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## G = C20×C3⋊D4order 480 = 25·3·5

### Direct product of C20 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C20×C3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C10×C3⋊D4 — C20×C3⋊D4
 Lower central C3 — C6 — C20×C3⋊D4
 Upper central C1 — C2×C20 — C22×C20

Generators and relations for C20×C3⋊D4
G = < a,b,c,d | a20=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 388 in 188 conjugacy classes, 90 normal (58 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C10 [×3], C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×2], C12, D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C20 [×2], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×6], C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3, C22×C6, C5×S3 [×2], C30 [×3], C30 [×2], C4×D4, C2×C20 [×2], C2×C20 [×7], C5×D4 [×4], C22×C10, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3 [×2], C5×Dic3 [×2], C60 [×2], C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, C22×C20, D4×C10, C4×C3⋊D4, S3×C20 [×2], C10×Dic3 [×3], C5×C3⋊D4 [×4], C2×C60 [×2], C2×C60 [×2], S3×C2×C10, C22×C30, D4×C20, Dic3×C20, C5×Dic3⋊C4, C5×D6⋊C4, C5×C6.D4, S3×C2×C20, C10×C3⋊D4, C22×C60, C20×C3⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×2], C23, C10 [×7], D6 [×3], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C5×S3, C4×D4, C2×C20 [×6], C5×D4 [×2], C22×C10, S3×C2×C4, C4○D12, C2×C3⋊D4, S3×C10 [×3], C22×C20, D4×C10, C5×C4○D4, C4×C3⋊D4, S3×C20 [×2], C5×C3⋊D4 [×2], S3×C2×C10, D4×C20, S3×C2×C20, C5×C4○D12, C10×C3⋊D4, C20×C3⋊D4

Smallest permutation representation of C20×C3⋊D4
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 67 146)(2 68 147)(3 69 148)(4 70 149)(5 71 150)(6 72 151)(7 73 152)(8 74 153)(9 75 154)(10 76 155)(11 77 156)(12 78 157)(13 79 158)(14 80 159)(15 61 160)(16 62 141)(17 63 142)(18 64 143)(19 65 144)(20 66 145)(21 207 221)(22 208 222)(23 209 223)(24 210 224)(25 211 225)(26 212 226)(27 213 227)(28 214 228)(29 215 229)(30 216 230)(31 217 231)(32 218 232)(33 219 233)(34 220 234)(35 201 235)(36 202 236)(37 203 237)(38 204 238)(39 205 239)(40 206 240)(41 135 88)(42 136 89)(43 137 90)(44 138 91)(45 139 92)(46 140 93)(47 121 94)(48 122 95)(49 123 96)(50 124 97)(51 125 98)(52 126 99)(53 127 100)(54 128 81)(55 129 82)(56 130 83)(57 131 84)(58 132 85)(59 133 86)(60 134 87)(101 189 171)(102 190 172)(103 191 173)(104 192 174)(105 193 175)(106 194 176)(107 195 177)(108 196 178)(109 197 179)(110 198 180)(111 199 161)(112 200 162)(113 181 163)(114 182 164)(115 183 165)(116 184 166)(117 185 167)(118 186 168)(119 187 169)(120 188 170)
(1 91 207 105)(2 92 208 106)(3 93 209 107)(4 94 210 108)(5 95 211 109)(6 96 212 110)(7 97 213 111)(8 98 214 112)(9 99 215 113)(10 100 216 114)(11 81 217 115)(12 82 218 116)(13 83 219 117)(14 84 220 118)(15 85 201 119)(16 86 202 120)(17 87 203 101)(18 88 204 102)(19 89 205 103)(20 90 206 104)(21 193 146 44)(22 194 147 45)(23 195 148 46)(24 196 149 47)(25 197 150 48)(26 198 151 49)(27 199 152 50)(28 200 153 51)(29 181 154 52)(30 182 155 53)(31 183 156 54)(32 184 157 55)(33 185 158 56)(34 186 159 57)(35 187 160 58)(36 188 141 59)(37 189 142 60)(38 190 143 41)(39 191 144 42)(40 192 145 43)(61 132 235 169)(62 133 236 170)(63 134 237 171)(64 135 238 172)(65 136 239 173)(66 137 240 174)(67 138 221 175)(68 139 222 176)(69 140 223 177)(70 121 224 178)(71 122 225 179)(72 123 226 180)(73 124 227 161)(74 125 228 162)(75 126 229 163)(76 127 230 164)(77 128 231 165)(78 129 232 166)(79 130 233 167)(80 131 234 168)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 231)(22 232)(23 233)(24 234)(25 235)(26 236)(27 237)(28 238)(29 239)(30 240)(31 221)(32 222)(33 223)(34 224)(35 225)(36 226)(37 227)(38 228)(39 229)(40 230)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 161)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 101)(98 102)(99 103)(100 104)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(127 192)(128 193)(129 194)(130 195)(131 196)(132 197)(133 198)(134 199)(135 200)(136 181)(137 182)(138 183)(139 184)(140 185)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 5A 5B 5C 5D 6A ··· 6G 10A ··· 10L 10M ··· 10T 10U ··· 10AB 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20X 20Y ··· 20AV 30A ··· 30AB 60A ··· 60AF order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 6 6 2 1 1 1 1 2 2 6 ··· 6 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 2 2 2 2 1 ··· 1 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 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C10 C10 C20 S3 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C5×S3 C5×D4 C4○D12 S3×C10 S3×C10 C5×C4○D4 C5×C3⋊D4 S3×C20 C5×C4○D12 kernel C20×C3⋊D4 Dic3×C20 C5×Dic3⋊C4 C5×D6⋊C4 C5×C6.D4 S3×C2×C20 C10×C3⋊D4 C22×C60 C5×C3⋊D4 C4×C3⋊D4 C4×Dic3 Dic3⋊C4 D6⋊C4 C6.D4 S3×C2×C4 C2×C3⋊D4 C22×C12 C3⋊D4 C22×C20 C60 C2×C20 C22×C10 C30 C20 C2×C10 C22×C4 C12 C10 C2×C4 C23 C6 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 8 4 4 4 4 4 4 4 4 32 1 2 2 1 2 4 4 4 8 4 8 4 8 16 16 16

Matrix representation of C20×C3⋊D4 in GL3(𝔽61) generated by

 11 0 0 0 41 0 0 0 41
,
 1 0 0 0 0 60 0 1 60
,
 60 0 0 0 18 52 0 9 43
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(61))| [11,0,0,0,41,0,0,0,41],[1,0,0,0,0,1,0,60,60],[60,0,0,0,18,9,0,52,43],[1,0,0,0,0,1,0,1,0] >;

C20×C3⋊D4 in GAP, Magma, Sage, TeX

C_{20}\times C_3\rtimes D_4
% in TeX

G:=Group("C20xC3:D4");
// GroupNames label

G:=SmallGroup(480,807);
// by ID

G=gap.SmallGroup(480,807);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,226,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^3=c^4=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

׿
×
𝔽