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## G = C20⋊2D12order 480 = 25·3·5

### 2nd semidirect product of C20 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C20⋊2D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C2×C5⋊D12 — C20⋊2D12
 Lower central C15 — C2×C30 — C20⋊2D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C202D12
G = < a,b,c | a60=b4=c2=1, bab-1=a19, cac=a29, cbc=b-1 >

Subgroups: 1100 in 188 conjugacy classes, 54 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C4⋊D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C3×Dic5, Dic15, C60, S3×C10, D30, D30, C2×C30, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C5⋊D4, D4×C10, C12⋊D4, C5⋊D12, C6×Dic5, C5×D12, C4×D15, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C202D4, D6⋊Dic5, C3×C4⋊Dic5, C2×C5⋊D12, C10×D12, C2×C4×D15, C202D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C4⋊D4, C5⋊D4, C22×D5, C2×D12, S3×D4, Q83S3, S3×D5, D4×D5, D42D5, C2×C5⋊D4, C12⋊D4, C5⋊D12, C2×S3×D5, C202D4, D12⋊D5, C20⋊D6, C2×C5⋊D12, C202D12

Smallest permutation representation of C202D12
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)
(1 179 110 189)(2 138 111 208)(3 157 112 227)(4 176 113 186)(5 135 114 205)(6 154 115 224)(7 173 116 183)(8 132 117 202)(9 151 118 221)(10 170 119 240)(11 129 120 199)(12 148 61 218)(13 167 62 237)(14 126 63 196)(15 145 64 215)(16 164 65 234)(17 123 66 193)(18 142 67 212)(19 161 68 231)(20 180 69 190)(21 139 70 209)(22 158 71 228)(23 177 72 187)(24 136 73 206)(25 155 74 225)(26 174 75 184)(27 133 76 203)(28 152 77 222)(29 171 78 181)(30 130 79 200)(31 149 80 219)(32 168 81 238)(33 127 82 197)(34 146 83 216)(35 165 84 235)(36 124 85 194)(37 143 86 213)(38 162 87 232)(39 121 88 191)(40 140 89 210)(41 159 90 229)(42 178 91 188)(43 137 92 207)(44 156 93 226)(45 175 94 185)(46 134 95 204)(47 153 96 223)(48 172 97 182)(49 131 98 201)(50 150 99 220)(51 169 100 239)(52 128 101 198)(53 147 102 217)(54 166 103 236)(55 125 104 195)(56 144 105 214)(57 163 106 233)(58 122 107 192)(59 141 108 211)(60 160 109 230)
(2 30)(3 59)(4 28)(5 57)(6 26)(7 55)(8 24)(9 53)(10 22)(11 51)(12 20)(13 49)(14 18)(15 47)(17 45)(19 43)(21 41)(23 39)(25 37)(27 35)(29 33)(32 60)(34 58)(36 56)(38 54)(40 52)(42 50)(44 48)(61 69)(62 98)(63 67)(64 96)(66 94)(68 92)(70 90)(71 119)(72 88)(73 117)(74 86)(75 115)(76 84)(77 113)(78 82)(79 111)(81 109)(83 107)(85 105)(87 103)(89 101)(91 99)(93 97)(100 120)(102 118)(104 116)(106 114)(108 112)(121 187)(122 216)(123 185)(124 214)(125 183)(126 212)(127 181)(128 210)(129 239)(130 208)(131 237)(132 206)(133 235)(134 204)(135 233)(136 202)(137 231)(138 200)(139 229)(140 198)(141 227)(142 196)(143 225)(144 194)(145 223)(146 192)(147 221)(148 190)(149 219)(150 188)(151 217)(152 186)(153 215)(154 184)(155 213)(156 182)(157 211)(158 240)(159 209)(160 238)(161 207)(162 236)(163 205)(164 234)(165 203)(166 232)(167 201)(168 230)(169 199)(170 228)(171 197)(172 226)(173 195)(174 224)(175 193)(176 222)(177 191)(178 220)(179 189)(180 218)

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

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

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

60 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 12F 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 30 30 2 2 2 20 20 30 30 2 2 2 2 2 2 ··· 2 12 ··· 12 4 4 20 20 20 20 4 4 4 4 4 4 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 D12 C5⋊D4 S3×D4 Q8⋊3S3 S3×D5 D4×D5 D4⋊2D5 C5⋊D12 C2×S3×D5 D12⋊D5 C20⋊D6 kernel C20⋊2D12 D6⋊Dic5 C3×C4⋊Dic5 C2×C5⋊D12 C10×D12 C2×C4×D15 C4⋊Dic5 C60 D30 C2×D12 C2×Dic5 C2×C20 C30 C2×C12 C22×S3 C20 C12 C10 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 1 1 1 2 2 2 2 1 2 2 4 4 8 1 1 2 2 2 4 2 4 4

Matrix representation of C202D12 in GL6(𝔽61)

 44 44 0 0 0 0 17 60 0 0 0 0 0 0 0 60 0 0 0 0 1 60 0 0 0 0 0 0 1 20 0 0 0 0 6 60
,
 1 0 0 0 0 0 17 60 0 0 0 0 0 0 23 15 0 0 0 0 46 38 0 0 0 0 0 0 1 20 0 0 0 0 0 60
,
 1 0 0 0 0 0 17 60 0 0 0 0 0 0 0 60 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [44,17,0,0,0,0,44,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,6,0,0,0,0,20,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,23,46,0,0,0,0,15,38,0,0,0,0,0,0,1,0,0,0,0,0,20,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C202D12 in GAP, Magma, Sage, TeX

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

G:=Group("C20:2D12");
// GroupNames label

G:=SmallGroup(480,542);
// by ID

G=gap.SmallGroup(480,542);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations

׿
×
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