmallGroup(480,139);
// by ID
G=gap.SmallGroup(480,139);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,136,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^6=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations