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## G = D20.38D6order 480 = 25·3·5

### 9th non-split extension by D20 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20.38D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — D5×Dic6 — D20.38D6
 Lower central C15 — C30 — D20.38D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D20.38D6
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a10c5 >

Subgroups: 1356 in 292 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, D15, C30, C30, 2- 1+4, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×Q8, C2×Dic6, C2×Dic6, C4○D12, D42S3, S3×Q8, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, D30, C2×C30, C4○D20, C4○D20, Q8×D5, Q82D5, Q8×C10, Q8○D12, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, C10×Dic3, Dic30, C4×D15, D60, C157D4, C2×C60, Q8.10D10, D5×Dic6, D20⋊S3, D15⋊Q8, C12.28D10, Dic5.D6, C3×C4○D20, C10×Dic6, D6011C2, D20.38D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2- 1+4, C22×D5, S3×C23, S3×D5, C23×D5, Q8○D12, C2×S3×D5, Q8.10D10, C22×S3×D5, D20.38D6

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

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

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

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

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 D10 D10 D10 2- 1+4 S3×D5 Q8○D12 C2×S3×D5 C2×S3×D5 Q8.10D10 D20.38D6 kernel D20.38D6 D5×Dic6 D20⋊S3 D15⋊Q8 C12.28D10 Dic5.D6 C3×C4○D20 C10×Dic6 D60⋊11C2 C4○D20 C2×Dic6 Dic10 C4×D5 D20 C5⋊D4 C2×C20 Dic6 C2×Dic3 C2×C12 C15 C2×C4 C5 C4 C22 C3 C1 # reps 1 2 2 2 2 4 1 1 1 1 2 1 2 1 2 1 8 4 2 1 2 2 4 2 4 8

Matrix representation of D20.38D6 in GL8(𝔽61)

 17 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 60 17 0 0 0 0 0 0 0 0 1 59 0 0 0 0 0 0 1 60 0 0 0 0 0 0 60 1 0 60 0 0 0 0 0 1 1 0
,
 17 24 59 33 0 0 0 0 40 44 33 51 0 0 0 0 37 55 41 33 0 0 0 0 55 44 59 20 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 1 0 1 1 0 0 0 0 60 1 0 60 0 0 0 0 0 0 0 60
,
 59 0 35 50 0 0 0 0 0 59 26 35 0 0 0 0 38 59 1 0 0 0 0 0 38 38 0 1 0 0 0 0 0 0 0 0 1 59 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 1 0 1 0 0 0 0 60 1 60 0
,
 25 44 41 52 0 0 0 0 17 9 44 41 0 0 0 0 52 27 52 17 0 0 0 0 3 52 44 36 0 0 0 0 0 0 0 0 52 19 0 0 0 0 0 0 31 9 0 0 0 0 0 0 0 21 31 21 0 0 0 0 30 21 21 30

`G:=sub<GL(8,GF(61))| [17,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,1,1,60,0,0,0,0,0,59,60,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0],[17,40,37,55,0,0,0,0,24,44,55,44,0,0,0,0,59,33,41,59,0,0,0,0,33,51,33,20,0,0,0,0,0,0,0,0,1,1,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,60,60],[59,0,38,38,0,0,0,0,0,59,59,38,0,0,0,0,35,26,1,0,0,0,0,0,50,35,0,1,0,0,0,0,0,0,0,0,1,1,0,60,0,0,0,0,59,60,1,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0],[25,17,52,3,0,0,0,0,44,9,27,52,0,0,0,0,41,44,52,44,0,0,0,0,52,41,17,36,0,0,0,0,0,0,0,0,52,31,0,30,0,0,0,0,19,9,21,21,0,0,0,0,0,0,31,21,0,0,0,0,0,0,21,30] >;`

D20.38D6 in GAP, Magma, Sage, TeX

`D_{20}._{38}D_6`
`% in TeX`

`G:=Group("D20.38D6");`
`// GroupNames label`

`G:=SmallGroup(480,1076);`
`// by ID`

`G=gap.SmallGroup(480,1076);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^6=d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^5>;`
`// generators/relations`

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