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## G = D12.D10order 480 = 25·3·5

### 16th non-split extension by D12 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D12.D10
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — D12⋊D5 — D12.D10
 Lower central C15 — C30 — C60 — D12.D10
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.D10
G = < a,b,c,d | a12=b2=1, c10=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c9 >

Subgroups: 780 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×3], C6, C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, D5 [×2], C10, C10, Dic3, C12, C12 [×2], D6 [×3], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20, C20, D10 [×2], C2×C10, C3⋊C8, C24, C4×S3 [×3], D12, D12 [×3], C3×Q8, C3×Q8, C5×S3, D15 [×2], C30, C4○D8, C52C8, C40, Dic10, C4×D5 [×2], D20 [×2], C2×Dic5, C5⋊D4, C5×D4, C5×Q8, S3×C8, D24, Q82S3, Q82S3, C3×Q16, Q83S3 [×2], C3×Dic5, Dic15, C60, C60, S3×C10, D30, D30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D42D5, Q82D5, D24⋊C2, C5×C3⋊C8, C3×C52C8, S3×Dic5, C5⋊D12, C3×Dic10, C5×D12, C4×D15, C4×D15, D60, D60, Q8×C15, SD163D5, D152C8, C5⋊D24, C15⋊SD16, C3×C5⋊Q16, C5×Q82S3, D12⋊D5, Q83D15, D12.D10
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, D24⋊C2, C2×S3×D5, SD163D5, D10⋊D6, D12.D10

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 12C 15A 15B 20A 20B 20C 20D 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 8 8 8 8 10 10 10 10 12 12 12 15 15 20 20 20 20 24 24 30 30 40 40 40 40 60 ··· 60 size 1 1 12 30 60 2 2 4 15 15 20 2 2 2 6 6 10 10 2 2 24 24 4 8 40 4 4 4 4 8 8 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 D24⋊C2 C2×S3×D5 SD16⋊3D5 D10⋊D6 D12.D10 kernel D12.D10 D15⋊2C8 C5⋊D24 C15⋊SD16 C3×C5⋊Q16 C5×Q8⋊2S3 D12⋊D5 Q8⋊3D15 C5⋊Q16 Dic15 D30 Q8⋊2S3 C5⋊2C8 Dic10 C5×Q8 C3⋊C8 D12 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 D12.D10 in GL6(𝔽241)

 64 0 0 0 0 0 131 177 0 0 0 0 0 0 239 187 0 0 0 0 67 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 74 16 0 0 0 0 185 167 0 0 0 0 0 0 239 187 0 0 0 0 67 2 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 51 239 0 0 0 0 96 190 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 189 189 0 0 0 0 52 1
,
 211 0 0 0 0 0 236 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 52 52 0 0 0 0 240 189

`G:=sub<GL(6,GF(241))| [64,131,0,0,0,0,0,177,0,0,0,0,0,0,239,67,0,0,0,0,187,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[74,185,0,0,0,0,16,167,0,0,0,0,0,0,239,67,0,0,0,0,187,2,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[51,96,0,0,0,0,239,190,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,52,0,0,0,0,189,1],[211,236,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,240,0,0,0,0,52,189] >;`

D12.D10 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(480,599);`
`// by ID`

`G=gap.SmallGroup(480,599);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,135,100,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^10=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^9>;`
`// generators/relations`

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