Copied to
clipboard

## G = D20.14D6order 480 = 25·3·5

### 14th non-split extension by D20 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.14D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D12⋊5D5 — D20.14D6
 Lower central C15 — C30 — C60 — D20.14D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D20.14D6
G = < a,b,c,d | a20=b2=c6=1, d2=a10, bab=dad-1=a-1, cac-1=a9, cbc-1=a18b, dbd-1=a13b, dcd-1=a10c-1 >

Subgroups: 636 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C4○D8, C52C8, C40, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, C3×Dic5, Dic15, C60, C60, C6×D5, C6×D5, S3×C10, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D42D5, Q82D5, Q8.13D6, C5×C3⋊C8, C153C8, S3×Dic5, C15⋊D4, D5×C12, D5×C12, C3×D20, C3×D20, C5×D12, Dic30, Q8×C15, SD163D5, D5×C3⋊C8, C15⋊D8, C6.D20, C5×Q82S3, C157Q16, D125D5, C3×Q82D5, D20.14D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4○D8, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.13D6, C2×S3×D5, SD163D5, D5×C3⋊D4, D20.14D6

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 30A 30B 40A 40B 40C 40D 60A ··· 60F order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 12 15 15 20 20 20 20 30 30 40 40 40 40 60 ··· 60 size 1 1 10 12 20 2 2 4 5 5 60 2 2 2 20 20 20 6 6 30 30 2 2 24 24 4 4 4 10 10 4 4 4 4 8 8 4 4 12 12 12 12 8 ··· 8

48 irreducible representations

 dim 1 1 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 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 C4○D8 S3×D5 D4×D5 Q8.13D6 C2×S3×D5 SD16⋊3D5 D5×C3⋊D4 D20.14D6 kernel D20.14D6 D5×C3⋊C8 C15⋊D8 C6.D20 C5×Q8⋊2S3 C15⋊7Q16 D12⋊5D5 C3×Q8⋊2D5 Q8⋊2D5 C3×Dic5 C6×D5 Q8⋊2S3 C4×D5 D20 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 4 2 2 2 2 4 4 2

Matrix representation of D20.14D6 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 190 1 0 0 0 0 50 240 0 0 0 0 0 0 64 0 0 0 0 0 43 177
,
 70 140 0 0 0 0 101 171 0 0 0 0 0 0 1 1 0 0 0 0 0 240 0 0 0 0 0 0 56 192 0 0 0 0 123 185
,
 240 240 0 0 0 0 1 0 0 0 0 0 0 0 51 52 0 0 0 0 191 190 0 0 0 0 0 0 1 0 0 0 0 0 140 240
,
 162 11 0 0 0 0 90 79 0 0 0 0 0 0 51 52 0 0 0 0 191 190 0 0 0 0 0 0 207 90 0 0 0 0 113 34

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,190,50,0,0,0,0,1,240,0,0,0,0,0,0,64,43,0,0,0,0,0,177],[70,101,0,0,0,0,140,171,0,0,0,0,0,0,1,0,0,0,0,0,1,240,0,0,0,0,0,0,56,123,0,0,0,0,192,185],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,51,191,0,0,0,0,52,190,0,0,0,0,0,0,1,140,0,0,0,0,0,240],[162,90,0,0,0,0,11,79,0,0,0,0,0,0,51,191,0,0,0,0,52,190,0,0,0,0,0,0,207,113,0,0,0,0,90,34] >;`

D20.14D6 in GAP, Magma, Sage, TeX

`D_{20}._{14}D_6`
`% in TeX`

`G:=Group("D20.14D6");`
`// GroupNames label`

`G:=SmallGroup(480,590);`
`// by ID`

`G=gap.SmallGroup(480,590);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,135,100,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^10,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d^-1=a^13*b,d*c*d^-1=a^10*c^-1>;`
`// generators/relations`

׿
×
𝔽