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## G = D12.27D10order 480 = 25·3·5

### 10th non-split extension by D12 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D12.27D10
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D12⋊5D5 — D12.27D10
 Lower central C15 — C30 — C60 — D12.27D10
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.27D10
G = < a,b,c,d | a12=b2=1, c10=d2=a6, bab=a-1, cac-1=dad-1=a7, cbc-1=a3b, dbd-1=a9b, dcd-1=c9 >

Subgroups: 588 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], D5, C10, C10, Dic3, C12, C12 [×3], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, Dic5 [×2], C20, C20, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C3×Q8, C3×Q8 [×2], C5×S3, C3×D5, C30, C8.C22, C52C8, C40, Dic10, Dic10 [×2], C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C4.Dic3, Q82S3, Q82S3, C3⋊Q16 [×2], C4○D12, C6×Q8, C3×Dic5, C3×Dic5, Dic15, C60, C60, C6×D5, S3×C10, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D42D5, Q8×D5, Q8.11D6, C5×C3⋊C8, C153C8, S3×Dic5, C15⋊D4, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C5×D12, Dic30, Q8×C15, SD16⋊D5, C20.32D6, C20.D6, C3⋊Dic20, C5×Q82S3, C157Q16, D125D5, C3×Q8×D5, D12.27D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8.C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.11D6, C2×S3×D5, SD16⋊D5, D5×C3⋊D4, D12.27D10

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 8A 8B 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 20C 20D 30A 30B 40A 40B 40C 40D 60A ··· 60F order 1 2 2 2 3 4 4 4 4 4 5 5 6 6 6 8 8 10 10 10 10 12 12 12 12 12 12 15 15 20 20 20 20 30 30 40 40 40 40 60 ··· 60 size 1 1 10 12 2 2 4 10 20 60 2 2 2 10 10 12 60 2 2 24 24 4 4 4 20 20 20 4 4 4 4 8 8 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 Q8.11D6 C2×S3×D5 SD16⋊D5 D5×C3⋊D4 D12.27D10 kernel D12.27D10 C20.32D6 C20.D6 C3⋊Dic20 C5×Q8⋊2S3 C15⋊7Q16 D12⋊5D5 C3×Q8×D5 Q8×D5 C3×Dic5 C6×D5 Q8⋊2S3 Dic10 C4×D5 C5×Q8 C3⋊C8 D12 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 D12.27D10 in GL6(𝔽241)

 1 1 0 0 0 0 240 0 0 0 0 0 0 0 240 0 69 76 0 0 0 240 69 69 0 0 138 89 1 0 0 0 103 138 0 1
,
 102 225 0 0 0 0 123 139 0 0 0 0 0 0 21 226 88 238 0 0 15 220 153 88 0 0 40 89 21 15 0 0 40 40 226 220
,
 171 101 0 0 0 0 140 70 0 0 0 0 0 0 204 204 44 158 0 0 37 4 158 44 0 0 51 52 237 37 0 0 239 51 204 37
,
 70 140 0 0 0 0 101 171 0 0 0 0 0 0 204 204 44 158 0 0 4 37 158 61 0 0 52 51 237 0 0 0 51 239 204 4

`G:=sub<GL(6,GF(241))| [1,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,138,103,0,0,0,240,89,138,0,0,69,69,1,0,0,0,76,69,0,1],[102,123,0,0,0,0,225,139,0,0,0,0,0,0,21,15,40,40,0,0,226,220,89,40,0,0,88,153,21,226,0,0,238,88,15,220],[171,140,0,0,0,0,101,70,0,0,0,0,0,0,204,37,51,239,0,0,204,4,52,51,0,0,44,158,237,204,0,0,158,44,37,37],[70,101,0,0,0,0,140,171,0,0,0,0,0,0,204,4,52,51,0,0,204,37,51,239,0,0,44,158,237,204,0,0,158,61,0,4] >;`

D12.27D10 in GAP, Magma, Sage, TeX

`D_{12}._{27}D_{10}`
`% in TeX`

`G:=Group("D12.27D10");`
`// GroupNames label`

`G:=SmallGroup(480,589);`
`// by ID`

`G=gap.SmallGroup(480,589);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,135,100,346,185,80,1356,18822]);`
`// Polycyclic`

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

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