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## G = D20.16D6order 480 = 25·3·5

### 16th non-split extension by D20 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.16D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — D20⋊S3 — D20.16D6
 Lower central C15 — C30 — C60 — D20.16D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D20.16D6
G = < a,b,c,d | a30=b2=d2=1, c4=a15, bab=a-1, cac-1=dad=a19, cbc-1=a18b, dbd=a3b, dcd=c3 >

Subgroups: 796 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, D5 [×3], C10, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10 [×3], C3⋊C8, C24, Dic6, C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×D5, D15 [×2], C30, C4○D8, C52C8, C40, C4×D5 [×3], D20, D20 [×3], C5×Q8, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D42S3, Q83S3, C5×Dic3, Dic15, C60, C60, C6×D5, D30, D30, C8×D5, D40, Q8⋊D5, Q8⋊D5, C5×Q16, Q82D5 [×2], Q8.7D6, C5×C3⋊C8, C3×C52C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C4×D15, C4×D15, D60, D60, Q8×C15, Q8.D10, D152C8, C3⋊D40, Dic6⋊D5, C3×Q8⋊D5, C5×C3⋊Q16, D20⋊S3, Q83D15, D20.16D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, Q8.7D6, C2×S3×D5, Q8.D10, D10⋊D6, D20.16D6

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 12A 12B 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 30A 30B 40A 40B 40C 40D 60A ··· 60F order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 8 8 8 8 10 10 12 12 15 15 20 20 20 20 20 20 24 24 30 30 40 40 40 40 60 ··· 60 size 1 1 20 30 60 2 2 4 12 15 15 2 2 2 40 6 6 10 10 2 2 4 8 4 4 4 4 8 8 24 24 20 20 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 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 C4○D8 S3×D4 S3×D5 D4×D5 Q8.7D6 C2×S3×D5 Q8.D10 D10⋊D6 D20.16D6 kernel D20.16D6 D15⋊2C8 C3⋊D40 Dic6⋊D5 C3×Q8⋊D5 C5×C3⋊Q16 D20⋊S3 Q8⋊3D15 Q8⋊D5 Dic15 D30 C3⋊Q16 C5⋊2C8 D20 C5×Q8 C3⋊C8 Dic6 C3×Q8 C15 C10 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 4 1 2 2 2 2 4 4 2

Matrix representation of D20.16D6 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 189 240 0 0 0 0 1 0 0 0 0 0 0 0 239 192 0 0 0 0 64 1
,
 240 0 0 0 0 0 0 1 0 0 0 0 0 0 189 240 0 0 0 0 52 52 0 0 0 0 0 0 240 0 0 0 0 0 64 1
,
 30 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 189 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 0 8 0 0 0 0 211 0 0 0 0 0 0 0 1 0 0 0 0 0 189 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,189,1,0,0,0,0,240,0,0,0,0,0,0,0,239,64,0,0,0,0,192,1],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,189,52,0,0,0,0,240,52,0,0,0,0,0,0,240,64,0,0,0,0,0,1],[30,0,0,0,0,0,0,8,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,211,0,0,0,0,8,0,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

D20.16D6 in GAP, Magma, Sage, TeX

`D_{20}._{16}D_6`
`% in TeX`

`G:=Group("D20.16D6");`
`// GroupNames label`

`G:=SmallGroup(480,597);`
`// by ID`

`G=gap.SmallGroup(480,597);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,100,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=d^2=1,c^4=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d=a^19,c*b*c^-1=a^18*b,d*b*d=a^3*b,d*c*d=c^3>;`
`// generators/relations`

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