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## G = D20.13D6order 480 = 25·3·5

### 13rd non-split extension by D20 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.13D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×Dic6 — D20.13D6
 Lower central C15 — C30 — C60 — D20.13D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D20.13D6
G = < a,b,c,d | a20=b2=c6=1, d2=a10, bab=dad-1=a-1, cac-1=a9, cbc-1=a18b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 604 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C8.C22, C52C8, C40, Dic10, C4×D5, C4×D5, D20, D20, C5×Q8, C5×Q8, C4.Dic3, D4.S3, C3⋊Q16, C3⋊Q16, C2×Dic6, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, C60, C6×D5, C6×D5, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, Q8.14D6, C5×C3⋊C8, C153C8, D5×Dic3, C15⋊Q8, D5×C12, D5×C12, C3×D20, C3×D20, C5×Dic6, Dic30, Q8×C15, Q16⋊D5, C20.32D6, C30.D4, C6.D20, C5×C3⋊Q16, C157Q16, D5×Dic6, C3×Q82D5, D20.13D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8.C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.14D6, C2×S3×D5, Q16⋊D5, D5×C3⋊D4, D20.13D6

Smallest permutation representation of D20.13D6
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 64)(2 63)(3 62)(4 61)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 88)(22 87)(23 86)(24 85)(25 84)(26 83)(27 82)(28 81)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 133)(42 132)(43 131)(44 130)(45 129)(46 128)(47 127)(48 126)(49 125)(50 124)(51 123)(52 122)(53 121)(54 140)(55 139)(56 138)(57 137)(58 136)(59 135)(60 134)(101 182)(102 181)(103 200)(104 199)(105 198)(106 197)(107 196)(108 195)(109 194)(110 193)(111 192)(112 191)(113 190)(114 189)(115 188)(116 187)(117 186)(118 185)(119 184)(120 183)(141 163)(142 162)(143 161)(144 180)(145 179)(146 178)(147 177)(148 176)(149 175)(150 174)(151 173)(152 172)(153 171)(154 170)(155 169)(156 168)(157 167)(158 166)(159 165)(160 164)(201 234)(202 233)(203 232)(204 231)(205 230)(206 229)(207 228)(208 227)(209 226)(210 225)(211 224)(212 223)(213 222)(214 221)(215 240)(216 239)(217 238)(218 237)(219 236)(220 235)
(1 95 186)(2 84 187 10 96 195)(3 93 188 19 97 184)(4 82 189 8 98 193)(5 91 190 17 99 182)(6 100 191)(7 89 192 15 81 200)(9 87 194 13 83 198)(11 85 196)(12 94 197 20 86 185)(14 92 199 18 88 183)(16 90 181)(21 102 71 39 104 69)(22 111 72 28 105 78)(23 120 73 37 106 67)(24 109 74 26 107 76)(25 118 75 35 108 65)(27 116 77 33 110 63)(29 114 79 31 112 61)(30 103 80 40 113 70)(32 101 62 38 115 68)(34 119 64 36 117 66)(41 225 175 43 223 177)(42 234 176 52 224 166)(44 232 178 50 226 164)(45 221 179 59 227 173)(46 230 180 48 228 162)(47 239 161 57 229 171)(49 237 163 55 231 169)(51 235 165 53 233 167)(54 222 168 60 236 174)(56 240 170 58 238 172)(121 204 157 125 220 141)(122 213 158 134 201 150)(123 202 159)(124 211 160 132 203 148)(126 209 142 130 205 146)(127 218 143 139 206 155)(128 207 144)(129 216 145 137 208 153)(131 214 147 135 210 151)(133 212 149)(136 219 152 140 215 156)(138 217 154)
(1 212 11 202)(2 211 12 201)(3 210 13 220)(4 209 14 219)(5 208 15 218)(6 207 16 217)(7 206 17 216)(8 205 18 215)(9 204 19 214)(10 203 20 213)(21 51 31 41)(22 50 32 60)(23 49 33 59)(24 48 34 58)(25 47 35 57)(26 46 36 56)(27 45 37 55)(28 44 38 54)(29 43 39 53)(30 42 40 52)(61 223 71 233)(62 222 72 232)(63 221 73 231)(64 240 74 230)(65 239 75 229)(66 238 76 228)(67 237 77 227)(68 236 78 226)(69 235 79 225)(70 234 80 224)(81 127 91 137)(82 126 92 136)(83 125 93 135)(84 124 94 134)(85 123 95 133)(86 122 96 132)(87 121 97 131)(88 140 98 130)(89 139 99 129)(90 138 100 128)(101 168 111 178)(102 167 112 177)(103 166 113 176)(104 165 114 175)(105 164 115 174)(106 163 116 173)(107 162 117 172)(108 161 118 171)(109 180 119 170)(110 179 120 169)(141 184 151 194)(142 183 152 193)(143 182 153 192)(144 181 154 191)(145 200 155 190)(146 199 156 189)(147 198 157 188)(148 197 158 187)(149 196 159 186)(150 195 160 185)```

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 20E 20F 30A 30B 40A 40B 40C 40D 60A ··· 60F order 1 2 2 2 3 4 4 4 4 4 5 5 6 6 6 6 8 8 10 10 12 12 12 12 12 15 15 20 20 20 20 20 20 30 30 40 40 40 40 60 ··· 60 size 1 1 10 20 2 2 4 10 12 60 2 2 2 20 20 20 12 60 2 2 4 4 4 10 10 4 4 4 4 8 8 24 24 4 4 12 12 12 12 8 ··· 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 C8.C22 S3×D5 D4×D5 Q8.14D6 C2×S3×D5 Q16⋊D5 D5×C3⋊D4 D20.13D6 kernel D20.13D6 C20.32D6 C30.D4 C6.D20 C5×C3⋊Q16 C15⋊7Q16 D5×Dic6 C3×Q8⋊2D5 Q8⋊2D5 C3×Dic5 C6×D5 C3⋊Q16 C4×D5 D20 C5×Q8 C3⋊C8 Dic6 C3×Q8 Dic5 D10 C15 Q8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of D20.13D6 in GL8(𝔽241)

 190 240 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 190 240 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 51 240 88 3 0 0 0 0 191 1 150 238 0 0 0 0 34 160 190 1 0 0 0 0 47 81 50 240
,
 0 171 0 140 0 0 0 0 171 0 140 0 0 0 0 0 0 101 0 70 0 0 0 0 101 0 70 0 0 0 0 0 0 0 0 0 154 218 40 138 0 0 0 0 174 87 161 201 0 0 0 0 58 176 87 23 0 0 0 0 125 183 67 154
,
 0 0 240 190 0 0 0 0 0 0 0 1 0 0 0 0 1 51 240 190 0 0 0 0 0 240 0 1 0 0 0 0 0 0 0 0 51 52 0 0 0 0 0 0 191 190 0 0 0 0 0 0 0 0 51 52 0 0 0 0 0 0 191 190
,
 101 90 92 113 0 0 0 0 0 140 0 149 0 0 0 0 193 203 140 151 0 0 0 0 0 48 0 101 0 0 0 0 0 0 0 0 70 95 204 208 0 0 0 0 196 171 41 37 0 0 0 0 92 56 171 146 0 0 0 0 113 149 45 70

`G:=sub<GL(8,GF(241))| [190,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,190,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,51,191,34,47,0,0,0,0,240,1,160,81,0,0,0,0,88,150,190,50,0,0,0,0,3,238,1,240],[0,171,0,101,0,0,0,0,171,0,101,0,0,0,0,0,0,140,0,70,0,0,0,0,140,0,70,0,0,0,0,0,0,0,0,0,154,174,58,125,0,0,0,0,218,87,176,183,0,0,0,0,40,161,87,67,0,0,0,0,138,201,23,154],[0,0,1,0,0,0,0,0,0,0,51,240,0,0,0,0,240,0,240,0,0,0,0,0,190,1,190,1,0,0,0,0,0,0,0,0,51,191,0,0,0,0,0,0,52,190,0,0,0,0,0,0,0,0,51,191,0,0,0,0,0,0,52,190],[101,0,193,0,0,0,0,0,90,140,203,48,0,0,0,0,92,0,140,0,0,0,0,0,113,149,151,101,0,0,0,0,0,0,0,0,70,196,92,113,0,0,0,0,95,171,56,149,0,0,0,0,204,41,171,45,0,0,0,0,208,37,146,70] >;`

D20.13D6 in GAP, Magma, Sage, TeX

`D_{20}._{13}D_6`
`% in TeX`

`G:=Group("D20.13D6");`
`// GroupNames label`

`G:=SmallGroup(480,584);`
`// by ID`

`G=gap.SmallGroup(480,584);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,100,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^10,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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