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## G = D40⋊5S3order 480 = 25·3·5

### 5th semidirect product of D40 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D40⋊5S3
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — D20⋊S3 — D40⋊5S3
 Lower central C15 — C30 — C60 — D40⋊5S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for D405S3
G = < a,b,c,d | a40=b2=c3=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a28b, dcd=c-1 >

Subgroups: 732 in 124 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×3], C10, Dic3 [×3], C12, D6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10 [×3], C3⋊C8, C24, Dic6 [×2], C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×D5 [×2], D15, C30, C4○D8, C52C8, C40, C4×D5 [×3], D20 [×2], D20 [×2], C5×Q8 [×2], S3×C8, Dic12, D4.S3 [×2], C3×D8, D42S3 [×2], C5×Dic3 [×2], Dic15, C60, C6×D5 [×2], D30, C8×D5, D40, Q8⋊D5 [×2], C5×Q16, Q82D5 [×2], D83S3, C153C8, C120, D5×Dic3 [×2], C3⋊D20 [×2], C3×D20 [×2], C5×Dic6 [×2], C4×D15, Q8.D10, C30.D4 [×2], C3×D40, C5×Dic12, C8×D15, D20⋊S3 [×2], D405S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, D83S3, C2×S3×D5, Q8.D10, C20⋊D6, D405S3

Smallest permutation representation of D405S3
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 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 84)(17 83)(18 82)(19 81)(20 120)(21 119)(22 118)(23 117)(24 116)(25 115)(26 114)(27 113)(28 112)(29 111)(30 110)(31 109)(32 108)(33 107)(34 106)(35 105)(36 104)(37 103)(38 102)(39 101)(40 100)(41 189)(42 188)(43 187)(44 186)(45 185)(46 184)(47 183)(48 182)(49 181)(50 180)(51 179)(52 178)(53 177)(54 176)(55 175)(56 174)(57 173)(58 172)(59 171)(60 170)(61 169)(62 168)(63 167)(64 166)(65 165)(66 164)(67 163)(68 162)(69 161)(70 200)(71 199)(72 198)(73 197)(74 196)(75 195)(76 194)(77 193)(78 192)(79 191)(80 190)(121 239)(122 238)(123 237)(124 236)(125 235)(126 234)(127 233)(128 232)(129 231)(130 230)(131 229)(132 228)(133 227)(134 226)(135 225)(136 224)(137 223)(138 222)(139 221)(140 220)(141 219)(142 218)(143 217)(144 216)(145 215)(146 214)(147 213)(148 212)(149 211)(150 210)(151 209)(152 208)(153 207)(154 206)(155 205)(156 204)(157 203)(158 202)(159 201)(160 240)
(1 140 46)(2 141 47)(3 142 48)(4 143 49)(5 144 50)(6 145 51)(7 146 52)(8 147 53)(9 148 54)(10 149 55)(11 150 56)(12 151 57)(13 152 58)(14 153 59)(15 154 60)(16 155 61)(17 156 62)(18 157 63)(19 158 64)(20 159 65)(21 160 66)(22 121 67)(23 122 68)(24 123 69)(25 124 70)(26 125 71)(27 126 72)(28 127 73)(29 128 74)(30 129 75)(31 130 76)(32 131 77)(33 132 78)(34 133 79)(35 134 80)(36 135 41)(37 136 42)(38 137 43)(39 138 44)(40 139 45)(81 202 166)(82 203 167)(83 204 168)(84 205 169)(85 206 170)(86 207 171)(87 208 172)(88 209 173)(89 210 174)(90 211 175)(91 212 176)(92 213 177)(93 214 178)(94 215 179)(95 216 180)(96 217 181)(97 218 182)(98 219 183)(99 220 184)(100 221 185)(101 222 186)(102 223 187)(103 224 188)(104 225 189)(105 226 190)(106 227 191)(107 228 192)(108 229 193)(109 230 194)(110 231 195)(111 232 196)(112 233 197)(113 234 198)(114 235 199)(115 236 200)(116 237 161)(117 238 162)(118 239 163)(119 240 164)(120 201 165)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(41 135)(42 144)(43 153)(44 122)(45 131)(46 140)(47 149)(48 158)(49 127)(50 136)(51 145)(52 154)(53 123)(54 132)(55 141)(56 150)(57 159)(58 128)(59 137)(60 146)(61 155)(62 124)(63 133)(64 142)(65 151)(66 160)(67 129)(68 138)(69 147)(70 156)(71 125)(72 134)(73 143)(74 152)(75 121)(76 130)(77 139)(78 148)(79 157)(80 126)(81 109)(82 118)(83 87)(84 96)(85 105)(86 114)(88 92)(89 101)(90 110)(91 119)(93 97)(94 106)(95 115)(98 102)(99 111)(100 120)(103 107)(104 116)(108 112)(113 117)(161 225)(162 234)(163 203)(164 212)(165 221)(166 230)(167 239)(168 208)(169 217)(170 226)(171 235)(172 204)(173 213)(174 222)(175 231)(176 240)(177 209)(178 218)(179 227)(180 236)(181 205)(182 214)(183 223)(184 232)(185 201)(186 210)(187 219)(188 228)(189 237)(190 206)(191 215)(192 224)(193 233)(194 202)(195 211)(196 220)(197 229)(198 238)(199 207)(200 216)```

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 C4○D8 S3×D4 S3×D5 D4×D5 D8⋊3S3 C2×S3×D5 Q8.D10 C20⋊D6 D40⋊5S3 kernel D40⋊5S3 C30.D4 C3×D40 C5×Dic12 C8×D15 D20⋊S3 D40 Dic15 D30 Dic12 C40 D20 C24 Dic6 C15 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 1 1 2 1 2 2 4 4 1 2 2 2 2 4 4 8

Matrix representation of D405S3 in GL6(𝔽241)

 189 190 0 0 0 0 52 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 233
,
 52 1 0 0 0 0 189 189 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 24 0 0 0 0 231 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 189 190 0 0 0 0 53 52 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 240

`G:=sub<GL(6,GF(241))| [189,52,0,0,0,0,190,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,233],[52,189,0,0,0,0,1,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,231,0,0,0,0,24,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[189,53,0,0,0,0,190,52,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240] >;`

D405S3 in GAP, Magma, Sage, TeX

`D_{40}\rtimes_5S_3`
`% in TeX`

`G:=Group("D40:5S3");`
`// GroupNames label`

`G:=SmallGroup(480,353);`
`// by ID`

`G=gap.SmallGroup(480,353);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,142,675,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^28*b,d*c*d=c^-1>;`
`// generators/relations`

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