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## G = C20×D12order 480 = 25·3·5

### Direct product of C20 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C20×D12
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C10×D12 — C20×D12
 Lower central C3 — C6 — C20×D12
 Upper central C1 — C2×C20 — C4×C20

Generators and relations for C20×D12
G = < a,b,c | a20=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 436 in 188 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], C10 [×3], C10 [×4], Dic3 [×2], C12 [×4], C12, D6 [×4], D6 [×4], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C20 [×4], C20 [×3], C2×C10, C2×C10 [×8], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C5×S3 [×4], C30 [×3], C4×D4, C2×C20 [×3], C2×C20 [×6], C5×D4 [×4], C22×C10 [×2], C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C5×Dic3 [×2], C60 [×4], C60, S3×C10 [×4], S3×C10 [×4], C2×C30, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, C4×D12, S3×C20 [×4], C5×D12 [×4], C10×Dic3 [×2], C2×C60 [×3], S3×C2×C10 [×2], D4×C20, C5×C4⋊Dic3, C5×D6⋊C4 [×2], C4×C60, S3×C2×C20 [×2], C10×D12, C20×D12
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], D12 [×2], C22×S3, C5×S3, C4×D4, C2×C20 [×6], C5×D4 [×2], C22×C10, S3×C2×C4, C2×D12, C4○D12, S3×C10 [×3], C22×C20, D4×C10, C5×C4○D4, C4×D12, S3×C20 [×2], C5×D12 [×2], S3×C2×C10, D4×C20, S3×C2×C20, C10×D12, C5×C4○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)
(1 124 54 227 208 64 145 28 170 184 88 106)(2 125 55 228 209 65 146 29 171 185 89 107)(3 126 56 229 210 66 147 30 172 186 90 108)(4 127 57 230 211 67 148 31 173 187 91 109)(5 128 58 231 212 68 149 32 174 188 92 110)(6 129 59 232 213 69 150 33 175 189 93 111)(7 130 60 233 214 70 151 34 176 190 94 112)(8 131 41 234 215 71 152 35 177 191 95 113)(9 132 42 235 216 72 153 36 178 192 96 114)(10 133 43 236 217 73 154 37 179 193 97 115)(11 134 44 237 218 74 155 38 180 194 98 116)(12 135 45 238 219 75 156 39 161 195 99 117)(13 136 46 239 220 76 157 40 162 196 100 118)(14 137 47 240 201 77 158 21 163 197 81 119)(15 138 48 221 202 78 159 22 164 198 82 120)(16 139 49 222 203 79 160 23 165 199 83 101)(17 140 50 223 204 80 141 24 166 200 84 102)(18 121 51 224 205 61 142 25 167 181 85 103)(19 122 52 225 206 62 143 26 168 182 86 104)(20 123 53 226 207 63 144 27 169 183 87 105)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 222)(102 223)(103 224)(104 225)(105 226)(106 227)(107 228)(108 229)(109 230)(110 231)(111 232)(112 233)(113 234)(114 235)(115 236)(116 237)(117 238)(118 239)(119 240)(120 221)(141 166)(142 167)(143 168)(144 169)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 161)(157 162)(158 163)(159 164)(160 165)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10AB 12A ··· 12L 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 20AG ··· 20AV 30A ··· 30L 60A ··· 60AV order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 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 6 6 6 6 2 1 1 1 1 2 2 2 2 6 6 6 6 1 1 1 1 2 2 2 1 ··· 1 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C20 S3 D4 D6 C4○D4 C4×S3 D12 C5×S3 C5×D4 C4○D12 S3×C10 C5×C4○D4 S3×C20 C5×D12 C5×C4○D12 kernel C20×D12 C5×C4⋊Dic3 C5×D6⋊C4 C4×C60 S3×C2×C20 C10×D12 C5×D12 C4×D12 C4⋊Dic3 D6⋊C4 C4×C12 S3×C2×C4 C2×D12 D12 C4×C20 C60 C2×C20 C30 C20 C20 C42 C12 C10 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 8 4 4 8 4 8 4 32 1 2 3 2 4 4 4 8 4 12 8 16 16 16

Matrix representation of C20×D12 in GL5(𝔽61)

 50 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 52 0 0 0 0 0 52
,
 1 0 0 0 0 0 1 59 0 0 0 1 60 0 0 0 0 0 0 60 0 0 0 1 60
,
 60 0 0 0 0 0 60 0 0 0 0 60 1 0 0 0 0 0 1 60 0 0 0 0 60

G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,1,1,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,60,60],[60,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,60,60] >;

C20×D12 in GAP, Magma, Sage, TeX

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

G:=Group("C20xD12");
// GroupNames label

G:=SmallGroup(480,752);
// by ID

G=gap.SmallGroup(480,752);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,226,15686]);
// Polycyclic

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

׿
×
𝔽