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## G = D12.37D10order 480 = 25·3·5

### 8th non-split extension by D12 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D12.37D10
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — C20.D6 — D12.37D10
 Lower central C15 — C30 — C60 — D12.37D10
 Upper central C1 — C2 — C2×C4

Generators and relations for D12.37D10
G = < a,b,c,d | a12=b2=c10=1, d2=a6, bab=a-1, ac=ca, dad-1=a7, cbc-1=a6b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 476 in 120 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C10, C10 [×2], Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8 [×3], C5×S3, C30, C30, C8.C22, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C4○D12, C6×Q8, C5×Dic3, C3×Dic5 [×2], C60 [×2], S3×C10, C2×C30, C4.Dic5, D4.D5 [×2], C5⋊Q16 [×2], C2×Dic10, C5×C4○D4, Q8.11D6, C153C8 [×2], C3×Dic10 [×2], C3×Dic10, C6×Dic5, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4.9D10, C20.D6 [×2], C15⋊Q16 [×2], C60.7C4, C6×Dic10, C5×C4○D12, D12.37D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8.C22, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, Q8.11D6, C15⋊D4 [×2], C2×S3×D5, D4.9D10, C2×C15⋊D4, D12.37D10

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

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 5 5 6 6 6 8 8 10 10 10 10 10 10 10 10 12 12 12 12 12 12 15 15 20 20 20 20 20 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 2 12 2 2 2 12 20 20 2 2 2 2 2 60 60 2 2 4 4 12 12 12 12 4 4 20 20 20 20 4 4 2 2 2 2 4 4 12 12 12 12 4 ··· 4 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 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 S3 D4 D4 D5 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 C5⋊D4 C5⋊D4 C8.C22 S3×D5 Q8.11D6 C15⋊D4 C2×S3×D5 C15⋊D4 D4.9D10 D12.37D10 kernel D12.37D10 C20.D6 C15⋊Q16 C60.7C4 C6×Dic10 C5×C4○D12 C2×Dic10 C60 C2×C30 C4○D12 Dic10 C2×C20 Dic6 D12 C2×C12 C20 C2×C10 C12 C2×C6 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 4 4 1 2 2 2 2 2 4 8

Matrix representation of D12.37D10 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 15 0 0 0 0 226 0 0 0 0 0 0 0 0 16 0 0 0 0 225 0
,
 27 138 0 0 0 0 103 214 0 0 0 0 0 0 0 0 0 16 0 0 0 0 225 0 0 0 0 15 0 0 0 0 226 0 0 0
,
 1 189 0 0 0 0 52 189 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 85 200 0 0 0 0 41 156 0 0 0 0 0 0 172 96 0 0 0 0 96 69 0 0 0 0 0 0 96 69 0 0 0 0 69 145

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,226,0,0,0,0,15,0,0,0,0,0,0,0,0,225,0,0,0,0,16,0],[27,103,0,0,0,0,138,214,0,0,0,0,0,0,0,0,0,226,0,0,0,0,15,0,0,0,0,225,0,0,0,0,16,0,0,0],[1,52,0,0,0,0,189,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[85,41,0,0,0,0,200,156,0,0,0,0,0,0,172,96,0,0,0,0,96,69,0,0,0,0,0,0,96,69,0,0,0,0,69,145] >;`

D12.37D10 in GAP, Magma, Sage, TeX

`D_{12}._{37}D_{10}`
`% in TeX`

`G:=Group("D12.37D10");`
`// GroupNames label`

`G:=SmallGroup(480,385);`
`// by ID`

`G=gap.SmallGroup(480,385);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,219,675,346,80,1356,18822]);`
`// Polycyclic`

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

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