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## G = C5×D6.C8order 480 = 25·3·5

### Direct product of C5 and D6.C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×D6.C8
 Chief series C1 — C3 — C6 — C12 — C24 — C120 — S3×C40 — C5×D6.C8
 Lower central C3 — C6 — C5×D6.C8
 Upper central C1 — C40 — C80

Generators and relations for C5×D6.C8
G = < a,b,c,d | a5=b6=c2=1, d8=b3, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Smallest permutation representation of C5×D6.C8
On 240 points
Generators in S240
(1 124 186 110 198)(2 125 187 111 199)(3 126 188 112 200)(4 127 189 97 201)(5 128 190 98 202)(6 113 191 99 203)(7 114 192 100 204)(8 115 177 101 205)(9 116 178 102 206)(10 117 179 103 207)(11 118 180 104 208)(12 119 181 105 193)(13 120 182 106 194)(14 121 183 107 195)(15 122 184 108 196)(16 123 185 109 197)(17 175 89 59 40)(18 176 90 60 41)(19 161 91 61 42)(20 162 92 62 43)(21 163 93 63 44)(22 164 94 64 45)(23 165 95 49 46)(24 166 96 50 47)(25 167 81 51 48)(26 168 82 52 33)(27 169 83 53 34)(28 170 84 54 35)(29 171 85 55 36)(30 172 86 56 37)(31 173 87 57 38)(32 174 88 58 39)(65 216 130 231 146)(66 217 131 232 147)(67 218 132 233 148)(68 219 133 234 149)(69 220 134 235 150)(70 221 135 236 151)(71 222 136 237 152)(72 223 137 238 153)(73 224 138 239 154)(74 209 139 240 155)(75 210 140 225 156)(76 211 141 226 157)(77 212 142 227 158)(78 213 143 228 159)(79 214 144 229 160)(80 215 129 230 145)
(1 163 153 9 171 145)(2 164 154 10 172 146)(3 165 155 11 173 147)(4 166 156 12 174 148)(5 167 157 13 175 149)(6 168 158 14 176 150)(7 169 159 15 161 151)(8 170 160 16 162 152)(17 234 202 25 226 194)(18 235 203 26 227 195)(19 236 204 27 228 196)(20 237 205 28 229 197)(21 238 206 29 230 198)(22 239 207 30 231 199)(23 240 208 31 232 200)(24 225 193 32 233 201)(33 142 107 41 134 99)(34 143 108 42 135 100)(35 144 109 43 136 101)(36 129 110 44 137 102)(37 130 111 45 138 103)(38 131 112 46 139 104)(39 132 97 47 140 105)(40 133 98 48 141 106)(49 209 180 57 217 188)(50 210 181 58 218 189)(51 211 182 59 219 190)(52 212 183 60 220 191)(53 213 184 61 221 192)(54 214 185 62 222 177)(55 215 186 63 223 178)(56 216 187 64 224 179)(65 125 94 73 117 86)(66 126 95 74 118 87)(67 127 96 75 119 88)(68 128 81 76 120 89)(69 113 82 77 121 90)(70 114 83 78 122 91)(71 115 84 79 123 92)(72 116 85 80 124 93)
(1 145)(2 154)(3 147)(4 156)(5 149)(6 158)(7 151)(8 160)(9 153)(10 146)(11 155)(12 148)(13 157)(14 150)(15 159)(16 152)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 117)(66 126)(67 119)(68 128)(69 121)(70 114)(71 123)(72 116)(73 125)(74 118)(75 127)(76 120)(77 113)(78 122)(79 115)(80 124)(81 89)(83 91)(85 93)(87 95)(97 140)(98 133)(99 142)(100 135)(101 144)(102 137)(103 130)(104 139)(105 132)(106 141)(107 134)(108 143)(109 136)(110 129)(111 138)(112 131)(161 169)(163 171)(165 173)(167 175)(177 214)(178 223)(179 216)(180 209)(181 218)(182 211)(183 220)(184 213)(185 222)(186 215)(187 224)(188 217)(189 210)(190 219)(191 212)(192 221)(193 233)(194 226)(195 235)(196 228)(197 237)(198 230)(199 239)(200 232)(201 225)(202 234)(203 227)(204 236)(205 229)(206 238)(207 231)(208 240)
(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)

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

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

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

180 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5A 5B 5C 5D 6 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 15A 15B 15C 15D 16A 16B 16C 16D 16E 16F 16G 16H 20A ··· 20H 20I 20J 20K 20L 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40P 40Q ··· 40X 48A ··· 48H 60A ··· 60H 80A ··· 80P 80Q ··· 80AF 120A ··· 120P 240A ··· 240AF order 1 2 2 3 4 4 4 5 5 5 5 6 8 8 8 8 8 8 10 10 10 10 10 10 10 10 12 12 15 15 15 15 16 16 16 16 16 16 16 16 20 ··· 20 20 20 20 20 24 24 24 24 30 30 30 30 40 ··· 40 40 ··· 40 48 ··· 48 60 ··· 60 80 ··· 80 80 ··· 80 120 ··· 120 240 ··· 240 size 1 1 6 2 1 1 6 1 1 1 1 2 1 1 1 1 6 6 1 1 1 1 6 6 6 6 2 2 2 2 2 2 2 2 2 2 6 6 6 6 1 ··· 1 6 6 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 type + + + + + + image C1 C2 C2 C2 C4 C4 C5 C8 C8 C10 C10 C10 C20 C20 C40 C40 S3 D6 C4×S3 C5×S3 M5(2) S3×C8 S3×C10 D6.C8 S3×C20 C5×M5(2) S3×C40 C5×D6.C8 kernel C5×D6.C8 C5×C3⋊C16 C240 S3×C40 C5×C3⋊C8 S3×C20 D6.C8 C5×Dic3 S3×C10 C3⋊C16 C48 S3×C8 C3⋊C8 C4×S3 Dic3 D6 C80 C40 C20 C16 C15 C10 C8 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 4 4 4 4 4 4 8 8 16 16 1 1 2 4 4 4 4 8 8 16 16 32

Matrix representation of C5×D6.C8 in GL2(𝔽41) generated by

 16 0 0 16
,
 1 33 36 0
,
 0 33 5 0
,
 32 21 8 9
G:=sub<GL(2,GF(41))| [16,0,0,16],[1,36,33,0],[0,5,33,0],[32,8,21,9] >;

C5×D6.C8 in GAP, Magma, Sage, TeX

C_5\times D_6.C_8
% in TeX

G:=Group("C5xD6.C8");
// GroupNames label

G:=SmallGroup(480,117);
// by ID

G=gap.SmallGroup(480,117);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,1149,148,80,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=c^2=1,d^8=b^3,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

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