direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D8.S3, C40.57D6, C30.54D8, C15⋊12SD32, C60.137D4, Dic12⋊3C10, C120.64C22, D8.(C5×S3), C3⋊C16⋊2C10, C6.9(C5×D8), C3⋊2(C5×SD32), C8.5(S3×C10), (C5×D8).2S3, C12.4(C5×D4), C24.3(C2×C10), (C3×D8).1C10, (C15×D8).3C2, C10.25(D4⋊S3), (C5×Dic12)⋊11C2, C20.67(C3⋊D4), (C5×C3⋊C16)⋊9C2, C2.5(C5×D4⋊S3), C4.2(C5×C3⋊D4), SmallGroup(480,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D8.S3
G = < a,b,c,d,e | a5=b8=c2=d3=1, e2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >
Smallest permutation representation of C5×D8.S3
►On 240 points
Generators in S240
(1 134 104 80 56)(2 135 97 73 49)(3 136 98 74 50)(4 129 99 75 51)(5 130 100 76 52)(6 131 101 77 53)(7 132 102 78 54)(8 133 103 79 55)(9 226 202 178 154)(10 227 203 179 155)(11 228 204 180 156)(12 229 205 181 157)(13 230 206 182 158)(14 231 207 183 159)(15 232 208 184 160)(16 225 201 177 153)(17 220 196 172 148)(18 221 197 173 149)(19 222 198 174 150)(20 223 199 175 151)(21 224 200 176 152)(22 217 193 169 145)(23 218 194 170 146)(24 219 195 171 147)(25 119 95 71 47)(26 120 96 72 48)(27 113 89 65 41)(28 114 90 66 42)(29 115 91 67 43)(30 116 92 68 44)(31 117 93 69 45)(32 118 94 70 46)(33 123 105 81 57)(34 124 106 82 58)(35 125 107 83 59)(36 126 108 84 60)(37 127 109 85 61)(38 128 110 86 62)(39 121 111 87 63)(40 122 112 88 64)(137 233 209 185 161)(138 234 210 186 162)(139 235 211 187 163)(140 236 212 188 164)(141 237 213 189 165)(142 238 214 190 166)(143 239 215 191 167)(144 240 216 192 168)
(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 8)(2 7)(3 6)(4 5)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(41 42)(43 48)(44 47)(45 46)(49 54)(50 53)(51 52)(55 56)(57 60)(58 59)(61 64)(62 63)(65 66)(67 72)(68 71)(69 70)(73 78)(74 77)(75 76)(79 80)(81 84)(82 83)(85 88)(86 87)(89 90)(91 96)(92 95)(93 94)(97 102)(98 101)(99 100)(103 104)(105 108)(106 107)(109 112)(110 111)(113 114)(115 120)(116 119)(117 118)(121 128)(122 127)(123 126)(124 125)(129 130)(131 136)(132 135)(133 134)(138 144)(139 143)(140 142)(145 147)(148 152)(149 151)(153 157)(154 156)(158 160)(162 168)(163 167)(164 166)(169 171)(172 176)(173 175)(177 181)(178 180)(182 184)(186 192)(187 191)(188 190)(193 195)(196 200)(197 199)(201 205)(202 204)(206 208)(210 216)(211 215)(212 214)(217 219)(220 224)(221 223)(225 229)(226 228)(230 232)(234 240)(235 239)(236 238)
(1 39 32)(2 40 25)(3 33 26)(4 34 27)(5 35 28)(6 36 29)(7 37 30)(8 38 31)(9 240 22)(10 233 23)(11 234 24)(12 235 17)(13 236 18)(14 237 19)(15 238 20)(16 239 21)(41 51 58)(42 52 59)(43 53 60)(44 54 61)(45 55 62)(46 56 63)(47 49 64)(48 50 57)(65 75 82)(66 76 83)(67 77 84)(68 78 85)(69 79 86)(70 80 87)(71 73 88)(72 74 81)(89 99 106)(90 100 107)(91 101 108)(92 102 109)(93 103 110)(94 104 111)(95 97 112)(96 98 105)(113 129 124)(114 130 125)(115 131 126)(116 132 127)(117 133 128)(118 134 121)(119 135 122)(120 136 123)(137 146 155)(138 147 156)(139 148 157)(140 149 158)(141 150 159)(142 151 160)(143 152 153)(144 145 154)(161 170 179)(162 171 180)(163 172 181)(164 173 182)(165 174 183)(166 175 184)(167 176 177)(168 169 178)(185 194 203)(186 195 204)(187 196 205)(188 197 206)(189 198 207)(190 199 208)(191 200 201)(192 193 202)(209 218 227)(210 219 228)(211 220 229)(212 221 230)(213 222 231)(214 223 232)(215 224 225)(216 217 226)
(1 160 5 156)(2 159 6 155)(3 158 7 154)(4 157 8 153)(9 136 13 132)(10 135 14 131)(11 134 15 130)(12 133 16 129)(17 128 21 124)(18 127 22 123)(19 126 23 122)(20 125 24 121)(25 141 29 137)(26 140 30 144)(27 139 31 143)(28 138 32 142)(33 149 37 145)(34 148 38 152)(35 147 39 151)(36 146 40 150)(41 163 45 167)(42 162 46 166)(43 161 47 165)(44 168 48 164)(49 183 53 179)(50 182 54 178)(51 181 55 177)(52 180 56 184)(57 173 61 169)(58 172 62 176)(59 171 63 175)(60 170 64 174)(65 187 69 191)(66 186 70 190)(67 185 71 189)(68 192 72 188)(73 207 77 203)(74 206 78 202)(75 205 79 201)(76 204 80 208)(81 197 85 193)(82 196 86 200)(83 195 87 199)(84 194 88 198)(89 211 93 215)(90 210 94 214)(91 209 95 213)(92 216 96 212)(97 231 101 227)(98 230 102 226)(99 229 103 225)(100 228 104 232)(105 221 109 217)(106 220 110 224)(107 219 111 223)(108 218 112 222)(113 235 117 239)(114 234 118 238)(115 233 119 237)(116 240 120 236)
G:=sub<Sym(240)| (1,134,104,80,56)(2,135,97,73,49)(3,136,98,74,50)(4,129,99,75,51)(5,130,100,76,52)(6,131,101,77,53)(7,132,102,78,54)(8,133,103,79,55)(9,226,202,178,154)(10,227,203,179,155)(11,228,204,180,156)(12,229,205,181,157)(13,230,206,182,158)(14,231,207,183,159)(15,232,208,184,160)(16,225,201,177,153)(17,220,196,172,148)(18,221,197,173,149)(19,222,198,174,150)(20,223,199,175,151)(21,224,200,176,152)(22,217,193,169,145)(23,218,194,170,146)(24,219,195,171,147)(25,119,95,71,47)(26,120,96,72,48)(27,113,89,65,41)(28,114,90,66,42)(29,115,91,67,43)(30,116,92,68,44)(31,117,93,69,45)(32,118,94,70,46)(33,123,105,81,57)(34,124,106,82,58)(35,125,107,83,59)(36,126,108,84,60)(37,127,109,85,61)(38,128,110,86,62)(39,121,111,87,63)(40,122,112,88,64)(137,233,209,185,161)(138,234,210,186,162)(139,235,211,187,163)(140,236,212,188,164)(141,237,213,189,165)(142,238,214,190,166)(143,239,215,191,167)(144,240,216,192,168), (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), (1,8)(2,7)(3,6)(4,5)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(41,42)(43,48)(44,47)(45,46)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,64)(62,63)(65,66)(67,72)(68,71)(69,70)(73,78)(74,77)(75,76)(79,80)(81,84)(82,83)(85,88)(86,87)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,108)(106,107)(109,112)(110,111)(113,114)(115,120)(116,119)(117,118)(121,128)(122,127)(123,126)(124,125)(129,130)(131,136)(132,135)(133,134)(138,144)(139,143)(140,142)(145,147)(148,152)(149,151)(153,157)(154,156)(158,160)(162,168)(163,167)(164,166)(169,171)(172,176)(173,175)(177,181)(178,180)(182,184)(186,192)(187,191)(188,190)(193,195)(196,200)(197,199)(201,205)(202,204)(206,208)(210,216)(211,215)(212,214)(217,219)(220,224)(221,223)(225,229)(226,228)(230,232)(234,240)(235,239)(236,238), (1,39,32)(2,40,25)(3,33,26)(4,34,27)(5,35,28)(6,36,29)(7,37,30)(8,38,31)(9,240,22)(10,233,23)(11,234,24)(12,235,17)(13,236,18)(14,237,19)(15,238,20)(16,239,21)(41,51,58)(42,52,59)(43,53,60)(44,54,61)(45,55,62)(46,56,63)(47,49,64)(48,50,57)(65,75,82)(66,76,83)(67,77,84)(68,78,85)(69,79,86)(70,80,87)(71,73,88)(72,74,81)(89,99,106)(90,100,107)(91,101,108)(92,102,109)(93,103,110)(94,104,111)(95,97,112)(96,98,105)(113,129,124)(114,130,125)(115,131,126)(116,132,127)(117,133,128)(118,134,121)(119,135,122)(120,136,123)(137,146,155)(138,147,156)(139,148,157)(140,149,158)(141,150,159)(142,151,160)(143,152,153)(144,145,154)(161,170,179)(162,171,180)(163,172,181)(164,173,182)(165,174,183)(166,175,184)(167,176,177)(168,169,178)(185,194,203)(186,195,204)(187,196,205)(188,197,206)(189,198,207)(190,199,208)(191,200,201)(192,193,202)(209,218,227)(210,219,228)(211,220,229)(212,221,230)(213,222,231)(214,223,232)(215,224,225)(216,217,226), (1,160,5,156)(2,159,6,155)(3,158,7,154)(4,157,8,153)(9,136,13,132)(10,135,14,131)(11,134,15,130)(12,133,16,129)(17,128,21,124)(18,127,22,123)(19,126,23,122)(20,125,24,121)(25,141,29,137)(26,140,30,144)(27,139,31,143)(28,138,32,142)(33,149,37,145)(34,148,38,152)(35,147,39,151)(36,146,40,150)(41,163,45,167)(42,162,46,166)(43,161,47,165)(44,168,48,164)(49,183,53,179)(50,182,54,178)(51,181,55,177)(52,180,56,184)(57,173,61,169)(58,172,62,176)(59,171,63,175)(60,170,64,174)(65,187,69,191)(66,186,70,190)(67,185,71,189)(68,192,72,188)(73,207,77,203)(74,206,78,202)(75,205,79,201)(76,204,80,208)(81,197,85,193)(82,196,86,200)(83,195,87,199)(84,194,88,198)(89,211,93,215)(90,210,94,214)(91,209,95,213)(92,216,96,212)(97,231,101,227)(98,230,102,226)(99,229,103,225)(100,228,104,232)(105,221,109,217)(106,220,110,224)(107,219,111,223)(108,218,112,222)(113,235,117,239)(114,234,118,238)(115,233,119,237)(116,240,120,236)>;
G:=Group( (1,134,104,80,56)(2,135,97,73,49)(3,136,98,74,50)(4,129,99,75,51)(5,130,100,76,52)(6,131,101,77,53)(7,132,102,78,54)(8,133,103,79,55)(9,226,202,178,154)(10,227,203,179,155)(11,228,204,180,156)(12,229,205,181,157)(13,230,206,182,158)(14,231,207,183,159)(15,232,208,184,160)(16,225,201,177,153)(17,220,196,172,148)(18,221,197,173,149)(19,222,198,174,150)(20,223,199,175,151)(21,224,200,176,152)(22,217,193,169,145)(23,218,194,170,146)(24,219,195,171,147)(25,119,95,71,47)(26,120,96,72,48)(27,113,89,65,41)(28,114,90,66,42)(29,115,91,67,43)(30,116,92,68,44)(31,117,93,69,45)(32,118,94,70,46)(33,123,105,81,57)(34,124,106,82,58)(35,125,107,83,59)(36,126,108,84,60)(37,127,109,85,61)(38,128,110,86,62)(39,121,111,87,63)(40,122,112,88,64)(137,233,209,185,161)(138,234,210,186,162)(139,235,211,187,163)(140,236,212,188,164)(141,237,213,189,165)(142,238,214,190,166)(143,239,215,191,167)(144,240,216,192,168), (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), (1,8)(2,7)(3,6)(4,5)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(41,42)(43,48)(44,47)(45,46)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,64)(62,63)(65,66)(67,72)(68,71)(69,70)(73,78)(74,77)(75,76)(79,80)(81,84)(82,83)(85,88)(86,87)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,108)(106,107)(109,112)(110,111)(113,114)(115,120)(116,119)(117,118)(121,128)(122,127)(123,126)(124,125)(129,130)(131,136)(132,135)(133,134)(138,144)(139,143)(140,142)(145,147)(148,152)(149,151)(153,157)(154,156)(158,160)(162,168)(163,167)(164,166)(169,171)(172,176)(173,175)(177,181)(178,180)(182,184)(186,192)(187,191)(188,190)(193,195)(196,200)(197,199)(201,205)(202,204)(206,208)(210,216)(211,215)(212,214)(217,219)(220,224)(221,223)(225,229)(226,228)(230,232)(234,240)(235,239)(236,238), (1,39,32)(2,40,25)(3,33,26)(4,34,27)(5,35,28)(6,36,29)(7,37,30)(8,38,31)(9,240,22)(10,233,23)(11,234,24)(12,235,17)(13,236,18)(14,237,19)(15,238,20)(16,239,21)(41,51,58)(42,52,59)(43,53,60)(44,54,61)(45,55,62)(46,56,63)(47,49,64)(48,50,57)(65,75,82)(66,76,83)(67,77,84)(68,78,85)(69,79,86)(70,80,87)(71,73,88)(72,74,81)(89,99,106)(90,100,107)(91,101,108)(92,102,109)(93,103,110)(94,104,111)(95,97,112)(96,98,105)(113,129,124)(114,130,125)(115,131,126)(116,132,127)(117,133,128)(118,134,121)(119,135,122)(120,136,123)(137,146,155)(138,147,156)(139,148,157)(140,149,158)(141,150,159)(142,151,160)(143,152,153)(144,145,154)(161,170,179)(162,171,180)(163,172,181)(164,173,182)(165,174,183)(166,175,184)(167,176,177)(168,169,178)(185,194,203)(186,195,204)(187,196,205)(188,197,206)(189,198,207)(190,199,208)(191,200,201)(192,193,202)(209,218,227)(210,219,228)(211,220,229)(212,221,230)(213,222,231)(214,223,232)(215,224,225)(216,217,226), (1,160,5,156)(2,159,6,155)(3,158,7,154)(4,157,8,153)(9,136,13,132)(10,135,14,131)(11,134,15,130)(12,133,16,129)(17,128,21,124)(18,127,22,123)(19,126,23,122)(20,125,24,121)(25,141,29,137)(26,140,30,144)(27,139,31,143)(28,138,32,142)(33,149,37,145)(34,148,38,152)(35,147,39,151)(36,146,40,150)(41,163,45,167)(42,162,46,166)(43,161,47,165)(44,168,48,164)(49,183,53,179)(50,182,54,178)(51,181,55,177)(52,180,56,184)(57,173,61,169)(58,172,62,176)(59,171,63,175)(60,170,64,174)(65,187,69,191)(66,186,70,190)(67,185,71,189)(68,192,72,188)(73,207,77,203)(74,206,78,202)(75,205,79,201)(76,204,80,208)(81,197,85,193)(82,196,86,200)(83,195,87,199)(84,194,88,198)(89,211,93,215)(90,210,94,214)(91,209,95,213)(92,216,96,212)(97,231,101,227)(98,230,102,226)(99,229,103,225)(100,228,104,232)(105,221,109,217)(106,220,110,224)(107,219,111,223)(108,218,112,222)(113,235,117,239)(114,234,118,238)(115,233,119,237)(116,240,120,236) );
G=PermutationGroup([[(1,134,104,80,56),(2,135,97,73,49),(3,136,98,74,50),(4,129,99,75,51),(5,130,100,76,52),(6,131,101,77,53),(7,132,102,78,54),(8,133,103,79,55),(9,226,202,178,154),(10,227,203,179,155),(11,228,204,180,156),(12,229,205,181,157),(13,230,206,182,158),(14,231,207,183,159),(15,232,208,184,160),(16,225,201,177,153),(17,220,196,172,148),(18,221,197,173,149),(19,222,198,174,150),(20,223,199,175,151),(21,224,200,176,152),(22,217,193,169,145),(23,218,194,170,146),(24,219,195,171,147),(25,119,95,71,47),(26,120,96,72,48),(27,113,89,65,41),(28,114,90,66,42),(29,115,91,67,43),(30,116,92,68,44),(31,117,93,69,45),(32,118,94,70,46),(33,123,105,81,57),(34,124,106,82,58),(35,125,107,83,59),(36,126,108,84,60),(37,127,109,85,61),(38,128,110,86,62),(39,121,111,87,63),(40,122,112,88,64),(137,233,209,185,161),(138,234,210,186,162),(139,235,211,187,163),(140,236,212,188,164),(141,237,213,189,165),(142,238,214,190,166),(143,239,215,191,167),(144,240,216,192,168)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24),(25,30),(26,29),(27,28),(31,32),(33,36),(34,35),(37,40),(38,39),(41,42),(43,48),(44,47),(45,46),(49,54),(50,53),(51,52),(55,56),(57,60),(58,59),(61,64),(62,63),(65,66),(67,72),(68,71),(69,70),(73,78),(74,77),(75,76),(79,80),(81,84),(82,83),(85,88),(86,87),(89,90),(91,96),(92,95),(93,94),(97,102),(98,101),(99,100),(103,104),(105,108),(106,107),(109,112),(110,111),(113,114),(115,120),(116,119),(117,118),(121,128),(122,127),(123,126),(124,125),(129,130),(131,136),(132,135),(133,134),(138,144),(139,143),(140,142),(145,147),(148,152),(149,151),(153,157),(154,156),(158,160),(162,168),(163,167),(164,166),(169,171),(172,176),(173,175),(177,181),(178,180),(182,184),(186,192),(187,191),(188,190),(193,195),(196,200),(197,199),(201,205),(202,204),(206,208),(210,216),(211,215),(212,214),(217,219),(220,224),(221,223),(225,229),(226,228),(230,232),(234,240),(235,239),(236,238)], [(1,39,32),(2,40,25),(3,33,26),(4,34,27),(5,35,28),(6,36,29),(7,37,30),(8,38,31),(9,240,22),(10,233,23),(11,234,24),(12,235,17),(13,236,18),(14,237,19),(15,238,20),(16,239,21),(41,51,58),(42,52,59),(43,53,60),(44,54,61),(45,55,62),(46,56,63),(47,49,64),(48,50,57),(65,75,82),(66,76,83),(67,77,84),(68,78,85),(69,79,86),(70,80,87),(71,73,88),(72,74,81),(89,99,106),(90,100,107),(91,101,108),(92,102,109),(93,103,110),(94,104,111),(95,97,112),(96,98,105),(113,129,124),(114,130,125),(115,131,126),(116,132,127),(117,133,128),(118,134,121),(119,135,122),(120,136,123),(137,146,155),(138,147,156),(139,148,157),(140,149,158),(141,150,159),(142,151,160),(143,152,153),(144,145,154),(161,170,179),(162,171,180),(163,172,181),(164,173,182),(165,174,183),(166,175,184),(167,176,177),(168,169,178),(185,194,203),(186,195,204),(187,196,205),(188,197,206),(189,198,207),(190,199,208),(191,200,201),(192,193,202),(209,218,227),(210,219,228),(211,220,229),(212,221,230),(213,222,231),(214,223,232),(215,224,225),(216,217,226)], [(1,160,5,156),(2,159,6,155),(3,158,7,154),(4,157,8,153),(9,136,13,132),(10,135,14,131),(11,134,15,130),(12,133,16,129),(17,128,21,124),(18,127,22,123),(19,126,23,122),(20,125,24,121),(25,141,29,137),(26,140,30,144),(27,139,31,143),(28,138,32,142),(33,149,37,145),(34,148,38,152),(35,147,39,151),(36,146,40,150),(41,163,45,167),(42,162,46,166),(43,161,47,165),(44,168,48,164),(49,183,53,179),(50,182,54,178),(51,181,55,177),(52,180,56,184),(57,173,61,169),(58,172,62,176),(59,171,63,175),(60,170,64,174),(65,187,69,191),(66,186,70,190),(67,185,71,189),(68,192,72,188),(73,207,77,203),(74,206,78,202),(75,205,79,201),(76,204,80,208),(81,197,85,193),(82,196,86,200),(83,195,87,199),(84,194,88,198),(89,211,93,215),(90,210,94,214),(91,209,95,213),(92,216,96,212),(97,231,101,227),(98,230,102,226),(99,229,103,225),(100,228,104,232),(105,221,109,217),(106,220,110,224),(107,219,111,223),(108,218,112,222),(113,235,117,239),(114,234,118,238),(115,233,119,237),(116,240,120,236)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12 | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 40A | ··· | 40H | 60A | 60B | 60C | 60D | 80A | ··· | 80P | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 8 | 2 | 2 | 24 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | D8 | C3⋊D4 | C5×S3 | SD32 | C5×D4 | S3×C10 | C5×D8 | C5×C3⋊D4 | C5×SD32 | D4⋊S3 | D8.S3 | C5×D4⋊S3 | C5×D8.S3 |
| kernel | C5×D8.S3 | C5×C3⋊C16 | C5×Dic12 | C15×D8 | D8.S3 | C3⋊C16 | Dic12 | C3×D8 | C5×D8 | C60 | C40 | C30 | C20 | D8 | C15 | C12 | C8 | C6 | C4 | C3 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×D8.S3 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 205 | 0 |
| 0 | 0 | 0 | 205 |
| 22 | 22 | 0 | 0 |
| 230 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 219 | 0 | 0 |
| 230 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 67 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 195 | 15 |
| 62 | 159 | 0 | 0 |
| 138 | 179 | 0 | 0 |
| 0 | 0 | 67 | 2 |
| 0 | 0 | 166 | 174 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[22,230,0,0,22,0,0,0,0,0,240,0,0,0,0,240],[0,230,0,0,219,0,0,0,0,0,240,67,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,225,195,0,0,0,15],[62,138,0,0,159,179,0,0,0,0,67,166,0,0,2,174] >;
C5×D8.S3 in GAP, Magma, Sage, TeX
C_5\times D_8.S_3
% in TeX
G:=Group("C5xD8.S3"); // GroupNames label
G:=SmallGroup(480,146);
// by ID
G=gap.SmallGroup(480,146);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,560,309,1683,850,192,4204,2111,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^3=1,e^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations
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