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## G = C20⋊D12order 480 = 25·3·5

### 1st semidirect product of C20 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C20⋊D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×C5⋊D12 — C20⋊D12
 Lower central C15 — C2×C30 — C20⋊D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C20⋊D12
G = < a,b,c | a20=b12=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >

Subgroups: 1484 in 216 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×12], C5, S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×2], D4 [×12], C23 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C12 [×2], C12 [×4], D6 [×12], C2×C6, C15, C42, C2×D4 [×6], Dic5 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], D12 [×12], C2×C12, C2×C12 [×2], C22×S3 [×2], C22×S3 [×2], C5×S3 [×2], D15 [×2], C30, C30 [×2], C41D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C4×C12, C2×D12, C2×D12 [×5], C3×Dic5 [×4], C60 [×2], S3×C10 [×6], D30 [×6], C2×C30, C4×Dic5, C2×D20, C2×C5⋊D4 [×4], D4×C10, C4⋊D12, C5⋊D12 [×8], C6×Dic5 [×2], C5×D12 [×2], D60 [×2], C2×C60, S3×C2×C10 [×2], C22×D15 [×2], C20⋊D4, C12×Dic5, C2×C5⋊D12 [×4], C10×D12, C2×D60, C20⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×6], C22×S3, C41D4, C5⋊D4 [×2], C22×D5, C2×D12 [×3], S3×D5, D4×D5 [×2], C2×C5⋊D4, C4⋊D12, C5⋊D12 [×2], C2×S3×D5, C20⋊D4, D5×D12 [×2], C2×C5⋊D12, C20⋊D12

Smallest permutation representation of C20⋊D12
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 87 107 135 47 72 40 185 162 207 149 235)(2 96 108 124 48 61 21 194 163 216 150 224)(3 85 109 133 49 70 22 183 164 205 151 233)(4 94 110 122 50 79 23 192 165 214 152 222)(5 83 111 131 51 68 24 181 166 203 153 231)(6 92 112 140 52 77 25 190 167 212 154 240)(7 81 113 129 53 66 26 199 168 201 155 229)(8 90 114 138 54 75 27 188 169 210 156 238)(9 99 115 127 55 64 28 197 170 219 157 227)(10 88 116 136 56 73 29 186 171 208 158 236)(11 97 117 125 57 62 30 195 172 217 159 225)(12 86 118 134 58 71 31 184 173 206 160 234)(13 95 119 123 59 80 32 193 174 215 141 223)(14 84 120 132 60 69 33 182 175 204 142 232)(15 93 101 121 41 78 34 191 176 213 143 221)(16 82 102 130 42 67 35 200 177 202 144 230)(17 91 103 139 43 76 36 189 178 211 145 239)(18 100 104 128 44 65 37 198 179 220 146 228)(19 89 105 137 45 74 38 187 180 209 147 237)(20 98 106 126 46 63 39 196 161 218 148 226)
(1 177)(2 176)(3 175)(4 174)(5 173)(6 172)(7 171)(8 170)(9 169)(10 168)(11 167)(12 166)(13 165)(14 164)(15 163)(16 162)(17 161)(18 180)(19 179)(20 178)(21 101)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 121)(62 140)(63 139)(64 138)(65 137)(66 136)(67 135)(68 134)(69 133)(70 132)(71 131)(72 130)(73 129)(74 128)(75 127)(76 126)(77 125)(78 124)(79 123)(80 122)(81 186)(82 185)(83 184)(84 183)(85 182)(86 181)(87 200)(88 199)(89 198)(90 197)(91 196)(92 195)(93 194)(94 193)(95 192)(96 191)(97 190)(98 189)(99 188)(100 187)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 160)(154 159)(155 158)(156 157)(201 236)(202 235)(203 234)(204 233)(205 232)(206 231)(207 230)(208 229)(209 228)(210 227)(211 226)(212 225)(213 224)(214 223)(215 222)(216 221)(217 240)(218 239)(219 238)(220 237)```

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 D12 D12 C5⋊D4 S3×D5 D4×D5 C5⋊D12 C2×S3×D5 D5×D12 kernel C20⋊D12 C12×Dic5 C2×C5⋊D12 C10×D12 C2×D60 C4×Dic5 C3×Dic5 C60 C2×D12 C2×Dic5 C2×C20 C2×C12 C22×S3 Dic5 C20 C12 C2×C4 C6 C4 C22 C2 # reps 1 1 4 1 1 1 4 2 2 2 1 2 4 8 4 8 2 4 4 2 8

Matrix representation of C20⋊D12 in GL4(𝔽61) generated by

 38 15 0 0 46 23 0 0 0 0 17 1 0 0 16 1
,
 15 38 0 0 23 38 0 0 0 0 39 30 0 0 55 22
,
 46 23 0 0 38 15 0 0 0 0 44 60 0 0 44 17
`G:=sub<GL(4,GF(61))| [38,46,0,0,15,23,0,0,0,0,17,16,0,0,1,1],[15,23,0,0,38,38,0,0,0,0,39,55,0,0,30,22],[46,38,0,0,23,15,0,0,0,0,44,44,0,0,60,17] >;`

C20⋊D12 in GAP, Magma, Sage, TeX

`C_{20}\rtimes D_{12}`
`% in TeX`

`G:=Group("C20:D12");`
`// GroupNames label`

`G:=SmallGroup(480,527);`
`// by ID`

`G=gap.SmallGroup(480,527);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,219,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^12=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

׿
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