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G = D10.16D12order 480 = 25·3·5

5th non-split extension by D10 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D10.16D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — D10.16D12
 Lower central C15 — C2×C30 — D10.16D12
 Upper central C1 — C22 — C2×C4

Generators and relations for D10.16D12
G = < a,b,c,d | a10=b2=c12=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 940 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C22.D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, D30, C2×C30, C10.D4, D10⋊C4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C23.21D6, D5×Dic3, C3⋊D20, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D10.13D4, D304C4, C30.Q8, C3×D10⋊C4, C5×C4⋊Dic3, D303C4, C2×D5×Dic3, C2×C3⋊D20, D10.16D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C22.D4, C22×D5, C2×D12, D42S3, S3×D5, C4○D20, D4×D5, Q82D5, C23.21D6, C2×S3×D5, D10.13D4, D20⋊S3, D5×D12, Dic5.D6, D10.16D12

Smallest permutation representation of D10.16D12
On 240 points
Generators in S240
```(1 75 67 180 47 185 131 155 25 85)(2 76 68 169 48 186 132 156 26 86)(3 77 69 170 37 187 121 145 27 87)(4 78 70 171 38 188 122 146 28 88)(5 79 71 172 39 189 123 147 29 89)(6 80 72 173 40 190 124 148 30 90)(7 81 61 174 41 191 125 149 31 91)(8 82 62 175 42 192 126 150 32 92)(9 83 63 176 43 181 127 151 33 93)(10 84 64 177 44 182 128 152 34 94)(11 73 65 178 45 183 129 153 35 95)(12 74 66 179 46 184 130 154 36 96)(13 205 194 226 143 117 52 107 160 237)(14 206 195 227 144 118 53 108 161 238)(15 207 196 228 133 119 54 97 162 239)(16 208 197 217 134 120 55 98 163 240)(17 209 198 218 135 109 56 99 164 229)(18 210 199 219 136 110 57 100 165 230)(19 211 200 220 137 111 58 101 166 231)(20 212 201 221 138 112 59 102 167 232)(21 213 202 222 139 113 60 103 168 233)(22 214 203 223 140 114 49 104 157 234)(23 215 204 224 141 115 50 105 158 235)(24 216 193 225 142 116 51 106 159 236)
(1 91)(2 42)(3 93)(4 44)(5 95)(6 46)(7 85)(8 48)(9 87)(10 38)(11 89)(12 40)(13 58)(14 212)(15 60)(16 214)(17 50)(18 216)(19 52)(20 206)(21 54)(22 208)(23 56)(24 210)(25 81)(26 126)(27 83)(28 128)(29 73)(30 130)(31 75)(32 132)(33 77)(34 122)(35 79)(36 124)(37 181)(39 183)(41 185)(43 187)(45 189)(47 191)(49 120)(51 110)(53 112)(55 114)(57 116)(59 118)(61 155)(62 68)(63 145)(64 70)(65 147)(66 72)(67 149)(69 151)(71 153)(74 173)(76 175)(78 177)(80 179)(82 169)(84 171)(86 192)(88 182)(90 184)(92 186)(94 188)(96 190)(97 233)(98 140)(99 235)(100 142)(101 237)(102 144)(103 239)(104 134)(105 229)(106 136)(107 231)(108 138)(109 215)(111 205)(113 207)(115 209)(117 211)(119 213)(121 176)(123 178)(125 180)(127 170)(129 172)(131 174)(133 202)(135 204)(137 194)(139 196)(141 198)(143 200)(146 152)(148 154)(150 156)(157 217)(158 164)(159 219)(160 166)(161 221)(162 168)(163 223)(165 225)(167 227)(193 230)(195 232)(197 234)(199 236)(201 238)(203 240)(218 224)(220 226)(222 228)
(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)
(1 211 185 58)(2 210 186 57)(3 209 187 56)(4 208 188 55)(5 207 189 54)(6 206 190 53)(7 205 191 52)(8 216 192 51)(9 215 181 50)(10 214 182 49)(11 213 183 60)(12 212 184 59)(13 41 117 91)(14 40 118 90)(15 39 119 89)(16 38 120 88)(17 37 109 87)(18 48 110 86)(19 47 111 85)(20 46 112 96)(21 45 113 95)(22 44 114 94)(23 43 115 93)(24 42 116 92)(25 231 180 137)(26 230 169 136)(27 229 170 135)(28 240 171 134)(29 239 172 133)(30 238 173 144)(31 237 174 143)(32 236 175 142)(33 235 176 141)(34 234 177 140)(35 233 178 139)(36 232 179 138)(61 226 149 160)(62 225 150 159)(63 224 151 158)(64 223 152 157)(65 222 153 168)(66 221 154 167)(67 220 155 166)(68 219 156 165)(69 218 145 164)(70 217 146 163)(71 228 147 162)(72 227 148 161)(73 202 129 103)(74 201 130 102)(75 200 131 101)(76 199 132 100)(77 198 121 99)(78 197 122 98)(79 196 123 97)(80 195 124 108)(81 194 125 107)(82 193 126 106)(83 204 127 105)(84 203 128 104)```

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

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

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

60 conjugacy classes

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

60 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 4 type + + + + + + + + + + + + + + + + + - + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 C4○D4 D10 D10 D12 C4○D20 D4⋊2S3 S3×D5 D4×D5 Q8⋊2D5 C2×S3×D5 D20⋊S3 D5×D12 Dic5.D6 kernel D10.16D12 D30⋊4C4 C30.Q8 C3×D10⋊C4 C5×C4⋊Dic3 D30⋊3C4 C2×D5×Dic3 C2×C3⋊D20 D10⋊C4 C6×D5 C4⋊Dic3 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 D10 C6 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 4 2 4 8 2 2 2 2 2 4 4 4

Matrix representation of D10.16D12 in GL4(𝔽61) generated by

 18 17 0 0 43 0 0 0 0 0 1 0 0 0 0 1
,
 43 60 0 0 18 18 0 0 0 0 60 0 0 0 0 60
,
 30 16 0 0 1 31 0 0 0 0 53 33 0 0 24 0
,
 25 54 0 0 11 36 0 0 0 0 19 12 0 0 31 42
`G:=sub<GL(4,GF(61))| [18,43,0,0,17,0,0,0,0,0,1,0,0,0,0,1],[43,18,0,0,60,18,0,0,0,0,60,0,0,0,0,60],[30,1,0,0,16,31,0,0,0,0,53,24,0,0,33,0],[25,11,0,0,54,36,0,0,0,0,19,31,0,0,12,42] >;`

D10.16D12 in GAP, Magma, Sage, TeX

`D_{10}._{16}D_{12}`
`% in TeX`

`G:=Group("D10.16D12");`
`// GroupNames label`

`G:=SmallGroup(480,489);`
`// by ID`

`G=gap.SmallGroup(480,489);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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