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## G = D20.39D6order 480 = 25·3·5

### The non-split extension by D20 of D6 acting through Inn(D20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20.39D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — D5×Dic6 — D20.39D6
 Lower central C15 — C30 — D20.39D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D20.39D6
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a10c5 >

Subgroups: 1292 in 292 conjugacy classes, 108 normal (38 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×8], C22, C22 [×4], C5, S3 [×2], C6, C6 [×3], C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], D5 [×2], C10, C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×2], C2×C6, C2×C6 [×2], C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], Dic6, Dic6 [×8], C4×S3 [×2], C4×S3 [×4], D12, C2×Dic3 [×6], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C5×S3 [×2], C3×D5 [×2], C30, C30, 2- 1+4, Dic10, Dic10 [×8], C4×D5 [×2], C4×D5 [×4], D20, C2×Dic5 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C2×Dic6 [×3], C4○D12, C4○D12 [×2], D42S3 [×6], S3×Q8 [×2], C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×4], C60 [×2], C6×D5 [×2], S3×C10 [×2], C2×C30, C2×Dic10 [×3], C4○D20, C4○D20 [×2], D42D5 [×6], Q8×D5 [×2], C5×C4○D4, Q8○D12, D5×Dic3 [×4], S3×Dic5 [×4], C15⋊D4 [×4], C15⋊Q8 [×4], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30 [×4], C2×Dic15 [×2], C2×C60, D4.10D10, D5×Dic6 [×2], D205S3 [×2], S3×Dic10 [×2], D125D5 [×2], C30.C23 [×4], C3×C4○D20, C5×C4○D12, C2×Dic30, D20.39D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8○D12, C2×S3×D5 [×3], D4.10D10, C22×S3×D5, D20.39D6

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

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 6D 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 10 10 10 10 10 10 10 10 12 12 12 12 12 15 15 20 20 20 20 20 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 2 6 6 10 10 2 2 2 6 6 10 10 30 30 30 30 2 2 2 4 20 20 2 2 4 4 12 12 12 12 2 2 4 20 20 4 4 2 2 2 2 4 4 12 12 12 12 4 ··· 4 4 ··· 4

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + - + - + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 D10 D10 D10 D10 D10 2- 1+4 S3×D5 Q8○D12 C2×S3×D5 C2×S3×D5 D4.10D10 D20.39D6 kernel D20.39D6 D5×Dic6 D20⋊5S3 S3×Dic10 D12⋊5D5 C30.C23 C3×C4○D20 C5×C4○D12 C2×Dic30 C4○D20 C4○D12 Dic10 C4×D5 D20 C5⋊D4 C2×C20 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C15 C2×C4 C5 C4 C22 C3 C1 # reps 1 2 2 2 2 4 1 1 1 1 2 1 2 1 2 1 2 4 2 4 2 1 2 2 4 2 4 8

Matrix representation of D20.39D6 in GL4(𝔽61) generated by

 7 32 0 0 29 2 0 0 0 0 7 32 0 0 29 2
,
 0 60 0 0 60 0 0 0 0 0 0 60 0 0 60 0
,
 23 0 38 0 0 23 0 38 23 0 46 0 0 23 0 46
,
 2 0 23 0 0 2 0 23 21 0 59 0 0 21 0 59
`G:=sub<GL(4,GF(61))| [7,29,0,0,32,2,0,0,0,0,7,29,0,0,32,2],[0,60,0,0,60,0,0,0,0,0,0,60,0,0,60,0],[23,0,23,0,0,23,0,23,38,0,46,0,0,38,0,46],[2,0,21,0,0,2,0,21,23,0,59,0,0,23,0,59] >;`

D20.39D6 in GAP, Magma, Sage, TeX

`D_{20}._{39}D_6`
`% in TeX`

`G:=Group("D20.39D6");`
`// GroupNames label`

`G:=SmallGroup(480,1077);`
`// by ID`

`G=gap.SmallGroup(480,1077);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,80,1356,18822]);`
`// Polycyclic`

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

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