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## G = C12⋊D20order 480 = 25·3·5

### 1st semidirect product of C12 and D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for C12⋊D20
G = < a,b,c | a12=b20=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 1564 in 216 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×12], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×2], D4 [×12], C23 [×4], D5 [×4], C10, C10 [×2], Dic3 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C15, C42, C2×D4 [×6], C20 [×2], C20 [×4], D10 [×12], C2×C10, D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C3×D5 [×2], D15 [×2], C30, C30 [×2], C41D4, D20 [×12], C2×C20, C2×C20 [×2], C22×D5 [×2], C22×D5 [×2], C4×Dic3, C2×D12, C2×C3⋊D4 [×4], C6×D4, C5×Dic3 [×4], C60 [×2], C6×D5 [×6], D30 [×6], C2×C30, C4×C20, C2×D20, C2×D20 [×5], C123D4, C3⋊D20 [×8], C3×D20 [×2], C10×Dic3 [×2], D60 [×2], C2×C60, D5×C2×C6 [×2], C22×D15 [×2], C204D4, Dic3×C20, C2×C3⋊D20 [×4], C6×D20, C2×D60, C12⋊D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, C41D4, D20 [×6], C22×D5, S3×D4 [×2], C2×C3⋊D4, S3×D5, C2×D20 [×3], C123D4, C3⋊D20 [×2], C2×S3×D5, C204D4, S3×D20 [×2], C2×C3⋊D20, C12⋊D20

Smallest permutation representation of C12⋊D20
On 240 points
Generators in S240
(1 33 150 101 134 205 167 66 56 191 97 229)(2 206 98 102 57 34 168 230 135 192 151 67)(3 35 152 103 136 207 169 68 58 193 99 231)(4 208 100 104 59 36 170 232 137 194 153 69)(5 37 154 105 138 209 171 70 60 195 81 233)(6 210 82 106 41 38 172 234 139 196 155 71)(7 39 156 107 140 211 173 72 42 197 83 235)(8 212 84 108 43 40 174 236 121 198 157 73)(9 21 158 109 122 213 175 74 44 199 85 237)(10 214 86 110 45 22 176 238 123 200 159 75)(11 23 160 111 124 215 177 76 46 181 87 239)(12 216 88 112 47 24 178 240 125 182 141 77)(13 25 142 113 126 217 179 78 48 183 89 221)(14 218 90 114 49 26 180 222 127 184 143 79)(15 27 144 115 128 219 161 80 50 185 91 223)(16 220 92 116 51 28 162 224 129 186 145 61)(17 29 146 117 130 201 163 62 52 187 93 225)(18 202 94 118 53 30 164 226 131 188 147 63)(19 31 148 119 132 203 165 64 54 189 95 227)(20 204 96 120 55 32 166 228 133 190 149 65)
(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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 120)(7 119)(8 118)(9 117)(10 116)(11 115)(12 114)(13 113)(14 112)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 146)(22 145)(23 144)(24 143)(25 142)(26 141)(27 160)(28 159)(29 158)(30 157)(31 156)(32 155)(33 154)(34 153)(35 152)(36 151)(37 150)(38 149)(39 148)(40 147)(41 65)(42 64)(43 63)(44 62)(45 61)(46 80)(47 79)(48 78)(49 77)(50 76)(51 75)(52 74)(53 73)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(81 205)(82 204)(83 203)(84 202)(85 201)(86 220)(87 219)(88 218)(89 217)(90 216)(91 215)(92 214)(93 213)(94 212)(95 211)(96 210)(97 209)(98 208)(99 207)(100 206)(121 226)(122 225)(123 224)(124 223)(125 222)(126 221)(127 240)(128 239)(129 238)(130 237)(131 236)(132 235)(133 234)(134 233)(135 232)(136 231)(137 230)(138 229)(139 228)(140 227)(161 181)(162 200)(163 199)(164 198)(165 197)(166 196)(167 195)(168 194)(169 193)(170 192)(171 191)(172 190)(173 189)(174 188)(175 187)(176 186)(177 185)(178 184)(179 183)(180 182)

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

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

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

72 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 ··· 20H 20I ··· 20X 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 30 ··· 30 60 ··· 60 size 1 1 1 1 20 20 60 60 2 2 2 6 6 6 6 2 2 2 2 2 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

72 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 C3⋊D4 D20 D20 S3×D4 S3×D5 C3⋊D20 C2×S3×D5 S3×D20 kernel C12⋊D20 Dic3×C20 C2×C3⋊D20 C6×D20 C2×D60 C2×D20 C5×Dic3 C60 C4×Dic3 C2×C20 C22×D5 C2×Dic3 C2×C12 C20 Dic3 C12 C10 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 1 4 2 2 1 2 4 2 4 16 8 2 2 4 2 8

Matrix representation of C12⋊D20 in GL4(𝔽61) generated by

 32 4 0 0 3 29 0 0 0 0 0 1 0 0 60 60
,
 1 60 0 0 45 17 0 0 0 0 52 43 0 0 52 9
,
 29 57 0 0 27 32 0 0 0 0 1 0 0 0 60 60
G:=sub<GL(4,GF(61))| [32,3,0,0,4,29,0,0,0,0,0,60,0,0,1,60],[1,45,0,0,60,17,0,0,0,0,52,52,0,0,43,9],[29,27,0,0,57,32,0,0,0,0,1,60,0,0,0,60] >;

C12⋊D20 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,534);
// by ID

G=gap.SmallGroup(480,534);
# by ID

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

G:=Group<a,b,c|a^12=b^20=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

׿
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