metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.15D6, C60.40C23, Dic6.28D10, D60.14C22, C3⋊D40⋊7C2, C3⋊C8.20D10, C3⋊Q16⋊7D5, C15⋊20(C4○D8), Q8⋊2D5⋊6S3, (C4×D5).50D6, (C6×D5).15D4, C6.153(D4×D5), Q8.19(S3×D5), (C5×Q8).25D6, C12.28D10⋊3C2, C3⋊5(Q8.D10), Q8⋊2D15⋊6C2, C30.202(C2×D4), C30.D4⋊7C2, C5⋊5(Q8.13D6), (C3×Q8).23D10, C20.40(C22×S3), (C3×Dic5).73D4, C12.40(C22×D5), D10.12(C3⋊D4), C15⋊3C8.14C22, (D5×C12).16C22, (C3×D20).14C22, (Q8×C15).10C22, Dic5.43(C3⋊D4), (C5×Dic6).14C22, (D5×C3⋊C8)⋊7C2, C4.40(C2×S3×D5), (C5×C3⋊Q16)⋊6C2, C2.35(D5×C3⋊D4), (C3×Q8⋊2D5)⋊3C2, C10.56(C2×C3⋊D4), (C5×C3⋊C8).14C22, SmallGroup(480,592)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a13b, dcd=a10c-1 >
Subgroups: 700 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, D10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×D5, D15, C30, C4○D8, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C5×Q8, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C5×Dic3, C3×Dic5, C60, C60, C6×D5, C6×D5, D30, C8×D5, D40, Q8⋊D5, C5×Q16, Q8⋊2D5, Q8⋊2D5, Q8.13D6, C5×C3⋊C8, C15⋊3C8, D30.C2, C3⋊D20, D5×C12, D5×C12, C3×D20, C3×D20, C5×Dic6, D60, Q8×C15, Q8.D10, D5×C3⋊C8, C3⋊D40, C30.D4, C5×C3⋊Q16, Q8⋊2D15, C12.28D10, C3×Q8⋊2D5, D20.D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4○D8, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.13D6, C2×S3×D5, Q8.D10, D5×C3⋊D4, D20.D6
►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)
(1 153)(2 152)(3 151)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 143)(12 142)(13 141)(14 160)(15 159)(16 158)(17 157)(18 156)(19 155)(20 154)(21 240)(22 239)(23 238)(24 237)(25 236)(26 235)(27 234)(28 233)(29 232)(30 231)(31 230)(32 229)(33 228)(34 227)(35 226)(36 225)(37 224)(38 223)(39 222)(40 221)(41 117)(42 116)(43 115)(44 114)(45 113)(46 112)(47 111)(48 110)(49 109)(50 108)(51 107)(52 106)(53 105)(54 104)(55 103)(56 102)(57 101)(58 120)(59 119)(60 118)(61 81)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(121 186)(122 185)(123 184)(124 183)(125 182)(126 181)(127 200)(128 199)(129 198)(130 197)(131 196)(132 195)(133 194)(134 193)(135 192)(136 191)(137 190)(138 189)(139 188)(140 187)(161 215)(162 214)(163 213)(164 212)(165 211)(166 210)(167 209)(168 208)(169 207)(170 206)(171 205)(172 204)(173 203)(174 202)(175 201)(176 220)(177 219)(178 218)(179 217)(180 216)
(1 77 25)(2 66 26 10 78 34)(3 75 27 19 79 23)(4 64 28 8 80 32)(5 73 29 17 61 21)(6 62 30)(7 71 31 15 63 39)(9 69 33 13 65 37)(11 67 35)(12 76 36 20 68 24)(14 74 38 18 70 22)(16 72 40)(41 121 201 45 137 205)(42 130 202 54 138 214)(43 139 203)(44 128 204 52 140 212)(46 126 206 50 122 210)(47 135 207 59 123 219)(48 124 208)(49 133 209 57 125 217)(51 131 211 55 127 215)(53 129 213)(56 136 216 60 132 220)(58 134 218)(81 222 149 91 232 159)(82 231 150 100 233 148)(83 240 151 89 234 157)(84 229 152 98 235 146)(85 238 153 87 236 155)(86 227 154 96 237 144)(88 225 156 94 239 142)(90 223 158 92 221 160)(93 230 141 99 224 147)(95 228 143 97 226 145)(101 184 179 111 194 169)(102 193 180 120 195 178)(103 182 161 109 196 167)(104 191 162 118 197 176)(105 200 163 107 198 165)(106 189 164 116 199 174)(108 187 166 114 181 172)(110 185 168 112 183 170)(113 192 171 119 186 177)(115 190 173 117 188 175)
(1 194)(2 193)(3 192)(4 191)(5 190)(6 189)(7 188)(8 187)(9 186)(10 185)(11 184)(12 183)(13 182)(14 181)(15 200)(16 199)(17 198)(18 197)(19 196)(20 195)(21 163)(22 162)(23 161)(24 180)(25 179)(26 178)(27 177)(28 176)(29 175)(30 174)(31 173)(32 172)(33 171)(34 170)(35 169)(36 168)(37 167)(38 166)(39 165)(40 164)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 100)(50 99)(51 98)(52 97)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 120)(79 119)(80 118)(121 152)(122 151)(123 150)(124 149)(125 148)(126 147)(127 146)(128 145)(129 144)(130 143)(131 142)(132 141)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 153)(201 239)(202 238)(203 237)(204 236)(205 235)(206 234)(207 233)(208 232)(209 231)(210 230)(211 229)(212 228)(213 227)(214 226)(215 225)(216 224)(217 223)(218 222)(219 221)(220 240)
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), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,240)(22,239)(23,238)(24,237)(25,236)(26,235)(27,234)(28,233)(29,232)(30,231)(31,230)(32,229)(33,228)(34,227)(35,226)(36,225)(37,224)(38,223)(39,222)(40,221)(41,117)(42,116)(43,115)(44,114)(45,113)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(57,101)(58,120)(59,119)(60,118)(61,81)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,200)(128,199)(129,198)(130,197)(131,196)(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)(139,188)(140,187)(161,215)(162,214)(163,213)(164,212)(165,211)(166,210)(167,209)(168,208)(169,207)(170,206)(171,205)(172,204)(173,203)(174,202)(175,201)(176,220)(177,219)(178,218)(179,217)(180,216), (1,77,25)(2,66,26,10,78,34)(3,75,27,19,79,23)(4,64,28,8,80,32)(5,73,29,17,61,21)(6,62,30)(7,71,31,15,63,39)(9,69,33,13,65,37)(11,67,35)(12,76,36,20,68,24)(14,74,38,18,70,22)(16,72,40)(41,121,201,45,137,205)(42,130,202,54,138,214)(43,139,203)(44,128,204,52,140,212)(46,126,206,50,122,210)(47,135,207,59,123,219)(48,124,208)(49,133,209,57,125,217)(51,131,211,55,127,215)(53,129,213)(56,136,216,60,132,220)(58,134,218)(81,222,149,91,232,159)(82,231,150,100,233,148)(83,240,151,89,234,157)(84,229,152,98,235,146)(85,238,153,87,236,155)(86,227,154,96,237,144)(88,225,156,94,239,142)(90,223,158,92,221,160)(93,230,141,99,224,147)(95,228,143,97,226,145)(101,184,179,111,194,169)(102,193,180,120,195,178)(103,182,161,109,196,167)(104,191,162,118,197,176)(105,200,163,107,198,165)(106,189,164,116,199,174)(108,187,166,114,181,172)(110,185,168,112,183,170)(113,192,171,119,186,177)(115,190,173,117,188,175), (1,194)(2,193)(3,192)(4,191)(5,190)(6,189)(7,188)(8,187)(9,186)(10,185)(11,184)(12,183)(13,182)(14,181)(15,200)(16,199)(17,198)(18,197)(19,196)(20,195)(21,163)(22,162)(23,161)(24,180)(25,179)(26,178)(27,177)(28,176)(29,175)(30,174)(31,173)(32,172)(33,171)(34,170)(35,169)(36,168)(37,167)(38,166)(39,165)(40,164)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,120)(79,119)(80,118)(121,152)(122,151)(123,150)(124,149)(125,148)(126,147)(127,146)(128,145)(129,144)(130,143)(131,142)(132,141)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(201,239)(202,238)(203,237)(204,236)(205,235)(206,234)(207,233)(208,232)(209,231)(210,230)(211,229)(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)(219,221)(220,240)>;
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), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,240)(22,239)(23,238)(24,237)(25,236)(26,235)(27,234)(28,233)(29,232)(30,231)(31,230)(32,229)(33,228)(34,227)(35,226)(36,225)(37,224)(38,223)(39,222)(40,221)(41,117)(42,116)(43,115)(44,114)(45,113)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(57,101)(58,120)(59,119)(60,118)(61,81)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,200)(128,199)(129,198)(130,197)(131,196)(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)(139,188)(140,187)(161,215)(162,214)(163,213)(164,212)(165,211)(166,210)(167,209)(168,208)(169,207)(170,206)(171,205)(172,204)(173,203)(174,202)(175,201)(176,220)(177,219)(178,218)(179,217)(180,216), (1,77,25)(2,66,26,10,78,34)(3,75,27,19,79,23)(4,64,28,8,80,32)(5,73,29,17,61,21)(6,62,30)(7,71,31,15,63,39)(9,69,33,13,65,37)(11,67,35)(12,76,36,20,68,24)(14,74,38,18,70,22)(16,72,40)(41,121,201,45,137,205)(42,130,202,54,138,214)(43,139,203)(44,128,204,52,140,212)(46,126,206,50,122,210)(47,135,207,59,123,219)(48,124,208)(49,133,209,57,125,217)(51,131,211,55,127,215)(53,129,213)(56,136,216,60,132,220)(58,134,218)(81,222,149,91,232,159)(82,231,150,100,233,148)(83,240,151,89,234,157)(84,229,152,98,235,146)(85,238,153,87,236,155)(86,227,154,96,237,144)(88,225,156,94,239,142)(90,223,158,92,221,160)(93,230,141,99,224,147)(95,228,143,97,226,145)(101,184,179,111,194,169)(102,193,180,120,195,178)(103,182,161,109,196,167)(104,191,162,118,197,176)(105,200,163,107,198,165)(106,189,164,116,199,174)(108,187,166,114,181,172)(110,185,168,112,183,170)(113,192,171,119,186,177)(115,190,173,117,188,175), (1,194)(2,193)(3,192)(4,191)(5,190)(6,189)(7,188)(8,187)(9,186)(10,185)(11,184)(12,183)(13,182)(14,181)(15,200)(16,199)(17,198)(18,197)(19,196)(20,195)(21,163)(22,162)(23,161)(24,180)(25,179)(26,178)(27,177)(28,176)(29,175)(30,174)(31,173)(32,172)(33,171)(34,170)(35,169)(36,168)(37,167)(38,166)(39,165)(40,164)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,120)(79,119)(80,118)(121,152)(122,151)(123,150)(124,149)(125,148)(126,147)(127,146)(128,145)(129,144)(130,143)(131,142)(132,141)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(201,239)(202,238)(203,237)(204,236)(205,235)(206,234)(207,233)(208,232)(209,231)(210,230)(211,229)(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)(219,221)(220,240) );
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)], [(1,153),(2,152),(3,151),(4,150),(5,149),(6,148),(7,147),(8,146),(9,145),(10,144),(11,143),(12,142),(13,141),(14,160),(15,159),(16,158),(17,157),(18,156),(19,155),(20,154),(21,240),(22,239),(23,238),(24,237),(25,236),(26,235),(27,234),(28,233),(29,232),(30,231),(31,230),(32,229),(33,228),(34,227),(35,226),(36,225),(37,224),(38,223),(39,222),(40,221),(41,117),(42,116),(43,115),(44,114),(45,113),(46,112),(47,111),(48,110),(49,109),(50,108),(51,107),(52,106),(53,105),(54,104),(55,103),(56,102),(57,101),(58,120),(59,119),(60,118),(61,81),(62,100),(63,99),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(121,186),(122,185),(123,184),(124,183),(125,182),(126,181),(127,200),(128,199),(129,198),(130,197),(131,196),(132,195),(133,194),(134,193),(135,192),(136,191),(137,190),(138,189),(139,188),(140,187),(161,215),(162,214),(163,213),(164,212),(165,211),(166,210),(167,209),(168,208),(169,207),(170,206),(171,205),(172,204),(173,203),(174,202),(175,201),(176,220),(177,219),(178,218),(179,217),(180,216)], [(1,77,25),(2,66,26,10,78,34),(3,75,27,19,79,23),(4,64,28,8,80,32),(5,73,29,17,61,21),(6,62,30),(7,71,31,15,63,39),(9,69,33,13,65,37),(11,67,35),(12,76,36,20,68,24),(14,74,38,18,70,22),(16,72,40),(41,121,201,45,137,205),(42,130,202,54,138,214),(43,139,203),(44,128,204,52,140,212),(46,126,206,50,122,210),(47,135,207,59,123,219),(48,124,208),(49,133,209,57,125,217),(51,131,211,55,127,215),(53,129,213),(56,136,216,60,132,220),(58,134,218),(81,222,149,91,232,159),(82,231,150,100,233,148),(83,240,151,89,234,157),(84,229,152,98,235,146),(85,238,153,87,236,155),(86,227,154,96,237,144),(88,225,156,94,239,142),(90,223,158,92,221,160),(93,230,141,99,224,147),(95,228,143,97,226,145),(101,184,179,111,194,169),(102,193,180,120,195,178),(103,182,161,109,196,167),(104,191,162,118,197,176),(105,200,163,107,198,165),(106,189,164,116,199,174),(108,187,166,114,181,172),(110,185,168,112,183,170),(113,192,171,119,186,177),(115,190,173,117,188,175)], [(1,194),(2,193),(3,192),(4,191),(5,190),(6,189),(7,188),(8,187),(9,186),(10,185),(11,184),(12,183),(13,182),(14,181),(15,200),(16,199),(17,198),(18,197),(19,196),(20,195),(21,163),(22,162),(23,161),(24,180),(25,179),(26,178),(27,177),(28,176),(29,175),(30,174),(31,173),(32,172),(33,171),(34,170),(35,169),(36,168),(37,167),(38,166),(39,165),(40,164),(41,88),(42,87),(43,86),(44,85),(45,84),(46,83),(47,82),(48,81),(49,100),(50,99),(51,98),(52,97),(53,96),(54,95),(55,94),(56,93),(57,92),(58,91),(59,90),(60,89),(61,117),(62,116),(63,115),(64,114),(65,113),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,120),(79,119),(80,118),(121,152),(122,151),(123,150),(124,149),(125,148),(126,147),(127,146),(128,145),(129,144),(130,143),(131,142),(132,141),(133,160),(134,159),(135,158),(136,157),(137,156),(138,155),(139,154),(140,153),(201,239),(202,238),(203,237),(204,236),(205,235),(206,234),(207,233),(208,232),(209,231),(210,230),(211,229),(212,228),(213,227),(214,226),(215,225),(216,224),(217,223),(218,222),(219,221),(220,240)]])
48 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 | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| 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 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 10 | 20 | 60 | 2 | 2 | 4 | 5 | 5 | 12 | 2 | 2 | 2 | 20 | 20 | 20 | 6 | 6 | 30 | 30 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | 4 | 8 | 8 | 24 | 24 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
48 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 | C3⋊D4 | C3⋊D4 | C4○D8 | S3×D5 | D4×D5 | Q8.13D6 | C2×S3×D5 | Q8.D10 | D5×C3⋊D4 | D20.D6 |
| kernel | D20.D6 | D5×C3⋊C8 | C3⋊D40 | C30.D4 | C5×C3⋊Q16 | Q8⋊2D15 | C12.28D10 | C3×Q8⋊2D5 | Q8⋊2D5 | C3×Dic5 | C6×D5 | C3⋊Q16 | C4×D5 | D20 | C5×Q8 | C3⋊C8 | Dic6 | C3×Q8 | Dic5 | D10 | C15 | Q8 | 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 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.D6 ►in GL6(𝔽241)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 143 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 9 | 238 | 0 | 0 | 0 | 0 |
| 107 | 232 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 210 | 112 |
| 0 | 0 | 0 | 0 | 198 | 31 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 49 |
| 0 | 0 | 0 | 0 | 177 | 239 |
| 212 | 90 | 0 | 0 | 0 | 0 |
| 71 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 52 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 121 | 76 |
| 0 | 0 | 0 | 0 | 23 | 120 |
G:=sub<GL(6,GF(241))| [64,143,0,0,0,0,0,177,0,0,0,0,0,0,0,240,0,0,0,0,1,189,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[9,107,0,0,0,0,238,232,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,210,198,0,0,0,0,112,31],[1,6,0,0,0,0,0,240,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,1,177,0,0,0,0,49,239],[212,71,0,0,0,0,90,29,0,0,0,0,0,0,240,52,0,0,0,0,0,1,0,0,0,0,0,0,121,23,0,0,0,0,76,120] >;
D20.D6 in GAP, Magma, Sage, TeX
D_{20}.D_6 % in TeX
G:=Group("D20.D6"); // GroupNames label
G:=SmallGroup(480,592);
// by ID
G=gap.SmallGroup(480,592);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,422,135,100,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d=a^13*b,d*c*d=a^10*c^-1>;
// generators/relations