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## G = C3×D40⋊7C2order 480 = 25·3·5

### Direct product of C3 and D40⋊7C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D40⋊7C2
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C3×C4○D20 — C3×D40⋊7C2
 Lower central C5 — C10 — C20 — C3×D40⋊7C2
 Upper central C1 — C12 — C2×C12 — C2×C24

Generators and relations for C3×D407C2
G = < a,b,c,d | a3=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b20c >

Subgroups: 464 in 124 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C40⋊C2, D40, Dic20, C2×C40, C4○D20, C3×C4○D8, C120, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D407C2, C3×C40⋊C2, C3×D40, C3×Dic20, C2×C120, C3×C4○D20, C3×D407C2
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C4○D8, D20, C22×D5, C6×D4, C6×D5, C2×D20, C3×C4○D8, C3×D20, D5×C2×C6, D407C2, C6×D20, C3×D407C2

Smallest permutation representation of C3×D407C2
On 240 points
Generators in S240
(1 85 71)(2 86 72)(3 87 73)(4 88 74)(5 89 75)(6 90 76)(7 91 77)(8 92 78)(9 93 79)(10 94 80)(11 95 41)(12 96 42)(13 97 43)(14 98 44)(15 99 45)(16 100 46)(17 101 47)(18 102 48)(19 103 49)(20 104 50)(21 105 51)(22 106 52)(23 107 53)(24 108 54)(25 109 55)(26 110 56)(27 111 57)(28 112 58)(29 113 59)(30 114 60)(31 115 61)(32 116 62)(33 117 63)(34 118 64)(35 119 65)(36 120 66)(37 81 67)(38 82 68)(39 83 69)(40 84 70)(121 182 230)(122 183 231)(123 184 232)(124 185 233)(125 186 234)(126 187 235)(127 188 236)(128 189 237)(129 190 238)(130 191 239)(131 192 240)(132 193 201)(133 194 202)(134 195 203)(135 196 204)(136 197 205)(137 198 206)(138 199 207)(139 200 208)(140 161 209)(141 162 210)(142 163 211)(143 164 212)(144 165 213)(145 166 214)(146 167 215)(147 168 216)(148 169 217)(149 170 218)(150 171 219)(151 172 220)(152 173 221)(153 174 222)(154 175 223)(155 176 224)(156 177 225)(157 178 226)(158 179 227)(159 180 228)(160 181 229)
(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)
(1 163)(2 162)(3 161)(4 200)(5 199)(6 198)(7 197)(8 196)(9 195)(10 194)(11 193)(12 192)(13 191)(14 190)(15 189)(16 188)(17 187)(18 186)(19 185)(20 184)(21 183)(22 182)(23 181)(24 180)(25 179)(26 178)(27 177)(28 176)(29 175)(30 174)(31 173)(32 172)(33 171)(34 170)(35 169)(36 168)(37 167)(38 166)(39 165)(40 164)(41 132)(42 131)(43 130)(44 129)(45 128)(46 127)(47 126)(48 125)(49 124)(50 123)(51 122)(52 121)(53 160)(54 159)(55 158)(56 157)(57 156)(58 155)(59 154)(60 153)(61 152)(62 151)(63 150)(64 149)(65 148)(66 147)(67 146)(68 145)(69 144)(70 143)(71 142)(72 141)(73 140)(74 139)(75 138)(76 137)(77 136)(78 135)(79 134)(80 133)(81 215)(82 214)(83 213)(84 212)(85 211)(86 210)(87 209)(88 208)(89 207)(90 206)(91 205)(92 204)(93 203)(94 202)(95 201)(96 240)(97 239)(98 238)(99 237)(100 236)(101 235)(102 234)(103 233)(104 232)(105 231)(106 230)(107 229)(108 228)(109 227)(110 226)(111 225)(112 224)(113 223)(114 222)(115 221)(116 220)(117 219)(118 218)(119 217)(120 216)
(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)(161 181)(162 182)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 189)(170 190)(171 191)(172 192)(173 193)(174 194)(175 195)(176 196)(177 197)(178 198)(179 199)(180 200)(201 221)(202 222)(203 223)(204 224)(205 225)(206 226)(207 227)(208 228)(209 229)(210 230)(211 231)(212 232)(213 233)(214 234)(215 235)(216 236)(217 237)(218 238)(219 239)(220 240)

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

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

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

138 conjugacy classes

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

138 irreducible representations

 dim 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 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C4○D8 D20 D20 C6×D5 C6×D5 C3×C4○D8 C3×D20 C3×D20 D40⋊7C2 C3×D40⋊7C2 kernel C3×D40⋊7C2 C3×C40⋊C2 C3×D40 C3×Dic20 C2×C120 C3×C4○D20 D40⋊7C2 C40⋊C2 D40 Dic20 C2×C40 C4○D20 C60 C2×C30 C2×C24 C24 C2×C12 C20 C2×C10 C2×C8 C15 C12 C2×C6 C8 C2×C4 C5 C4 C22 C3 C1 # reps 1 2 1 1 1 2 2 4 2 2 2 4 1 1 2 4 2 2 2 4 4 4 4 8 4 8 8 8 16 32

Matrix representation of C3×D407C2 in GL2(𝔽241) generated by

 225 0 0 225
,
 27 0 75 125
,
 23 210 87 218
,
 1 0 87 240
G:=sub<GL(2,GF(241))| [225,0,0,225],[27,75,0,125],[23,87,210,218],[1,87,0,240] >;

C3×D407C2 in GAP, Magma, Sage, TeX

C_3\times D_{40}\rtimes_7C_2
% in TeX

G:=Group("C3xD40:7C2");
// GroupNames label

G:=SmallGroup(480,697);
// by ID

G=gap.SmallGroup(480,697);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,142,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^40=c^2=d^2=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=b^20*c>;
// generators/relations

׿
×
𝔽