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## G = D20.10D6order 480 = 25·3·5

### 10th non-split extension by D20 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.10D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — D20⋊S3 — D20.10D6
 Lower central C15 — C30 — C60 — D20.10D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D20.10D6
G = < a,b,c,d | a20=b2=c6=1, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a15b, dcd-1=a15c-1 >

Subgroups: 716 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C8 [×2], C2×C4 [×3], D4, D4 [×3], Q8 [×2], D5 [×2], C10, C10, Dic3 [×3], C12, D6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20, C20, D10 [×2], C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C3×D4, C3×D5, D15, C30, C30, C4○D8, C52C8, C40, Dic10, C4×D5 [×2], D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, S3×C8, Dic12, D4.S3, D4.S3, C3×D8, D42S3 [×2], C5×Dic3, Dic15, Dic15, C60, C6×D5, D30, C2×C30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D42D5, Q82D5, D83S3, C5×C3⋊C8, C3×C52C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, Dic30, C4×D15, C2×Dic15, C157D4, D4×C15, SD163D5, D152C8, C6.D20, C5⋊Dic12, C3×D4⋊D5, C5×D4.S3, D20⋊S3, D42D15, D20.10D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, D83S3, C2×S3×D5, SD163D5, D10⋊D6, D20.10D6

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 10D 12 15A 15B 20A 20B 20C 20D 24A 24B 30A 30B 30C 30D 30E 30F 40A 40B 40C 40D 60A 60B order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 10 10 10 12 15 15 20 20 20 20 24 24 30 30 30 30 30 30 40 40 40 40 60 60 size 1 1 4 20 30 2 2 12 15 15 60 2 2 2 8 40 6 6 10 10 2 2 8 8 4 4 4 4 4 24 24 20 20 4 4 8 8 8 8 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 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 C4○D8 S3×D4 S3×D5 D4×D5 D8⋊3S3 C2×S3×D5 SD16⋊3D5 D10⋊D6 D20.10D6 kernel D20.10D6 D15⋊2C8 C6.D20 C5⋊Dic12 C3×D4⋊D5 C5×D4.S3 D20⋊S3 D4⋊2D15 D4⋊D5 Dic15 D30 D4.S3 C5⋊2C8 D20 C5×D4 C3⋊C8 Dic6 C3×D4 C15 C10 D4 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 4 1 2 2 2 2 4 4 2

Matrix representation of D20.10D6 in GL6(𝔽241)

 177 0 0 0 0 0 110 64 0 0 0 0 0 0 51 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 70 191 0 0 0 0 151 171 0 0 0 0 0 0 1 51 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 69 54 0 0 0 0 135 172 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 240 0 0 0 0 1 240
,
 30 0 0 0 0 0 79 8 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 240

`G:=sub<GL(6,GF(241))| [177,110,0,0,0,0,0,64,0,0,0,0,0,0,51,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[70,151,0,0,0,0,191,171,0,0,0,0,0,0,1,0,0,0,0,0,51,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[69,135,0,0,0,0,54,172,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[30,79,0,0,0,0,0,8,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;`

D20.10D6 in GAP, Magma, Sage, TeX

`D_{20}._{10}D_6`
`% in TeX`

`G:=Group("D20.10D6");`
`// GroupNames label`

`G:=SmallGroup(480,573);`
`// by ID`

`G=gap.SmallGroup(480,573);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;`
`// generators/relations`

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