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## G = D20.24D6order 480 = 25·3·5

### 7th non-split extension by D20 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.24D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — D20⋊5S3 — D20.24D6
 Lower central C15 — C30 — C60 — D20.24D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D20.24D6
G = < a,b,c,d | a20=b2=c6=1, d2=a10, bab=a-1, cac-1=dad-1=a11, cbc-1=dbd-1=a15b, dcd-1=c-1 >

Subgroups: 588 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C5×S3, C3×D5, C30, C30, C4○D8, C52C8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, C5×Q8, S3×C8, Dic12, D4.S3, C3×D8, D42S3, D42S3, C5×Dic3, C5×Dic3, Dic15, C60, C6×D5, S3×C10, C2×C30, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, D83S3, C3×C52C8, C153C8, D5×Dic3, C15⋊D4, C3×D20, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, Dic30, D4×C15, D4.8D10, S3×C52C8, C30.D4, C5⋊Dic12, C3×D4⋊D5, D4.D15, D205S3, C5×D42S3, D20.24D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C5⋊D4, C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D83S3, C2×S3×D5, D4.8D10, S3×C5⋊D4, D20.24D6

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

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

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

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

51 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 10E 10F 10G 10H 12 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 24A 24B 30A 30B 30C 30D 30E 30F 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 10 10 10 10 12 15 15 20 20 20 20 20 20 20 20 20 20 24 24 30 30 30 30 30 30 60 60 size 1 1 4 6 20 2 2 3 3 12 60 2 2 2 8 40 10 10 30 30 2 2 4 4 4 4 12 12 4 4 4 4 4 6 6 6 6 12 12 12 12 20 20 4 4 8 8 8 8 8 8

51 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 C4○D8 C5⋊D4 C5⋊D4 S3×D4 S3×D5 D8⋊3S3 C2×S3×D5 D4.8D10 S3×C5⋊D4 D20.24D6 kernel D20.24D6 S3×C5⋊2C8 C30.D4 C5⋊Dic12 C3×D4⋊D5 D4.D15 D20⋊5S3 C5×D4⋊2S3 D4⋊D5 C5×Dic3 S3×C10 D4⋊2S3 C5⋊2C8 D20 C5×D4 Dic6 C4×S3 C3×D4 C15 Dic3 D6 C10 D4 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 4 4 1 2 2 2 4 4 2

Matrix representation of D20.24D6 in GL6(𝔽241)

 64 0 0 0 0 0 0 177 0 0 0 0 0 0 189 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 211 0 0 0 0 8 0 0 0 0 0 0 0 211 177 0 0 0 0 63 30 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 0 1 0 0 0 0 1 0 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 1
,
 0 177 0 0 0 0 177 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 1 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [64,0,0,0,0,0,0,177,0,0,0,0,0,0,189,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,211,0,0,0,0,0,0,0,211,63,0,0,0,0,177,30,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[0,177,0,0,0,0,177,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,1,1] >;`

D20.24D6 in GAP, Magma, Sage, TeX

`D_{20}._{24}D_6`
`% in TeX`

`G:=Group("D20.24D6");`
`// GroupNames label`

`G:=SmallGroup(480,569);`
`// by ID`

`G=gap.SmallGroup(480,569);`
`# by ID`

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

`G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^10,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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