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## G = C2×C40⋊S3order 480 = 25·3·5

### Direct product of C2 and C40⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C40⋊S3
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — C2×C4×D15 — C2×C40⋊S3
 Lower central C15 — C30 — C2×C40⋊S3
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C2×C40⋊S3
G = < a,b,c,d | a2=b40=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b29, dcd=c-1 >

Subgroups: 692 in 136 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, C2×C8, M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30, C30 [×2], C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C8⋊D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, C2×C8⋊S3, C153C8 [×2], C120 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C2×C8⋊D5, C40⋊S3 [×4], C2×C153C8, C2×C120, C2×C4×D15, C2×C40⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, D15, C2×M4(2), C4×D5 [×2], C22×D5, C8⋊S3 [×2], S3×C2×C4, D30 [×3], C8⋊D5 [×2], C2×C4×D5, C2×C8⋊S3, C4×D15 [×2], C22×D15, C2×C8⋊D5, C40⋊S3 [×2], C2×C4×D15, C2×C40⋊S3

Smallest permutation representation of C2×C40⋊S3
On 240 points
Generators in S240
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 81)(41 224)(42 225)(43 226)(44 227)(45 228)(46 229)(47 230)(48 231)(49 232)(50 233)(51 234)(52 235)(53 236)(54 237)(55 238)(56 239)(57 240)(58 201)(59 202)(60 203)(61 204)(62 205)(63 206)(64 207)(65 208)(66 209)(67 210)(68 211)(69 212)(70 213)(71 214)(72 215)(73 216)(74 217)(75 218)(76 219)(77 220)(78 221)(79 222)(80 223)(121 189)(122 190)(123 191)(124 192)(125 193)(126 194)(127 195)(128 196)(129 197)(130 198)(131 199)(132 200)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(141 169)(142 170)(143 171)(144 172)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 181)(154 182)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)
(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 57 143)(2 58 144)(3 59 145)(4 60 146)(5 61 147)(6 62 148)(7 63 149)(8 64 150)(9 65 151)(10 66 152)(11 67 153)(12 68 154)(13 69 155)(14 70 156)(15 71 157)(16 72 158)(17 73 159)(18 74 160)(19 75 121)(20 76 122)(21 77 123)(22 78 124)(23 79 125)(24 80 126)(25 41 127)(26 42 128)(27 43 129)(28 44 130)(29 45 131)(30 46 132)(31 47 133)(32 48 134)(33 49 135)(34 50 136)(35 51 137)(36 52 138)(37 53 139)(38 54 140)(39 55 141)(40 56 142)(81 239 170)(82 240 171)(83 201 172)(84 202 173)(85 203 174)(86 204 175)(87 205 176)(88 206 177)(89 207 178)(90 208 179)(91 209 180)(92 210 181)(93 211 182)(94 212 183)(95 213 184)(96 214 185)(97 215 186)(98 216 187)(99 217 188)(100 218 189)(101 219 190)(102 220 191)(103 221 192)(104 222 193)(105 223 194)(106 224 195)(107 225 196)(108 226 197)(109 227 198)(110 228 199)(111 229 200)(112 230 161)(113 231 162)(114 232 163)(115 233 164)(116 234 165)(117 235 166)(118 236 167)(119 237 168)(120 238 169)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 159)(42 148)(43 137)(44 126)(45 155)(46 144)(47 133)(48 122)(49 151)(50 140)(51 129)(52 158)(53 147)(54 136)(55 125)(56 154)(57 143)(58 132)(59 121)(60 150)(61 139)(62 128)(63 157)(64 146)(65 135)(66 124)(67 153)(68 142)(69 131)(70 160)(71 149)(72 138)(73 127)(74 156)(75 145)(76 134)(77 123)(78 152)(79 141)(80 130)(81 93)(83 111)(84 100)(85 89)(86 118)(87 107)(88 96)(90 114)(91 103)(94 110)(95 99)(97 117)(98 106)(101 113)(104 120)(105 109)(108 116)(115 119)(161 230)(162 219)(163 208)(164 237)(165 226)(166 215)(167 204)(168 233)(169 222)(170 211)(171 240)(172 229)(173 218)(174 207)(175 236)(176 225)(177 214)(178 203)(179 232)(180 221)(181 210)(182 239)(183 228)(184 217)(185 206)(186 235)(187 224)(188 213)(189 202)(190 231)(191 220)(192 209)(193 238)(194 227)(195 216)(196 205)(197 234)(198 223)(199 212)(200 201)

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

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

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

132 conjugacy classes

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

132 irreducible representations

 dim 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 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D5 D6 D6 M4(2) D10 D10 C4×S3 C4×S3 D15 C4×D5 C4×D5 C8⋊S3 D30 D30 C8⋊D5 C4×D15 C4×D15 C40⋊S3 kernel C2×C40⋊S3 C40⋊S3 C2×C15⋊3C8 C2×C120 C2×C4×D15 C4×D15 C2×Dic15 C22×D15 C2×C40 C2×C24 C40 C2×C20 C30 C24 C2×C12 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 1 2 2 1 4 4 2 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C2×C40⋊S3 in GL3(𝔽241) generated by

 240 0 0 0 240 0 0 0 240
,
 64 0 0 0 196 45 0 196 25
,
 1 0 0 0 63 211 0 30 177
,
 240 0 0 0 189 240 0 52 52
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[64,0,0,0,196,196,0,45,25],[1,0,0,0,63,30,0,211,177],[240,0,0,0,189,52,0,240,52] >;

C2×C40⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_{40}\rtimes S_3
% in TeX

G:=Group("C2xC40:S3");
// GroupNames label

G:=SmallGroup(480,865);
// by ID

G=gap.SmallGroup(480,865);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,58,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^40=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^29,d*c*d=c^-1>;
// generators/relations

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