direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C8⋊S3, C40⋊35D6, C120⋊48C22, C30⋊12M4(2), C60.284C23, C8⋊9(S3×C10), (C2×C40)⋊14S3, (C2×C120)⋊27C2, (C2×C24)⋊11C10, C24⋊12(C2×C10), (C4×S3).3C20, C4.24(S3×C20), D6.5(C2×C20), C6⋊1(C5×M4(2)), C3⋊1(C10×M4(2)), (S3×C20).14C4, C20.126(C4×S3), C12.28(C2×C20), C60.224(C2×C4), (C2×C20).452D6, C15⋊25(C2×M4(2)), (C22×S3).3C20, C22.14(S3×C20), C6.13(C22×C20), Dic3.7(C2×C20), (C2×Dic3).5C20, (S3×C20).63C22, C12.36(C22×C10), C20.242(C22×S3), (C2×C60).564C22, C30.204(C22×C4), (C10×Dic3).22C4, (C2×C8)⋊6(C5×S3), (C2×C3⋊C8)⋊11C10, (C10×C3⋊C8)⋊25C2, C3⋊C8⋊10(C2×C10), C2.14(S3×C2×C20), C4.36(S3×C2×C10), (C5×C3⋊C8)⋊43C22, (S3×C2×C10).13C4, (S3×C2×C4).10C10, (S3×C2×C20).23C2, C10.140(S3×C2×C4), (C2×C10).86(C4×S3), (C2×C6).15(C2×C20), (C2×C4).99(S3×C10), (C4×S3).14(C2×C10), (S3×C10).41(C2×C4), (C2×C30).160(C2×C4), (C2×C12).117(C2×C10), (C5×Dic3).49(C2×C4), SmallGroup(480,779)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C8⋊S3
G = < a,b,c,d | a10=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 260 in 136 conjugacy classes, 82 normal (38 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C2×M4(2), C40, C40, C2×C20, C2×C20, C22×C10, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C40, C2×C40, C5×M4(2), C22×C20, C2×C8⋊S3, C5×C3⋊C8, C120, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C10×M4(2), C5×C8⋊S3, C10×C3⋊C8, C2×C120, S3×C2×C20, C10×C8⋊S3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, M4(2), C22×C4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C2×M4(2), C2×C20, C22×C10, C8⋊S3, S3×C2×C4, S3×C10, C5×M4(2), C22×C20, C2×C8⋊S3, S3×C20, S3×C2×C10, C10×M4(2), C5×C8⋊S3, S3×C2×C20, C10×C8⋊S3
►On 240 points
Generators in S240
(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 13 211 139 158 182 207 114)(2 14 212 140 159 183 208 115)(3 15 213 131 160 184 209 116)(4 16 214 132 151 185 210 117)(5 17 215 133 152 186 201 118)(6 18 216 134 153 187 202 119)(7 19 217 135 154 188 203 120)(8 20 218 136 155 189 204 111)(9 11 219 137 156 190 205 112)(10 12 220 138 157 181 206 113)(21 225 64 101 58 126 84 38)(22 226 65 102 59 127 85 39)(23 227 66 103 60 128 86 40)(24 228 67 104 51 129 87 31)(25 229 68 105 52 130 88 32)(26 230 69 106 53 121 89 33)(27 221 70 107 54 122 90 34)(28 222 61 108 55 123 81 35)(29 223 62 109 56 124 82 36)(30 224 63 110 57 125 83 37)(41 237 76 95 192 175 141 170)(42 238 77 96 193 176 142 161)(43 239 78 97 194 177 143 162)(44 240 79 98 195 178 144 163)(45 231 80 99 196 179 145 164)(46 232 71 100 197 180 146 165)(47 233 72 91 198 171 147 166)(48 234 73 92 199 172 148 167)(49 235 74 93 200 173 149 168)(50 236 75 94 191 174 150 169)
(1 177 28)(2 178 29)(3 179 30)(4 180 21)(5 171 22)(6 172 23)(7 173 24)(8 174 25)(9 175 26)(10 176 27)(11 141 230)(12 142 221)(13 143 222)(14 144 223)(15 145 224)(16 146 225)(17 147 226)(18 148 227)(19 149 228)(20 150 229)(31 120 200)(32 111 191)(33 112 192)(34 113 193)(35 114 194)(36 115 195)(37 116 196)(38 117 197)(39 118 198)(40 119 199)(41 106 137)(42 107 138)(43 108 139)(44 109 140)(45 110 131)(46 101 132)(47 102 133)(48 103 134)(49 104 135)(50 105 136)(51 154 235)(52 155 236)(53 156 237)(54 157 238)(55 158 239)(56 159 240)(57 160 231)(58 151 232)(59 152 233)(60 153 234)(61 211 162)(62 212 163)(63 213 164)(64 214 165)(65 215 166)(66 216 167)(67 217 168)(68 218 169)(69 219 170)(70 220 161)(71 126 185)(72 127 186)(73 128 187)(74 129 188)(75 130 189)(76 121 190)(77 122 181)(78 123 182)(79 124 183)(80 125 184)(81 207 97)(82 208 98)(83 209 99)(84 210 100)(85 201 91)(86 202 92)(87 203 93)(88 204 94)(89 205 95)(90 206 96)
(11 190)(12 181)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 188)(20 189)(21 180)(22 171)(23 172)(24 173)(25 174)(26 175)(27 176)(28 177)(29 178)(30 179)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 235)(52 236)(53 237)(54 238)(55 239)(56 240)(57 231)(58 232)(59 233)(60 234)(61 162)(62 163)(63 164)(64 165)(65 166)(66 167)(67 168)(68 169)(69 170)(70 161)(71 225)(72 226)(73 227)(74 228)(75 229)(76 230)(77 221)(78 222)(79 223)(80 224)(81 97)(82 98)(83 99)(84 100)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(101 197)(102 198)(103 199)(104 200)(105 191)(106 192)(107 193)(108 194)(109 195)(110 196)(111 136)(112 137)(113 138)(114 139)(115 140)(116 131)(117 132)(118 133)(119 134)(120 135)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)
G:=sub<Sym(240)| (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), (1,13,211,139,158,182,207,114)(2,14,212,140,159,183,208,115)(3,15,213,131,160,184,209,116)(4,16,214,132,151,185,210,117)(5,17,215,133,152,186,201,118)(6,18,216,134,153,187,202,119)(7,19,217,135,154,188,203,120)(8,20,218,136,155,189,204,111)(9,11,219,137,156,190,205,112)(10,12,220,138,157,181,206,113)(21,225,64,101,58,126,84,38)(22,226,65,102,59,127,85,39)(23,227,66,103,60,128,86,40)(24,228,67,104,51,129,87,31)(25,229,68,105,52,130,88,32)(26,230,69,106,53,121,89,33)(27,221,70,107,54,122,90,34)(28,222,61,108,55,123,81,35)(29,223,62,109,56,124,82,36)(30,224,63,110,57,125,83,37)(41,237,76,95,192,175,141,170)(42,238,77,96,193,176,142,161)(43,239,78,97,194,177,143,162)(44,240,79,98,195,178,144,163)(45,231,80,99,196,179,145,164)(46,232,71,100,197,180,146,165)(47,233,72,91,198,171,147,166)(48,234,73,92,199,172,148,167)(49,235,74,93,200,173,149,168)(50,236,75,94,191,174,150,169), (1,177,28)(2,178,29)(3,179,30)(4,180,21)(5,171,22)(6,172,23)(7,173,24)(8,174,25)(9,175,26)(10,176,27)(11,141,230)(12,142,221)(13,143,222)(14,144,223)(15,145,224)(16,146,225)(17,147,226)(18,148,227)(19,149,228)(20,150,229)(31,120,200)(32,111,191)(33,112,192)(34,113,193)(35,114,194)(36,115,195)(37,116,196)(38,117,197)(39,118,198)(40,119,199)(41,106,137)(42,107,138)(43,108,139)(44,109,140)(45,110,131)(46,101,132)(47,102,133)(48,103,134)(49,104,135)(50,105,136)(51,154,235)(52,155,236)(53,156,237)(54,157,238)(55,158,239)(56,159,240)(57,160,231)(58,151,232)(59,152,233)(60,153,234)(61,211,162)(62,212,163)(63,213,164)(64,214,165)(65,215,166)(66,216,167)(67,217,168)(68,218,169)(69,219,170)(70,220,161)(71,126,185)(72,127,186)(73,128,187)(74,129,188)(75,130,189)(76,121,190)(77,122,181)(78,123,182)(79,124,183)(80,125,184)(81,207,97)(82,208,98)(83,209,99)(84,210,100)(85,201,91)(86,202,92)(87,203,93)(88,204,94)(89,205,95)(90,206,96), (11,190)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,180)(22,171)(23,172)(24,173)(25,174)(26,175)(27,176)(28,177)(29,178)(30,179)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,235)(52,236)(53,237)(54,238)(55,239)(56,240)(57,231)(58,232)(59,233)(60,234)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,161)(71,225)(72,226)(73,227)(74,228)(75,229)(76,230)(77,221)(78,222)(79,223)(80,224)(81,97)(82,98)(83,99)(84,100)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(101,197)(102,198)(103,199)(104,200)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,136)(112,137)(113,138)(114,139)(115,140)(116,131)(117,132)(118,133)(119,134)(120,135)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)>;
G:=Group( (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), (1,13,211,139,158,182,207,114)(2,14,212,140,159,183,208,115)(3,15,213,131,160,184,209,116)(4,16,214,132,151,185,210,117)(5,17,215,133,152,186,201,118)(6,18,216,134,153,187,202,119)(7,19,217,135,154,188,203,120)(8,20,218,136,155,189,204,111)(9,11,219,137,156,190,205,112)(10,12,220,138,157,181,206,113)(21,225,64,101,58,126,84,38)(22,226,65,102,59,127,85,39)(23,227,66,103,60,128,86,40)(24,228,67,104,51,129,87,31)(25,229,68,105,52,130,88,32)(26,230,69,106,53,121,89,33)(27,221,70,107,54,122,90,34)(28,222,61,108,55,123,81,35)(29,223,62,109,56,124,82,36)(30,224,63,110,57,125,83,37)(41,237,76,95,192,175,141,170)(42,238,77,96,193,176,142,161)(43,239,78,97,194,177,143,162)(44,240,79,98,195,178,144,163)(45,231,80,99,196,179,145,164)(46,232,71,100,197,180,146,165)(47,233,72,91,198,171,147,166)(48,234,73,92,199,172,148,167)(49,235,74,93,200,173,149,168)(50,236,75,94,191,174,150,169), (1,177,28)(2,178,29)(3,179,30)(4,180,21)(5,171,22)(6,172,23)(7,173,24)(8,174,25)(9,175,26)(10,176,27)(11,141,230)(12,142,221)(13,143,222)(14,144,223)(15,145,224)(16,146,225)(17,147,226)(18,148,227)(19,149,228)(20,150,229)(31,120,200)(32,111,191)(33,112,192)(34,113,193)(35,114,194)(36,115,195)(37,116,196)(38,117,197)(39,118,198)(40,119,199)(41,106,137)(42,107,138)(43,108,139)(44,109,140)(45,110,131)(46,101,132)(47,102,133)(48,103,134)(49,104,135)(50,105,136)(51,154,235)(52,155,236)(53,156,237)(54,157,238)(55,158,239)(56,159,240)(57,160,231)(58,151,232)(59,152,233)(60,153,234)(61,211,162)(62,212,163)(63,213,164)(64,214,165)(65,215,166)(66,216,167)(67,217,168)(68,218,169)(69,219,170)(70,220,161)(71,126,185)(72,127,186)(73,128,187)(74,129,188)(75,130,189)(76,121,190)(77,122,181)(78,123,182)(79,124,183)(80,125,184)(81,207,97)(82,208,98)(83,209,99)(84,210,100)(85,201,91)(86,202,92)(87,203,93)(88,204,94)(89,205,95)(90,206,96), (11,190)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,180)(22,171)(23,172)(24,173)(25,174)(26,175)(27,176)(28,177)(29,178)(30,179)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,235)(52,236)(53,237)(54,238)(55,239)(56,240)(57,231)(58,232)(59,233)(60,234)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,161)(71,225)(72,226)(73,227)(74,228)(75,229)(76,230)(77,221)(78,222)(79,223)(80,224)(81,97)(82,98)(83,99)(84,100)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(101,197)(102,198)(103,199)(104,200)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,136)(112,137)(113,138)(114,139)(115,140)(116,131)(117,132)(118,133)(119,134)(120,135)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150) );
G=PermutationGroup([[(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)], [(1,13,211,139,158,182,207,114),(2,14,212,140,159,183,208,115),(3,15,213,131,160,184,209,116),(4,16,214,132,151,185,210,117),(5,17,215,133,152,186,201,118),(6,18,216,134,153,187,202,119),(7,19,217,135,154,188,203,120),(8,20,218,136,155,189,204,111),(9,11,219,137,156,190,205,112),(10,12,220,138,157,181,206,113),(21,225,64,101,58,126,84,38),(22,226,65,102,59,127,85,39),(23,227,66,103,60,128,86,40),(24,228,67,104,51,129,87,31),(25,229,68,105,52,130,88,32),(26,230,69,106,53,121,89,33),(27,221,70,107,54,122,90,34),(28,222,61,108,55,123,81,35),(29,223,62,109,56,124,82,36),(30,224,63,110,57,125,83,37),(41,237,76,95,192,175,141,170),(42,238,77,96,193,176,142,161),(43,239,78,97,194,177,143,162),(44,240,79,98,195,178,144,163),(45,231,80,99,196,179,145,164),(46,232,71,100,197,180,146,165),(47,233,72,91,198,171,147,166),(48,234,73,92,199,172,148,167),(49,235,74,93,200,173,149,168),(50,236,75,94,191,174,150,169)], [(1,177,28),(2,178,29),(3,179,30),(4,180,21),(5,171,22),(6,172,23),(7,173,24),(8,174,25),(9,175,26),(10,176,27),(11,141,230),(12,142,221),(13,143,222),(14,144,223),(15,145,224),(16,146,225),(17,147,226),(18,148,227),(19,149,228),(20,150,229),(31,120,200),(32,111,191),(33,112,192),(34,113,193),(35,114,194),(36,115,195),(37,116,196),(38,117,197),(39,118,198),(40,119,199),(41,106,137),(42,107,138),(43,108,139),(44,109,140),(45,110,131),(46,101,132),(47,102,133),(48,103,134),(49,104,135),(50,105,136),(51,154,235),(52,155,236),(53,156,237),(54,157,238),(55,158,239),(56,159,240),(57,160,231),(58,151,232),(59,152,233),(60,153,234),(61,211,162),(62,212,163),(63,213,164),(64,214,165),(65,215,166),(66,216,167),(67,217,168),(68,218,169),(69,219,170),(70,220,161),(71,126,185),(72,127,186),(73,128,187),(74,129,188),(75,130,189),(76,121,190),(77,122,181),(78,123,182),(79,124,183),(80,125,184),(81,207,97),(82,208,98),(83,209,99),(84,210,100),(85,201,91),(86,202,92),(87,203,93),(88,204,94),(89,205,95),(90,206,96)], [(11,190),(12,181),(13,182),(14,183),(15,184),(16,185),(17,186),(18,187),(19,188),(20,189),(21,180),(22,171),(23,172),(24,173),(25,174),(26,175),(27,176),(28,177),(29,178),(30,179),(31,49),(32,50),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(51,235),(52,236),(53,237),(54,238),(55,239),(56,240),(57,231),(58,232),(59,233),(60,234),(61,162),(62,163),(63,164),(64,165),(65,166),(66,167),(67,168),(68,169),(69,170),(70,161),(71,225),(72,226),(73,227),(74,228),(75,229),(76,230),(77,221),(78,222),(79,223),(80,224),(81,97),(82,98),(83,99),(84,100),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(101,197),(102,198),(103,199),(104,200),(105,191),(106,192),(107,193),(108,194),(109,195),(110,196),(111,136),(112,137),(113,138),(114,139),(115,140),(116,131),(117,132),(118,133),(119,134),(120,135),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | C5×S3 | C8⋊S3 | S3×C10 | S3×C10 | C5×M4(2) | S3×C20 | S3×C20 | C5×C8⋊S3 |
| kernel | C10×C8⋊S3 | C5×C8⋊S3 | C10×C3⋊C8 | C2×C120 | S3×C2×C20 | S3×C20 | C10×Dic3 | S3×C2×C10 | C2×C8⋊S3 | C8⋊S3 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | C2×C40 | C40 | C2×C20 | C30 | C20 | C2×C10 | C2×C8 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 16 | 4 | 4 | 4 | 16 | 8 | 8 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C10×C8⋊S3 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 | 0 |
| 0 | 0 | 87 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 69 | 138 |
| 0 | 0 | 0 | 103 | 172 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 240 | 240 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 | 240 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,69,103,0,0,0,138,172],[1,0,0,0,0,0,0,240,0,0,0,1,240,0,0,0,0,0,0,240,0,0,0,1,240],[240,0,0,0,0,0,1,240,0,0,0,0,240,0,0,0,0,0,1,240,0,0,0,0,240] >;
C10×C8⋊S3 in GAP, Magma, Sage, TeX
C_{10}\times C_8\rtimes S_3 % in TeX
G:=Group("C10xC8:S3"); // GroupNames label
G:=SmallGroup(480,779);
// by ID
G=gap.SmallGroup(480,779);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,226,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations