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## G = C2×C3⋊D40order 480 = 25·3·5

### Direct product of C2 and C3⋊D40

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C3⋊D40
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C3⋊D40 — C2×C3⋊D40
 Lower central C15 — C30 — C60 — C2×C3⋊D40
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C3⋊D40
G = < a,b,c,d | a2=b3=c40=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1148 in 152 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C15, C2×C8, D8 [×4], C2×D4 [×2], C20 [×2], D10 [×8], C2×C10, C3⋊C8 [×2], D12 [×3], C2×C12, C3×D4 [×3], C22×S3, C22×C6, C3×D5 [×2], D15 [×2], C30, C30 [×2], C2×D8, C40 [×2], D20 [×2], D20 [×4], C2×C20, C22×D5 [×2], C2×C3⋊C8, D4⋊S3 [×4], C2×D12, C6×D4, C60 [×2], C6×D5 [×4], D30 [×4], C2×C30, D40 [×4], C2×C40, C2×D20, C2×D20, C2×D4⋊S3, C5×C3⋊C8 [×2], C3×D20 [×2], C3×D20, D60 [×2], D60, C2×C60, D5×C2×C6, C22×D15, C2×D40, C3⋊D40 [×4], C10×C3⋊C8, C6×D20, C2×D60, C2×C3⋊D40
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C2×D8, D20 [×2], C22×D5, D4⋊S3 [×2], C2×C3⋊D4, S3×D5, D40 [×2], C2×D20, C2×D4⋊S3, C3⋊D20 [×2], C2×S3×D5, C2×D40, C3⋊D40 [×2], C2×C3⋊D20, C2×C3⋊D40

Smallest permutation representation of C2×C3⋊D40
On 240 points
Generators in S240
(1 217)(2 218)(3 219)(4 220)(5 221)(6 222)(7 223)(8 224)(9 225)(10 226)(11 227)(12 228)(13 229)(14 230)(15 231)(16 232)(17 233)(18 234)(19 235)(20 236)(21 237)(22 238)(23 239)(24 240)(25 201)(26 202)(27 203)(28 204)(29 205)(30 206)(31 207)(32 208)(33 209)(34 210)(35 211)(36 212)(37 213)(38 214)(39 215)(40 216)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(121 187)(122 188)(123 189)(124 190)(125 191)(126 192)(127 193)(128 194)(129 195)(130 196)(131 197)(132 198)(133 199)(134 200)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 167)(142 168)(143 169)(144 170)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(151 177)(152 178)(153 179)(154 180)(155 181)(156 182)(157 183)(158 184)(159 185)(160 186)
(1 126 111)(2 112 127)(3 128 113)(4 114 129)(5 130 115)(6 116 131)(7 132 117)(8 118 133)(9 134 119)(10 120 135)(11 136 81)(12 82 137)(13 138 83)(14 84 139)(15 140 85)(16 86 141)(17 142 87)(18 88 143)(19 144 89)(20 90 145)(21 146 91)(22 92 147)(23 148 93)(24 94 149)(25 150 95)(26 96 151)(27 152 97)(28 98 153)(29 154 99)(30 100 155)(31 156 101)(32 102 157)(33 158 103)(34 104 159)(35 160 105)(36 106 121)(37 122 107)(38 108 123)(39 124 109)(40 110 125)(41 205 180)(42 181 206)(43 207 182)(44 183 208)(45 209 184)(46 185 210)(47 211 186)(48 187 212)(49 213 188)(50 189 214)(51 215 190)(52 191 216)(53 217 192)(54 193 218)(55 219 194)(56 195 220)(57 221 196)(58 197 222)(59 223 198)(60 199 224)(61 225 200)(62 161 226)(63 227 162)(64 163 228)(65 229 164)(66 165 230)(67 231 166)(68 167 232)(69 233 168)(70 169 234)(71 235 170)(72 171 236)(73 237 172)(74 173 238)(75 239 174)(76 175 240)(77 201 176)(78 177 202)(79 203 178)(80 179 204)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 178)(42 177)(43 176)(44 175)(45 174)(46 173)(47 172)(48 171)(49 170)(50 169)(51 168)(52 167)(53 166)(54 165)(55 164)(56 163)(57 162)(58 161)(59 200)(60 199)(61 198)(62 197)(63 196)(64 195)(65 194)(66 193)(67 192)(68 191)(69 190)(70 189)(71 188)(72 187)(73 186)(74 185)(75 184)(76 183)(77 182)(78 181)(79 180)(80 179)(81 130)(82 129)(83 128)(84 127)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 160)(92 159)(93 158)(94 157)(95 156)(96 155)(97 154)(98 153)(99 152)(100 151)(101 150)(102 149)(103 148)(104 147)(105 146)(106 145)(107 144)(108 143)(109 142)(110 141)(111 140)(112 139)(113 138)(114 137)(115 136)(116 135)(117 134)(118 133)(119 132)(120 131)(201 207)(202 206)(203 205)(208 240)(209 239)(210 238)(211 237)(212 236)(213 235)(214 234)(215 233)(216 232)(217 231)(218 230)(219 229)(220 228)(221 227)(222 226)(223 225)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D8 D10 D10 C3⋊D4 C3⋊D4 D20 D20 D40 D4⋊S3 S3×D5 C3⋊D20 C2×S3×D5 C3⋊D20 C3⋊D40 kernel C2×C3⋊D40 C3⋊D40 C10×C3⋊C8 C6×D20 C2×D60 C2×D20 C60 C2×C30 C2×C3⋊C8 D20 C2×C20 C30 C3⋊C8 C2×C12 C20 C2×C10 C12 C2×C6 C6 C10 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 16 2 2 2 2 2 8

Matrix representation of C2×C3⋊D40 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 54 0 0 174 239 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 67 1 0 0 0 0 228 214 0 0 27 56
,
 1 0 0 0 174 240 0 0 0 0 197 78 0 0 3 44
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,174,0,0,54,239,0,0,0,0,1,0,0,0,0,1],[240,67,0,0,0,1,0,0,0,0,228,27,0,0,214,56],[1,174,0,0,0,240,0,0,0,0,197,3,0,0,78,44] >;

C2×C3⋊D40 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{40}
% in TeX

G:=Group("C2xC3:D40");
// GroupNames label

G:=SmallGroup(480,376);
// by ID

G=gap.SmallGroup(480,376);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,675,80,1356,18822]);
// Polycyclic

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

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