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## G = C12⋊2D20order 480 = 25·3·5

### 2nd semidirect product of C12 and D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C12⋊2D20
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C3⋊D20 — C12⋊2D20
 Lower central C15 — C2×C30 — C12⋊2D20
 Upper central C1 — C22 — C2×C4

Generators and relations for C122D20
G = < a,b,c | a60=b4=c2=1, bab-1=a11, cac=a29, cbc=b-1 >

Subgroups: 1180 in 188 conjugacy classes, 54 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×4], C10 [×3], Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×6], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×10], C2×C10, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C3×D5 [×2], D15 [×2], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5 [×2], C22×D5, C4⋊Dic3, C6.D4 [×2], S3×C2×C4, C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], Dic15, C60 [×2], C6×D5 [×6], D30 [×2], D30 [×2], C2×C30, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20 [×2], D63D4, C3⋊D20 [×4], C3×D20 [×2], C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, D5×C2×C6 [×2], C22×D15, C4⋊D20, D10⋊Dic3 [×2], C5×C4⋊Dic3, C2×C3⋊D20 [×2], C6×D20, C2×C4×D15, C122D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, D20 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, Q82D5, D63D4, C3⋊D20 [×2], C2×S3×D5, C4⋊D20, D20⋊S3, C20⋊D6, C2×C3⋊D20, C122D20

Smallest permutation representation of C122D20
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)
(1 147 202 104)(2 158 203 115)(3 169 204 66)(4 180 205 77)(5 131 206 88)(6 142 207 99)(7 153 208 110)(8 164 209 61)(9 175 210 72)(10 126 211 83)(11 137 212 94)(12 148 213 105)(13 159 214 116)(14 170 215 67)(15 121 216 78)(16 132 217 89)(17 143 218 100)(18 154 219 111)(19 165 220 62)(20 176 221 73)(21 127 222 84)(22 138 223 95)(23 149 224 106)(24 160 225 117)(25 171 226 68)(26 122 227 79)(27 133 228 90)(28 144 229 101)(29 155 230 112)(30 166 231 63)(31 177 232 74)(32 128 233 85)(33 139 234 96)(34 150 235 107)(35 161 236 118)(36 172 237 69)(37 123 238 80)(38 134 239 91)(39 145 240 102)(40 156 181 113)(41 167 182 64)(42 178 183 75)(43 129 184 86)(44 140 185 97)(45 151 186 108)(46 162 187 119)(47 173 188 70)(48 124 189 81)(49 135 190 92)(50 146 191 103)(51 157 192 114)(52 168 193 65)(53 179 194 76)(54 130 195 87)(55 141 196 98)(56 152 197 109)(57 163 198 120)(58 174 199 71)(59 125 200 82)(60 136 201 93)
(1 202)(2 231)(3 200)(4 229)(5 198)(6 227)(7 196)(8 225)(9 194)(10 223)(11 192)(12 221)(13 190)(14 219)(15 188)(16 217)(17 186)(18 215)(19 184)(20 213)(21 182)(22 211)(23 240)(24 209)(25 238)(26 207)(27 236)(28 205)(29 234)(30 203)(31 232)(32 201)(33 230)(34 199)(35 228)(36 197)(37 226)(38 195)(39 224)(40 193)(41 222)(42 191)(43 220)(44 189)(45 218)(46 187)(47 216)(48 185)(49 214)(50 183)(51 212)(52 181)(53 210)(54 239)(55 208)(56 237)(57 206)(58 235)(59 204)(60 233)(61 117)(62 86)(63 115)(64 84)(65 113)(66 82)(67 111)(68 80)(69 109)(70 78)(71 107)(72 76)(73 105)(75 103)(77 101)(79 99)(81 97)(83 95)(85 93)(87 91)(88 120)(90 118)(92 116)(94 114)(96 112)(98 110)(100 108)(102 106)(121 173)(122 142)(123 171)(124 140)(125 169)(126 138)(127 167)(128 136)(129 165)(130 134)(131 163)(133 161)(135 159)(137 157)(139 155)(141 153)(143 151)(144 180)(145 149)(146 178)(148 176)(150 174)(152 172)(154 170)(156 168)(158 166)(160 164)(175 179)```

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 C4○D4 D10 D10 C3⋊D4 D20 S3×D4 D4⋊2S3 S3×D5 D4×D5 Q8⋊2D5 C3⋊D20 C2×S3×D5 D20⋊S3 C20⋊D6 kernel C12⋊2D20 D10⋊Dic3 C5×C4⋊Dic3 C2×C3⋊D20 C6×D20 C2×C4×D15 C2×D20 C60 D30 C4⋊Dic3 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C20 C12 C10 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 1 1 1 2 2 2 1 2 2 4 2 4 8 1 1 2 2 2 4 2 4 4

Matrix representation of C122D20 in GL6(𝔽61)

 1 60 0 0 0 0 1 0 0 0 0 0 0 0 50 30 0 0 0 0 0 11 0 0 0 0 0 0 18 18 0 0 0 0 43 60
,
 9 9 0 0 0 0 18 52 0 0 0 0 0 0 35 37 0 0 0 0 46 26 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 1 0 0 0 0 0 1 0 0 0 0 0 0 60 36 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 43 60

`G:=sub<GL(6,GF(61))| [1,1,0,0,0,0,60,0,0,0,0,0,0,0,50,0,0,0,0,0,30,11,0,0,0,0,0,0,18,43,0,0,0,0,18,60],[9,18,0,0,0,0,9,52,0,0,0,0,0,0,35,46,0,0,0,0,37,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,1,1,0,0,0,0,0,0,60,0,0,0,0,0,36,1,0,0,0,0,0,0,1,43,0,0,0,0,0,60] >;`

C122D20 in GAP, Magma, Sage, TeX

`C_{12}\rtimes_2D_{20}`
`% in TeX`

`G:=Group("C12:2D20");`
`// GroupNames label`

`G:=SmallGroup(480,541);`
`// by ID`

`G=gap.SmallGroup(480,541);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,422,219,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^11,c*a*c=a^29,c*b*c=b^-1>;`
`// generators/relations`

׿
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