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## G = D20.34D6order 480 = 25·3·5

### 5th non-split extension by D20 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.34D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C15⋊D8 — D20.34D6
 Lower central C15 — C30 — C60 — D20.34D6
 Upper central C1 — C4 — C2×C4

Generators and relations for D20.34D6
G = < a,b,c,d | a20=b2=1, c6=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=c5 >

Subgroups: 524 in 124 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C30, C4○D8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, C5×Dic3, C3×Dic5, C60, C6×D5, S3×C10, C2×C30, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, Q8.13D6, C153C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4.8D10, C15⋊D8, C30.D4, C20.D6, C15⋊Q16, C2×C153C8, C3×C4○D20, C5×C4○D12, D20.34D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4○D8, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, Q8.13D6, C15⋊D4, C2×S3×D5, D4.8D10, C2×C15⋊D4, D20.34D6

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

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

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

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

60 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 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 3 4 4 4 4 4 5 5 6 6 6 6 8 8 8 8 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 12 20 2 1 1 2 12 20 2 2 2 4 20 20 30 30 30 30 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

60 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 4 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 C5⋊D4 C5⋊D4 S3×D5 Q8.13D6 C15⋊D4 C2×S3×D5 C15⋊D4 D4.8D10 D20.34D6 kernel D20.34D6 C15⋊D8 C30.D4 C20.D6 C15⋊Q16 C2×C15⋊3C8 C3×C4○D20 C5×C4○D12 C4○D20 C60 C2×C30 C4○D12 Dic10 D20 C2×C20 Dic6 D12 C2×C12 C20 C2×C10 C15 C12 C2×C6 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 4 4 4 2 2 2 2 2 4 8

Matrix representation of D20.34D6 in GL6(𝔽241)

 0 1 0 0 0 0 240 51 0 0 0 0 0 0 64 0 0 0 0 0 159 177 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 125 144 0 0 0 0 12 116 0 0 0 0 0 0 30 135 0 0 0 0 179 211 0 0 0 0 0 0 171 140 0 0 0 0 101 70
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 177 0 0 0 0 0 0 177 0 0 0 0 0 0 0 1 0 0 0 0 240 1
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 64 47 0 0 0 0 0 177 0 0 0 0 0 0 89 71 0 0 0 0 160 152

`G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,51,0,0,0,0,0,0,64,159,0,0,0,0,0,177,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[125,12,0,0,0,0,144,116,0,0,0,0,0,0,30,179,0,0,0,0,135,211,0,0,0,0,0,0,171,101,0,0,0,0,140,70],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,47,177,0,0,0,0,0,0,89,160,0,0,0,0,71,152] >;`

D20.34D6 in GAP, Magma, Sage, TeX

`D_{20}._{34}D_6`
`% in TeX`

`G:=Group("D20.34D6");`
`// GroupNames label`

`G:=SmallGroup(480,373);`
`// by ID`

`G=gap.SmallGroup(480,373);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,100,675,346,80,1356,18822]);`
`// 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,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^5>;`
`// generators/relations`

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