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G = D6.1D20order 480 = 25·3·5

1st non-split extension by D6 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6.1D20, C40.47D6, D20.20D6, C24.31D10, C60.97C23, C120.30C22, Dic3.12D20, Dic10.20D6, D60.28C22, Dic30.29C22, (S3×C8)⋊5D5, (S3×C40)⋊5C2, C40⋊C27S3, C154(C4○D8), C3⋊C8.28D10, C6.7(C2×D20), C8.14(S3×D5), C10.7(S3×D4), C3⋊D4012C2, C30.22(C2×D4), C2.12(S3×D20), C24⋊D512C2, D60⋊C29C2, D205S39C2, C31(D407C2), (S3×C10).20D4, (C4×S3).39D10, C3⋊Dic2011C2, C51(Q8.7D6), (C5×Dic3).23D4, C12.75(C22×D5), (S3×C20).45C22, C20.147(C22×S3), (C3×D20).22C22, (C3×Dic10).24C22, C4.96(C2×S3×D5), (C3×C40⋊C2)⋊4C2, (C5×C3⋊C8).32C22, SmallGroup(480,348)

Series: Derived Chief Lower central Upper central

C1C60 — D6.1D20
C1C5C15C30C60C3×D20D205S3 — D6.1D20
C15C30C60 — D6.1D20
C1C2C4C8

Generators and relations for D6.1D20
 G = < a,b,c,d | a6=b2=1, c20=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c19 >

Subgroups: 780 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20, C20, D10 [×2], C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12 [×2], C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C3×D5, D15, C30, C4○D8, C40, C40, Dic10, Dic10, C4×D5 [×2], D20, D20, C5⋊D4 [×2], C2×C20, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D42S3, Q83S3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C40⋊C2, C40⋊C2, D40, Dic20, C2×C40, C4○D20 [×2], Q8.7D6, C5×C3⋊C8, C120, D5×Dic3, D30.C2, C15⋊D4, C5⋊D12, C3×Dic10, C3×D20, S3×C20, Dic30, D60, D407C2, C3⋊D40, C3⋊Dic20, C3×C40⋊C2, S3×C40, C24⋊D5, D205S3, D60⋊C2, D6.1D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, D20 [×2], C22×D5, S3×D4, S3×D5, C2×D20, Q8.7D6, C2×S3×D5, D407C2, S3×D20, D6.1D20

Smallest permutation representation of D6.1D20
On 240 points
Generators in S240
(1 225 102 21 205 82)(2 226 103 22 206 83)(3 227 104 23 207 84)(4 228 105 24 208 85)(5 229 106 25 209 86)(6 230 107 26 210 87)(7 231 108 27 211 88)(8 232 109 28 212 89)(9 233 110 29 213 90)(10 234 111 30 214 91)(11 235 112 31 215 92)(12 236 113 32 216 93)(13 237 114 33 217 94)(14 238 115 34 218 95)(15 239 116 35 219 96)(16 240 117 36 220 97)(17 201 118 37 221 98)(18 202 119 38 222 99)(19 203 120 39 223 100)(20 204 81 40 224 101)(41 200 122 61 180 142)(42 161 123 62 181 143)(43 162 124 63 182 144)(44 163 125 64 183 145)(45 164 126 65 184 146)(46 165 127 66 185 147)(47 166 128 67 186 148)(48 167 129 68 187 149)(49 168 130 69 188 150)(50 169 131 70 189 151)(51 170 132 71 190 152)(52 171 133 72 191 153)(53 172 134 73 192 154)(54 173 135 74 193 155)(55 174 136 75 194 156)(56 175 137 76 195 157)(57 176 138 77 196 158)(58 177 139 78 197 159)(59 178 140 79 198 160)(60 179 141 80 199 121)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 81)(41 180)(42 181)(43 182)(44 183)(45 184)(46 185)(47 186)(48 187)(49 188)(50 189)(51 190)(52 191)(53 192)(54 193)(55 194)(56 195)(57 196)(58 197)(59 198)(60 199)(61 200)(62 161)(63 162)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 169)(71 170)(72 171)(73 172)(74 173)(75 174)(76 175)(77 176)(78 177)(79 178)(80 179)(201 221)(202 222)(203 223)(204 224)(205 225)(206 226)(207 227)(208 228)(209 229)(210 230)(211 231)(212 232)(213 233)(214 234)(215 235)(216 236)(217 237)(218 238)(219 239)(220 240)
(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)
(1 189 21 169)(2 168 22 188)(3 187 23 167)(4 166 24 186)(5 185 25 165)(6 164 26 184)(7 183 27 163)(8 162 28 182)(9 181 29 161)(10 200 30 180)(11 179 31 199)(12 198 32 178)(13 177 33 197)(14 196 34 176)(15 175 35 195)(16 194 36 174)(17 173 37 193)(18 192 38 172)(19 171 39 191)(20 190 40 170)(41 111 61 91)(42 90 62 110)(43 109 63 89)(44 88 64 108)(45 107 65 87)(46 86 66 106)(47 105 67 85)(48 84 68 104)(49 103 69 83)(50 82 70 102)(51 101 71 81)(52 120 72 100)(53 99 73 119)(54 118 74 98)(55 97 75 117)(56 116 76 96)(57 95 77 115)(58 114 78 94)(59 93 79 113)(60 112 80 92)(121 235 141 215)(122 214 142 234)(123 233 143 213)(124 212 144 232)(125 231 145 211)(126 210 146 230)(127 229 147 209)(128 208 148 228)(129 227 149 207)(130 206 150 226)(131 225 151 205)(132 204 152 224)(133 223 153 203)(134 202 154 222)(135 221 155 201)(136 240 156 220)(137 219 157 239)(138 238 158 218)(139 217 159 237)(140 236 160 216)

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

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

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

69 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B8A8B8C8D10A10B10C10D10E10F12A12B15A15B20A20B20C20D20E20F20G20H24A24B30A30B40A···40H40I···40P60A60B60C60D120A···120H
order12222344444556688881010101010101212151520202020202020202424303040···4040···4060606060120···120
size116206022332060222402266226666440442222666644442···26···644444···4

69 irreducible representations

dim1111111122222222222222444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C4○D8D20D20D407C2S3×D4S3×D5Q8.7D6C2×S3×D5S3×D20D6.1D20
kernelD6.1D20C3⋊D40C3⋊Dic20C3×C40⋊C2S3×C40C24⋊D5D205S3D60⋊C2C40⋊C2C5×Dic3S3×C10S3×C8C40Dic10D20C3⋊C8C24C4×S3C15Dic3D6C3C10C8C5C4C2C1
# reps11111111111211122244416122248

Matrix representation of D6.1D20 in GL6(𝔽241)

24000000
02400000
001000
000100
0000240240
000010
,
24000000
16910000
001000
000100
0000240240
000001
,
21100000
1722330000
000100
002405100
00002400
00000240
,
2011350000
222400000
00240000
00190100
000010
000001

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0],[240,169,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,240,1],[211,172,0,0,0,0,0,233,0,0,0,0,0,0,0,240,0,0,0,0,1,51,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[201,222,0,0,0,0,135,40,0,0,0,0,0,0,240,190,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D6.1D20 in GAP, Magma, Sage, TeX

D_6._1D_{20}
% in TeX

G:=Group("D6.1D20");
// GroupNames label

G:=SmallGroup(480,348);
// by ID

G=gap.SmallGroup(480,348);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,58,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=1,c^20=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^19>;
// generators/relations

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