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## G = C3×D8.D5order 480 = 25·3·5

### Direct product of C3 and D8.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C3×D8.D5
 Chief series C1 — C5 — C10 — C20 — C40 — C120 — C3×Dic20 — C3×D8.D5
 Lower central C5 — C10 — C20 — C40 — C3×D8.D5
 Upper central C1 — C6 — C12 — C24 — C3×D8

Generators and relations for C3×D8.D5
G = < a,b,c,d,e | a3=b8=c2=d5=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 C3×D8.D5
On 240 points
Generators in S240
(1 142 87)(2 143 88)(3 144 81)(4 137 82)(5 138 83)(6 139 84)(7 140 85)(8 141 86)(9 228 188)(10 229 189)(11 230 190)(12 231 191)(13 232 192)(14 225 185)(15 226 186)(16 227 187)(17 224 184)(18 217 177)(19 218 178)(20 219 179)(21 220 180)(22 221 181)(23 222 182)(24 223 183)(25 212 172)(26 213 173)(27 214 174)(28 215 175)(29 216 176)(30 209 169)(31 210 170)(32 211 171)(33 202 162)(34 203 163)(35 204 164)(36 205 165)(37 206 166)(38 207 167)(39 208 168)(40 201 161)(41 113 73)(42 114 74)(43 115 75)(44 116 76)(45 117 77)(46 118 78)(47 119 79)(48 120 80)(49 127 89)(50 128 90)(51 121 91)(52 122 92)(53 123 93)(54 124 94)(55 125 95)(56 126 96)(57 135 97)(58 136 98)(59 129 99)(60 130 100)(61 131 101)(62 132 102)(63 133 103)(64 134 104)(65 150 105)(66 151 106)(67 152 107)(68 145 108)(69 146 109)(70 147 110)(71 148 111)(72 149 112)(153 233 193)(154 234 194)(155 235 195)(156 236 196)(157 237 197)(158 238 198)(159 239 199)(160 240 200)
(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 13)(10 12)(14 16)(17 19)(20 24)(21 23)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(41 46)(42 45)(43 44)(47 48)(49 54)(50 53)(51 52)(55 56)(57 60)(58 59)(61 64)(62 63)(65 70)(66 69)(67 68)(71 72)(73 78)(74 77)(75 76)(79 80)(81 84)(82 83)(85 88)(86 87)(89 94)(90 93)(91 92)(95 96)(97 100)(98 99)(101 104)(102 103)(105 110)(106 109)(107 108)(111 112)(113 118)(114 117)(115 116)(119 120)(121 122)(123 128)(124 127)(125 126)(129 136)(130 135)(131 134)(132 133)(137 138)(139 144)(140 143)(141 142)(145 152)(146 151)(147 150)(148 149)(153 157)(154 156)(158 160)(161 165)(162 164)(166 168)(169 171)(172 176)(173 175)(178 184)(179 183)(180 182)(185 187)(188 192)(189 191)(193 197)(194 196)(198 200)(201 205)(202 204)(206 208)(209 211)(212 216)(213 215)(218 224)(219 223)(220 222)(225 227)(228 232)(229 231)(233 237)(234 236)(238 240)
(1 72 44 56 63)(2 65 45 49 64)(3 66 46 50 57)(4 67 47 51 58)(5 68 48 52 59)(6 69 41 53 60)(7 70 42 54 61)(8 71 43 55 62)(9 24 25 40 233)(10 17 26 33 234)(11 18 27 34 235)(12 19 28 35 236)(13 20 29 36 237)(14 21 30 37 238)(15 22 31 38 239)(16 23 32 39 240)(73 93 100 84 109)(74 94 101 85 110)(75 95 102 86 111)(76 96 103 87 112)(77 89 104 88 105)(78 90 97 81 106)(79 91 98 82 107)(80 92 99 83 108)(113 123 130 139 146)(114 124 131 140 147)(115 125 132 141 148)(116 126 133 142 149)(117 127 134 143 150)(118 128 135 144 151)(119 121 136 137 152)(120 122 129 138 145)(153 188 183 172 161)(154 189 184 173 162)(155 190 177 174 163)(156 191 178 175 164)(157 192 179 176 165)(158 185 180 169 166)(159 186 181 170 167)(160 187 182 171 168)(193 228 223 212 201)(194 229 224 213 202)(195 230 217 214 203)(196 231 218 215 204)(197 232 219 216 205)(198 225 220 209 206)(199 226 221 210 207)(200 227 222 211 208)
(1 187 5 191)(2 186 6 190)(3 185 7 189)(4 192 8 188)(9 137 13 141)(10 144 14 140)(11 143 15 139)(12 142 16 138)(17 135 21 131)(18 134 22 130)(19 133 23 129)(20 132 24 136)(25 121 29 125)(26 128 30 124)(27 127 31 123)(28 126 32 122)(33 118 37 114)(34 117 38 113)(35 116 39 120)(36 115 40 119)(41 163 45 167)(42 162 46 166)(43 161 47 165)(44 168 48 164)(49 170 53 174)(50 169 54 173)(51 176 55 172)(52 175 56 171)(57 180 61 184)(58 179 62 183)(59 178 63 182)(60 177 64 181)(65 159 69 155)(66 158 70 154)(67 157 71 153)(68 156 72 160)(73 203 77 207)(74 202 78 206)(75 201 79 205)(76 208 80 204)(81 225 85 229)(82 232 86 228)(83 231 87 227)(84 230 88 226)(89 210 93 214)(90 209 94 213)(91 216 95 212)(92 215 96 211)(97 220 101 224)(98 219 102 223)(99 218 103 222)(100 217 104 221)(105 199 109 195)(106 198 110 194)(107 197 111 193)(108 196 112 200)(145 236 149 240)(146 235 150 239)(147 234 151 238)(148 233 152 237)

G:=sub<Sym(240)| (1,142,87)(2,143,88)(3,144,81)(4,137,82)(5,138,83)(6,139,84)(7,140,85)(8,141,86)(9,228,188)(10,229,189)(11,230,190)(12,231,191)(13,232,192)(14,225,185)(15,226,186)(16,227,187)(17,224,184)(18,217,177)(19,218,178)(20,219,179)(21,220,180)(22,221,181)(23,222,182)(24,223,183)(25,212,172)(26,213,173)(27,214,174)(28,215,175)(29,216,176)(30,209,169)(31,210,170)(32,211,171)(33,202,162)(34,203,163)(35,204,164)(36,205,165)(37,206,166)(38,207,167)(39,208,168)(40,201,161)(41,113,73)(42,114,74)(43,115,75)(44,116,76)(45,117,77)(46,118,78)(47,119,79)(48,120,80)(49,127,89)(50,128,90)(51,121,91)(52,122,92)(53,123,93)(54,124,94)(55,125,95)(56,126,96)(57,135,97)(58,136,98)(59,129,99)(60,130,100)(61,131,101)(62,132,102)(63,133,103)(64,134,104)(65,150,105)(66,151,106)(67,152,107)(68,145,108)(69,146,109)(70,147,110)(71,148,111)(72,149,112)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (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)(9,13)(10,12)(14,16)(17,19)(20,24)(21,23)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,46)(42,45)(43,44)(47,48)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,64)(62,63)(65,70)(66,69)(67,68)(71,72)(73,78)(74,77)(75,76)(79,80)(81,84)(82,83)(85,88)(86,87)(89,94)(90,93)(91,92)(95,96)(97,100)(98,99)(101,104)(102,103)(105,110)(106,109)(107,108)(111,112)(113,118)(114,117)(115,116)(119,120)(121,122)(123,128)(124,127)(125,126)(129,136)(130,135)(131,134)(132,133)(137,138)(139,144)(140,143)(141,142)(145,152)(146,151)(147,150)(148,149)(153,157)(154,156)(158,160)(161,165)(162,164)(166,168)(169,171)(172,176)(173,175)(178,184)(179,183)(180,182)(185,187)(188,192)(189,191)(193,197)(194,196)(198,200)(201,205)(202,204)(206,208)(209,211)(212,216)(213,215)(218,224)(219,223)(220,222)(225,227)(228,232)(229,231)(233,237)(234,236)(238,240), (1,72,44,56,63)(2,65,45,49,64)(3,66,46,50,57)(4,67,47,51,58)(5,68,48,52,59)(6,69,41,53,60)(7,70,42,54,61)(8,71,43,55,62)(9,24,25,40,233)(10,17,26,33,234)(11,18,27,34,235)(12,19,28,35,236)(13,20,29,36,237)(14,21,30,37,238)(15,22,31,38,239)(16,23,32,39,240)(73,93,100,84,109)(74,94,101,85,110)(75,95,102,86,111)(76,96,103,87,112)(77,89,104,88,105)(78,90,97,81,106)(79,91,98,82,107)(80,92,99,83,108)(113,123,130,139,146)(114,124,131,140,147)(115,125,132,141,148)(116,126,133,142,149)(117,127,134,143,150)(118,128,135,144,151)(119,121,136,137,152)(120,122,129,138,145)(153,188,183,172,161)(154,189,184,173,162)(155,190,177,174,163)(156,191,178,175,164)(157,192,179,176,165)(158,185,180,169,166)(159,186,181,170,167)(160,187,182,171,168)(193,228,223,212,201)(194,229,224,213,202)(195,230,217,214,203)(196,231,218,215,204)(197,232,219,216,205)(198,225,220,209,206)(199,226,221,210,207)(200,227,222,211,208), (1,187,5,191)(2,186,6,190)(3,185,7,189)(4,192,8,188)(9,137,13,141)(10,144,14,140)(11,143,15,139)(12,142,16,138)(17,135,21,131)(18,134,22,130)(19,133,23,129)(20,132,24,136)(25,121,29,125)(26,128,30,124)(27,127,31,123)(28,126,32,122)(33,118,37,114)(34,117,38,113)(35,116,39,120)(36,115,40,119)(41,163,45,167)(42,162,46,166)(43,161,47,165)(44,168,48,164)(49,170,53,174)(50,169,54,173)(51,176,55,172)(52,175,56,171)(57,180,61,184)(58,179,62,183)(59,178,63,182)(60,177,64,181)(65,159,69,155)(66,158,70,154)(67,157,71,153)(68,156,72,160)(73,203,77,207)(74,202,78,206)(75,201,79,205)(76,208,80,204)(81,225,85,229)(82,232,86,228)(83,231,87,227)(84,230,88,226)(89,210,93,214)(90,209,94,213)(91,216,95,212)(92,215,96,211)(97,220,101,224)(98,219,102,223)(99,218,103,222)(100,217,104,221)(105,199,109,195)(106,198,110,194)(107,197,111,193)(108,196,112,200)(145,236,149,240)(146,235,150,239)(147,234,151,238)(148,233,152,237)>;

G:=Group( (1,142,87)(2,143,88)(3,144,81)(4,137,82)(5,138,83)(6,139,84)(7,140,85)(8,141,86)(9,228,188)(10,229,189)(11,230,190)(12,231,191)(13,232,192)(14,225,185)(15,226,186)(16,227,187)(17,224,184)(18,217,177)(19,218,178)(20,219,179)(21,220,180)(22,221,181)(23,222,182)(24,223,183)(25,212,172)(26,213,173)(27,214,174)(28,215,175)(29,216,176)(30,209,169)(31,210,170)(32,211,171)(33,202,162)(34,203,163)(35,204,164)(36,205,165)(37,206,166)(38,207,167)(39,208,168)(40,201,161)(41,113,73)(42,114,74)(43,115,75)(44,116,76)(45,117,77)(46,118,78)(47,119,79)(48,120,80)(49,127,89)(50,128,90)(51,121,91)(52,122,92)(53,123,93)(54,124,94)(55,125,95)(56,126,96)(57,135,97)(58,136,98)(59,129,99)(60,130,100)(61,131,101)(62,132,102)(63,133,103)(64,134,104)(65,150,105)(66,151,106)(67,152,107)(68,145,108)(69,146,109)(70,147,110)(71,148,111)(72,149,112)(153,233,193)(154,234,194)(155,235,195)(156,236,196)(157,237,197)(158,238,198)(159,239,199)(160,240,200), (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)(9,13)(10,12)(14,16)(17,19)(20,24)(21,23)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,46)(42,45)(43,44)(47,48)(49,54)(50,53)(51,52)(55,56)(57,60)(58,59)(61,64)(62,63)(65,70)(66,69)(67,68)(71,72)(73,78)(74,77)(75,76)(79,80)(81,84)(82,83)(85,88)(86,87)(89,94)(90,93)(91,92)(95,96)(97,100)(98,99)(101,104)(102,103)(105,110)(106,109)(107,108)(111,112)(113,118)(114,117)(115,116)(119,120)(121,122)(123,128)(124,127)(125,126)(129,136)(130,135)(131,134)(132,133)(137,138)(139,144)(140,143)(141,142)(145,152)(146,151)(147,150)(148,149)(153,157)(154,156)(158,160)(161,165)(162,164)(166,168)(169,171)(172,176)(173,175)(178,184)(179,183)(180,182)(185,187)(188,192)(189,191)(193,197)(194,196)(198,200)(201,205)(202,204)(206,208)(209,211)(212,216)(213,215)(218,224)(219,223)(220,222)(225,227)(228,232)(229,231)(233,237)(234,236)(238,240), (1,72,44,56,63)(2,65,45,49,64)(3,66,46,50,57)(4,67,47,51,58)(5,68,48,52,59)(6,69,41,53,60)(7,70,42,54,61)(8,71,43,55,62)(9,24,25,40,233)(10,17,26,33,234)(11,18,27,34,235)(12,19,28,35,236)(13,20,29,36,237)(14,21,30,37,238)(15,22,31,38,239)(16,23,32,39,240)(73,93,100,84,109)(74,94,101,85,110)(75,95,102,86,111)(76,96,103,87,112)(77,89,104,88,105)(78,90,97,81,106)(79,91,98,82,107)(80,92,99,83,108)(113,123,130,139,146)(114,124,131,140,147)(115,125,132,141,148)(116,126,133,142,149)(117,127,134,143,150)(118,128,135,144,151)(119,121,136,137,152)(120,122,129,138,145)(153,188,183,172,161)(154,189,184,173,162)(155,190,177,174,163)(156,191,178,175,164)(157,192,179,176,165)(158,185,180,169,166)(159,186,181,170,167)(160,187,182,171,168)(193,228,223,212,201)(194,229,224,213,202)(195,230,217,214,203)(196,231,218,215,204)(197,232,219,216,205)(198,225,220,209,206)(199,226,221,210,207)(200,227,222,211,208), (1,187,5,191)(2,186,6,190)(3,185,7,189)(4,192,8,188)(9,137,13,141)(10,144,14,140)(11,143,15,139)(12,142,16,138)(17,135,21,131)(18,134,22,130)(19,133,23,129)(20,132,24,136)(25,121,29,125)(26,128,30,124)(27,127,31,123)(28,126,32,122)(33,118,37,114)(34,117,38,113)(35,116,39,120)(36,115,40,119)(41,163,45,167)(42,162,46,166)(43,161,47,165)(44,168,48,164)(49,170,53,174)(50,169,54,173)(51,176,55,172)(52,175,56,171)(57,180,61,184)(58,179,62,183)(59,178,63,182)(60,177,64,181)(65,159,69,155)(66,158,70,154)(67,157,71,153)(68,156,72,160)(73,203,77,207)(74,202,78,206)(75,201,79,205)(76,208,80,204)(81,225,85,229)(82,232,86,228)(83,231,87,227)(84,230,88,226)(89,210,93,214)(90,209,94,213)(91,216,95,212)(92,215,96,211)(97,220,101,224)(98,219,102,223)(99,218,103,222)(100,217,104,221)(105,199,109,195)(106,198,110,194)(107,197,111,193)(108,196,112,200)(145,236,149,240)(146,235,150,239)(147,234,151,238)(148,233,152,237) );

G=PermutationGroup([[(1,142,87),(2,143,88),(3,144,81),(4,137,82),(5,138,83),(6,139,84),(7,140,85),(8,141,86),(9,228,188),(10,229,189),(11,230,190),(12,231,191),(13,232,192),(14,225,185),(15,226,186),(16,227,187),(17,224,184),(18,217,177),(19,218,178),(20,219,179),(21,220,180),(22,221,181),(23,222,182),(24,223,183),(25,212,172),(26,213,173),(27,214,174),(28,215,175),(29,216,176),(30,209,169),(31,210,170),(32,211,171),(33,202,162),(34,203,163),(35,204,164),(36,205,165),(37,206,166),(38,207,167),(39,208,168),(40,201,161),(41,113,73),(42,114,74),(43,115,75),(44,116,76),(45,117,77),(46,118,78),(47,119,79),(48,120,80),(49,127,89),(50,128,90),(51,121,91),(52,122,92),(53,123,93),(54,124,94),(55,125,95),(56,126,96),(57,135,97),(58,136,98),(59,129,99),(60,130,100),(61,131,101),(62,132,102),(63,133,103),(64,134,104),(65,150,105),(66,151,106),(67,152,107),(68,145,108),(69,146,109),(70,147,110),(71,148,111),(72,149,112),(153,233,193),(154,234,194),(155,235,195),(156,236,196),(157,237,197),(158,238,198),(159,239,199),(160,240,200)], [(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),(9,13),(10,12),(14,16),(17,19),(20,24),(21,23),(25,29),(26,28),(30,32),(33,35),(36,40),(37,39),(41,46),(42,45),(43,44),(47,48),(49,54),(50,53),(51,52),(55,56),(57,60),(58,59),(61,64),(62,63),(65,70),(66,69),(67,68),(71,72),(73,78),(74,77),(75,76),(79,80),(81,84),(82,83),(85,88),(86,87),(89,94),(90,93),(91,92),(95,96),(97,100),(98,99),(101,104),(102,103),(105,110),(106,109),(107,108),(111,112),(113,118),(114,117),(115,116),(119,120),(121,122),(123,128),(124,127),(125,126),(129,136),(130,135),(131,134),(132,133),(137,138),(139,144),(140,143),(141,142),(145,152),(146,151),(147,150),(148,149),(153,157),(154,156),(158,160),(161,165),(162,164),(166,168),(169,171),(172,176),(173,175),(178,184),(179,183),(180,182),(185,187),(188,192),(189,191),(193,197),(194,196),(198,200),(201,205),(202,204),(206,208),(209,211),(212,216),(213,215),(218,224),(219,223),(220,222),(225,227),(228,232),(229,231),(233,237),(234,236),(238,240)], [(1,72,44,56,63),(2,65,45,49,64),(3,66,46,50,57),(4,67,47,51,58),(5,68,48,52,59),(6,69,41,53,60),(7,70,42,54,61),(8,71,43,55,62),(9,24,25,40,233),(10,17,26,33,234),(11,18,27,34,235),(12,19,28,35,236),(13,20,29,36,237),(14,21,30,37,238),(15,22,31,38,239),(16,23,32,39,240),(73,93,100,84,109),(74,94,101,85,110),(75,95,102,86,111),(76,96,103,87,112),(77,89,104,88,105),(78,90,97,81,106),(79,91,98,82,107),(80,92,99,83,108),(113,123,130,139,146),(114,124,131,140,147),(115,125,132,141,148),(116,126,133,142,149),(117,127,134,143,150),(118,128,135,144,151),(119,121,136,137,152),(120,122,129,138,145),(153,188,183,172,161),(154,189,184,173,162),(155,190,177,174,163),(156,191,178,175,164),(157,192,179,176,165),(158,185,180,169,166),(159,186,181,170,167),(160,187,182,171,168),(193,228,223,212,201),(194,229,224,213,202),(195,230,217,214,203),(196,231,218,215,204),(197,232,219,216,205),(198,225,220,209,206),(199,226,221,210,207),(200,227,222,211,208)], [(1,187,5,191),(2,186,6,190),(3,185,7,189),(4,192,8,188),(9,137,13,141),(10,144,14,140),(11,143,15,139),(12,142,16,138),(17,135,21,131),(18,134,22,130),(19,133,23,129),(20,132,24,136),(25,121,29,125),(26,128,30,124),(27,127,31,123),(28,126,32,122),(33,118,37,114),(34,117,38,113),(35,116,39,120),(36,115,40,119),(41,163,45,167),(42,162,46,166),(43,161,47,165),(44,168,48,164),(49,170,53,174),(50,169,54,173),(51,176,55,172),(52,175,56,171),(57,180,61,184),(58,179,62,183),(59,178,63,182),(60,177,64,181),(65,159,69,155),(66,158,70,154),(67,157,71,153),(68,156,72,160),(73,203,77,207),(74,202,78,206),(75,201,79,205),(76,208,80,204),(81,225,85,229),(82,232,86,228),(83,231,87,227),(84,230,88,226),(89,210,93,214),(90,209,94,213),(91,216,95,212),(92,215,96,211),(97,220,101,224),(98,219,102,223),(99,218,103,222),(100,217,104,221),(105,199,109,195),(106,198,110,194),(107,197,111,193),(108,196,112,200),(145,236,149,240),(146,235,150,239),(147,234,151,238),(148,233,152,237)]])

75 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 15C 15D 16A 16B 16C 16D 20A 20B 24A 24B 24C 24D 30A 30B 30C 30D 30E ··· 30L 40A 40B 40C 40D 48A ··· 48H 60A 60B 60C 60D 120A ··· 120H order 1 2 2 3 3 4 4 5 5 6 6 6 6 8 8 10 10 10 10 10 10 12 12 12 12 15 15 15 15 16 16 16 16 20 20 24 24 24 24 30 30 30 30 30 ··· 30 40 40 40 40 48 ··· 48 60 60 60 60 120 ··· 120 size 1 1 8 1 1 2 40 2 2 1 1 8 8 2 2 2 2 8 8 8 8 2 2 40 40 2 2 2 2 10 10 10 10 4 4 2 2 2 2 2 2 2 2 8 ··· 8 4 4 4 4 10 ··· 10 4 4 4 4 4 ··· 4

75 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 C3 C6 C6 C6 D4 D5 D8 D10 C3×D4 C3×D5 SD32 C5⋊D4 C3×D8 C6×D5 C3×SD32 C3×C5⋊D4 D4⋊D5 D8.D5 C3×D4⋊D5 C3×D8.D5 kernel C3×D8.D5 C3×C5⋊2C16 C3×Dic20 C15×D8 D8.D5 C5⋊2C16 Dic20 C5×D8 C60 C3×D8 C30 C24 C20 D8 C15 C12 C10 C8 C5 C4 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 2 4 4 4 4 4 8 8 2 4 4 8

Matrix representation of C3×D8.D5 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 15 0 0 0 0 15
,
 1 0 0 0 0 1 0 0 0 0 0 129 0 0 170 22
,
 1 0 0 0 0 1 0 0 0 0 0 129 0 0 71 0
,
 91 2 0 0 0 98 0 0 0 0 1 0 0 0 0 1
,
 8 58 0 0 28 233 0 0 0 0 62 19 0 0 64 179
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,15,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,0,170,0,0,129,22],[1,0,0,0,0,1,0,0,0,0,0,71,0,0,129,0],[91,0,0,0,2,98,0,0,0,0,1,0,0,0,0,1],[8,28,0,0,58,233,0,0,0,0,62,64,0,0,19,179] >;

C3×D8.D5 in GAP, Magma, Sage, TeX

C_3\times D_8.D_5
% in TeX

G:=Group("C3xD8.D5");
// GroupNames label

G:=SmallGroup(480,105);
// by ID

G=gap.SmallGroup(480,105);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,336,197,1011,514,192,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^5=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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