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## G = D20.31D6order 480 = 25·3·5

### 2nd non-split extension by D20 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.31D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C6.D20 — D20.31D6
 Lower central C15 — C30 — C60 — D20.31D6
 Upper central C1 — C4 — C2×C4

Generators and relations for D20.31D6
G = < a,b,c,d | a20=b2=1, c6=a10, d2=a15, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a15b, dcd-1=c5 >

Subgroups: 716 in 124 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5 [×2], C10, C10, Dic3, C12 [×2], C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4 [×2], C3×Q8, C3×D5, D15, C30, C30, C4○D8, C40 [×2], Dic10, Dic10, C4×D5 [×2], D20, D20, C5⋊D4 [×2], C2×C20, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, C3×Dic5, Dic15, C60 [×2], C6×D5, D30, C2×C30, C40⋊C2 [×2], D40, Dic20, C2×C40, C4○D20, C4○D20, Q8.13D6, C5×C3⋊C8 [×2], C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, Dic30, C4×D15, D60, C157D4, C2×C60, D407C2, C3⋊D40, C6.D20, C15⋊SD16, C3⋊Dic20, C10×C3⋊C8, C3×C4○D20, D6011C2, D20.31D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4○D8, D20 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×D20, Q8.13D6, C3⋊D20 [×2], C2×S3×D5, D407C2, C2×C3⋊D20, D20.31D6

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

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

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

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

72 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 ··· 10F 12A 12B 12C 12D 12E 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 20 60 2 1 1 2 20 60 2 2 2 4 20 20 6 6 6 6 2 ··· 2 2 2 4 20 20 4 4 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 C3⋊D4 C3⋊D4 C4○D8 D20 D20 D40⋊7C2 S3×D5 Q8.13D6 C3⋊D20 C2×S3×D5 C3⋊D20 D20.31D6 kernel D20.31D6 C3⋊D40 C6.D20 C15⋊SD16 C3⋊Dic20 C10×C3⋊C8 C3×C4○D20 D60⋊11C2 C4○D20 C60 C2×C30 C2×C3⋊C8 Dic10 D20 C2×C20 C3⋊C8 C2×C12 C20 C2×C10 C15 C12 C2×C6 C3 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 4 16 2 2 2 2 2 8

Matrix representation of D20.31D6 in GL4(𝔽241) generated by

 6 0 0 0 26 201 0 0 0 0 1 0 0 0 0 1
,
 91 80 0 0 17 150 0 0 0 0 1 0 0 0 0 1
,
 177 0 0 0 0 177 0 0 0 0 240 240 0 0 1 0
,
 30 0 0 0 11 233 0 0 0 0 127 136 0 0 9 114
`G:=sub<GL(4,GF(241))| [6,26,0,0,0,201,0,0,0,0,1,0,0,0,0,1],[91,17,0,0,80,150,0,0,0,0,1,0,0,0,0,1],[177,0,0,0,0,177,0,0,0,0,240,1,0,0,240,0],[30,11,0,0,0,233,0,0,0,0,127,9,0,0,136,114] >;`

D20.31D6 in GAP, Magma, Sage, TeX

`D_{20}._{31}D_6`
`% in TeX`

`G:=Group("D20.31D6");`
`// GroupNames label`

`G:=SmallGroup(480,387);`
`// by ID`

`G=gap.SmallGroup(480,387);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,100,675,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^6=a^10,d^2=a^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^5>;`
`// generators/relations`

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