metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.31D6, C12.60D20, C60.110D4, C60.104C23, Dic10.32D6, D60.44C22, Dic30.46C22, C4○D20⋊1S3, C3⋊C8.31D10, (C2×C6).6D20, C15⋊12(C4○D8), C3⋊D40⋊15C2, (C2×C30).54D4, C30.86(C2×D4), C6.49(C2×D20), C3⋊5(D40⋊7C2), C3⋊Dic20⋊15C2, (C2×C20).314D6, (C2×C12).97D10, C5⋊1(Q8.13D6), D60⋊11C2⋊8C2, C15⋊SD16⋊15C2, C6.D20⋊15C2, C20.65(C3⋊D4), C4.32(C3⋊D20), (C2×C60).42C22, C12.95(C22×D5), C20.154(C22×S3), (C3×D20).36C22, C22.1(C3⋊D20), (C3×Dic10).37C22, (C2×C3⋊C8)⋊7D5, (C10×C3⋊C8)⋊3C2, C4.103(C2×S3×D5), (C3×C4○D20)⋊4C2, (C2×C4).96(S3×D5), C2.8(C2×C3⋊D20), C10.4(C2×C3⋊D4), (C5×C3⋊C8).35C22, (C2×C10).32(C3⋊D4), SmallGroup(480,387)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.31D6
G = < a,b,c,d | a20=b2=1, c6=a10, d2=a15, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a15b, dcd-1=c5 >
Subgroups: 716 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, D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, D15, C30, C30, C4○D8, C40, Dic10, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C3×Dic5, Dic15, C60, C6×D5, D30, C2×C30, C40⋊C2, D40, Dic20, C2×C40, C4○D20, C4○D20, Q8.13D6, C5×C3⋊C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D40⋊7C2, C3⋊D40, C6.D20, C15⋊SD16, C3⋊Dic20, C10×C3⋊C8, C3×C4○D20, D60⋊11C2, D20.31D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4○D8, D20, C22×D5, C2×C3⋊D4, S3×D5, C2×D20, Q8.13D6, C3⋊D20, C2×S3×D5, D40⋊7C2, C2×C3⋊D20, D20.31D6
►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)
(1 132)(2 131)(3 130)(4 129)(5 128)(6 127)(7 126)(8 125)(9 124)(10 123)(11 122)(12 121)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 152)(22 151)(23 150)(24 149)(25 148)(26 147)(27 146)(28 145)(29 144)(30 143)(31 142)(32 141)(33 160)(34 159)(35 158)(36 157)(37 156)(38 155)(39 154)(40 153)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 100)(59 99)(60 98)(61 223)(62 222)(63 221)(64 240)(65 239)(66 238)(67 237)(68 236)(69 235)(70 234)(71 233)(72 232)(73 231)(74 230)(75 229)(76 228)(77 227)(78 226)(79 225)(80 224)(101 212)(102 211)(103 210)(104 209)(105 208)(106 207)(107 206)(108 205)(109 204)(110 203)(111 202)(112 201)(113 220)(114 219)(115 218)(116 217)(117 216)(118 215)(119 214)(120 213)(161 189)(162 188)(163 187)(164 186)(165 185)(166 184)(167 183)(168 182)(169 181)(170 200)(171 199)(172 198)(173 197)(174 196)(175 195)(176 194)(177 193)(178 192)(179 191)(180 190)
(1 210 153 6 215 158 11 220 143 16 205 148)(2 211 154 7 216 159 12 201 144 17 206 149)(3 212 155 8 217 160 13 202 145 18 207 150)(4 213 156 9 218 141 14 203 146 19 208 151)(5 214 157 10 219 142 15 204 147 20 209 152)(21 128 119 36 123 114 31 138 109 26 133 104)(22 129 120 37 124 115 32 139 110 27 134 105)(23 130 101 38 125 116 33 140 111 28 135 106)(24 131 102 39 126 117 34 121 112 29 136 107)(25 132 103 40 127 118 35 122 113 30 137 108)(41 64 187 46 69 192 51 74 197 56 79 182)(42 65 188 47 70 193 52 75 198 57 80 183)(43 66 189 48 71 194 53 76 199 58 61 184)(44 67 190 49 72 195 54 77 200 59 62 185)(45 68 191 50 73 196 55 78 181 60 63 186)(81 224 167 96 239 162 91 234 177 86 229 172)(82 225 168 97 240 163 92 235 178 87 230 173)(83 226 169 98 221 164 93 236 179 88 231 174)(84 227 170 99 222 165 94 237 180 89 232 175)(85 228 171 100 223 166 95 238 161 90 233 176)
(1 41 16 56 11 51 6 46)(2 42 17 57 12 52 7 47)(3 43 18 58 13 53 8 48)(4 44 19 59 14 54 9 49)(5 45 20 60 15 55 10 50)(21 221 36 236 31 231 26 226)(22 222 37 237 32 232 27 227)(23 223 38 238 33 233 28 228)(24 224 39 239 34 234 29 229)(25 225 40 240 35 235 30 230)(61 150 76 145 71 160 66 155)(62 151 77 146 72 141 67 156)(63 152 78 147 73 142 68 157)(64 153 79 148 74 143 69 158)(65 154 80 149 75 144 70 159)(81 136 96 131 91 126 86 121)(82 137 97 132 92 127 87 122)(83 138 98 133 93 128 88 123)(84 139 99 134 94 129 89 124)(85 140 100 135 95 130 90 125)(101 171 116 166 111 161 106 176)(102 172 117 167 112 162 107 177)(103 173 118 168 113 163 108 178)(104 174 119 169 114 164 109 179)(105 175 120 170 115 165 110 180)(181 214 196 209 191 204 186 219)(182 215 197 210 192 205 187 220)(183 216 198 211 193 206 188 201)(184 217 199 212 194 207 189 202)(185 218 200 213 195 208 190 203)
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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,152)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,160)(34,159)(35,158)(36,157)(37,156)(38,155)(39,154)(40,153)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,100)(59,99)(60,98)(61,223)(62,222)(63,221)(64,240)(65,239)(66,238)(67,237)(68,236)(69,235)(70,234)(71,233)(72,232)(73,231)(74,230)(75,229)(76,228)(77,227)(78,226)(79,225)(80,224)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201)(113,220)(114,219)(115,218)(116,217)(117,216)(118,215)(119,214)(120,213)(161,189)(162,188)(163,187)(164,186)(165,185)(166,184)(167,183)(168,182)(169,181)(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190), (1,210,153,6,215,158,11,220,143,16,205,148)(2,211,154,7,216,159,12,201,144,17,206,149)(3,212,155,8,217,160,13,202,145,18,207,150)(4,213,156,9,218,141,14,203,146,19,208,151)(5,214,157,10,219,142,15,204,147,20,209,152)(21,128,119,36,123,114,31,138,109,26,133,104)(22,129,120,37,124,115,32,139,110,27,134,105)(23,130,101,38,125,116,33,140,111,28,135,106)(24,131,102,39,126,117,34,121,112,29,136,107)(25,132,103,40,127,118,35,122,113,30,137,108)(41,64,187,46,69,192,51,74,197,56,79,182)(42,65,188,47,70,193,52,75,198,57,80,183)(43,66,189,48,71,194,53,76,199,58,61,184)(44,67,190,49,72,195,54,77,200,59,62,185)(45,68,191,50,73,196,55,78,181,60,63,186)(81,224,167,96,239,162,91,234,177,86,229,172)(82,225,168,97,240,163,92,235,178,87,230,173)(83,226,169,98,221,164,93,236,179,88,231,174)(84,227,170,99,222,165,94,237,180,89,232,175)(85,228,171,100,223,166,95,238,161,90,233,176), (1,41,16,56,11,51,6,46)(2,42,17,57,12,52,7,47)(3,43,18,58,13,53,8,48)(4,44,19,59,14,54,9,49)(5,45,20,60,15,55,10,50)(21,221,36,236,31,231,26,226)(22,222,37,237,32,232,27,227)(23,223,38,238,33,233,28,228)(24,224,39,239,34,234,29,229)(25,225,40,240,35,235,30,230)(61,150,76,145,71,160,66,155)(62,151,77,146,72,141,67,156)(63,152,78,147,73,142,68,157)(64,153,79,148,74,143,69,158)(65,154,80,149,75,144,70,159)(81,136,96,131,91,126,86,121)(82,137,97,132,92,127,87,122)(83,138,98,133,93,128,88,123)(84,139,99,134,94,129,89,124)(85,140,100,135,95,130,90,125)(101,171,116,166,111,161,106,176)(102,172,117,167,112,162,107,177)(103,173,118,168,113,163,108,178)(104,174,119,169,114,164,109,179)(105,175,120,170,115,165,110,180)(181,214,196,209,191,204,186,219)(182,215,197,210,192,205,187,220)(183,216,198,211,193,206,188,201)(184,217,199,212,194,207,189,202)(185,218,200,213,195,208,190,203)>;
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), (1,132)(2,131)(3,130)(4,129)(5,128)(6,127)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,152)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,160)(34,159)(35,158)(36,157)(37,156)(38,155)(39,154)(40,153)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,100)(59,99)(60,98)(61,223)(62,222)(63,221)(64,240)(65,239)(66,238)(67,237)(68,236)(69,235)(70,234)(71,233)(72,232)(73,231)(74,230)(75,229)(76,228)(77,227)(78,226)(79,225)(80,224)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201)(113,220)(114,219)(115,218)(116,217)(117,216)(118,215)(119,214)(120,213)(161,189)(162,188)(163,187)(164,186)(165,185)(166,184)(167,183)(168,182)(169,181)(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190), (1,210,153,6,215,158,11,220,143,16,205,148)(2,211,154,7,216,159,12,201,144,17,206,149)(3,212,155,8,217,160,13,202,145,18,207,150)(4,213,156,9,218,141,14,203,146,19,208,151)(5,214,157,10,219,142,15,204,147,20,209,152)(21,128,119,36,123,114,31,138,109,26,133,104)(22,129,120,37,124,115,32,139,110,27,134,105)(23,130,101,38,125,116,33,140,111,28,135,106)(24,131,102,39,126,117,34,121,112,29,136,107)(25,132,103,40,127,118,35,122,113,30,137,108)(41,64,187,46,69,192,51,74,197,56,79,182)(42,65,188,47,70,193,52,75,198,57,80,183)(43,66,189,48,71,194,53,76,199,58,61,184)(44,67,190,49,72,195,54,77,200,59,62,185)(45,68,191,50,73,196,55,78,181,60,63,186)(81,224,167,96,239,162,91,234,177,86,229,172)(82,225,168,97,240,163,92,235,178,87,230,173)(83,226,169,98,221,164,93,236,179,88,231,174)(84,227,170,99,222,165,94,237,180,89,232,175)(85,228,171,100,223,166,95,238,161,90,233,176), (1,41,16,56,11,51,6,46)(2,42,17,57,12,52,7,47)(3,43,18,58,13,53,8,48)(4,44,19,59,14,54,9,49)(5,45,20,60,15,55,10,50)(21,221,36,236,31,231,26,226)(22,222,37,237,32,232,27,227)(23,223,38,238,33,233,28,228)(24,224,39,239,34,234,29,229)(25,225,40,240,35,235,30,230)(61,150,76,145,71,160,66,155)(62,151,77,146,72,141,67,156)(63,152,78,147,73,142,68,157)(64,153,79,148,74,143,69,158)(65,154,80,149,75,144,70,159)(81,136,96,131,91,126,86,121)(82,137,97,132,92,127,87,122)(83,138,98,133,93,128,88,123)(84,139,99,134,94,129,89,124)(85,140,100,135,95,130,90,125)(101,171,116,166,111,161,106,176)(102,172,117,167,112,162,107,177)(103,173,118,168,113,163,108,178)(104,174,119,169,114,164,109,179)(105,175,120,170,115,165,110,180)(181,214,196,209,191,204,186,219)(182,215,197,210,192,205,187,220)(183,216,198,211,193,206,188,201)(184,217,199,212,194,207,189,202)(185,218,200,213,195,208,190,203) );
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)], [(1,132),(2,131),(3,130),(4,129),(5,128),(6,127),(7,126),(8,125),(9,124),(10,123),(11,122),(12,121),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,152),(22,151),(23,150),(24,149),(25,148),(26,147),(27,146),(28,145),(29,144),(30,143),(31,142),(32,141),(33,160),(34,159),(35,158),(36,157),(37,156),(38,155),(39,154),(40,153),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,100),(59,99),(60,98),(61,223),(62,222),(63,221),(64,240),(65,239),(66,238),(67,237),(68,236),(69,235),(70,234),(71,233),(72,232),(73,231),(74,230),(75,229),(76,228),(77,227),(78,226),(79,225),(80,224),(101,212),(102,211),(103,210),(104,209),(105,208),(106,207),(107,206),(108,205),(109,204),(110,203),(111,202),(112,201),(113,220),(114,219),(115,218),(116,217),(117,216),(118,215),(119,214),(120,213),(161,189),(162,188),(163,187),(164,186),(165,185),(166,184),(167,183),(168,182),(169,181),(170,200),(171,199),(172,198),(173,197),(174,196),(175,195),(176,194),(177,193),(178,192),(179,191),(180,190)], [(1,210,153,6,215,158,11,220,143,16,205,148),(2,211,154,7,216,159,12,201,144,17,206,149),(3,212,155,8,217,160,13,202,145,18,207,150),(4,213,156,9,218,141,14,203,146,19,208,151),(5,214,157,10,219,142,15,204,147,20,209,152),(21,128,119,36,123,114,31,138,109,26,133,104),(22,129,120,37,124,115,32,139,110,27,134,105),(23,130,101,38,125,116,33,140,111,28,135,106),(24,131,102,39,126,117,34,121,112,29,136,107),(25,132,103,40,127,118,35,122,113,30,137,108),(41,64,187,46,69,192,51,74,197,56,79,182),(42,65,188,47,70,193,52,75,198,57,80,183),(43,66,189,48,71,194,53,76,199,58,61,184),(44,67,190,49,72,195,54,77,200,59,62,185),(45,68,191,50,73,196,55,78,181,60,63,186),(81,224,167,96,239,162,91,234,177,86,229,172),(82,225,168,97,240,163,92,235,178,87,230,173),(83,226,169,98,221,164,93,236,179,88,231,174),(84,227,170,99,222,165,94,237,180,89,232,175),(85,228,171,100,223,166,95,238,161,90,233,176)], [(1,41,16,56,11,51,6,46),(2,42,17,57,12,52,7,47),(3,43,18,58,13,53,8,48),(4,44,19,59,14,54,9,49),(5,45,20,60,15,55,10,50),(21,221,36,236,31,231,26,226),(22,222,37,237,32,232,27,227),(23,223,38,238,33,233,28,228),(24,224,39,239,34,234,29,229),(25,225,40,240,35,235,30,230),(61,150,76,145,71,160,66,155),(62,151,77,146,72,141,67,156),(63,152,78,147,73,142,68,157),(64,153,79,148,74,143,69,158),(65,154,80,149,75,144,70,159),(81,136,96,131,91,126,86,121),(82,137,97,132,92,127,87,122),(83,138,98,133,93,128,88,123),(84,139,99,134,94,129,89,124),(85,140,100,135,95,130,90,125),(101,171,116,166,111,161,106,176),(102,172,117,167,112,162,107,177),(103,173,118,168,113,163,108,178),(104,174,119,169,114,164,109,179),(105,175,120,170,115,165,110,180),(181,214,196,209,191,204,186,219),(182,215,197,210,192,205,187,220),(183,216,198,211,193,206,188,201),(184,217,199,212,194,207,189,202),(185,218,200,213,195,208,190,203)]])
72 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 | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 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 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 60 | 2 | 1 | 1 | 2 | 20 | 60 | 2 | 2 | 2 | 4 | 20 | 20 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | C4○D8 | D20 | D20 | D40⋊7C2 | S3×D5 | Q8.13D6 | C3⋊D20 | C2×S3×D5 | C3⋊D20 | D20.31D6 |
| kernel | D20.31D6 | C3⋊D40 | C6.D20 | C15⋊SD16 | C3⋊Dic20 | C10×C3⋊C8 | C3×C4○D20 | D60⋊11C2 | C4○D20 | C60 | C2×C30 | C2×C3⋊C8 | Dic10 | D20 | C2×C20 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C15 | C12 | C2×C6 | C3 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of D20.31D6 ►in GL4(𝔽241) generated by
| 6 | 0 | 0 | 0 |
| 26 | 201 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 91 | 80 | 0 | 0 |
| 17 | 150 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 177 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 240 | 240 |
| 0 | 0 | 1 | 0 |
| 30 | 0 | 0 | 0 |
| 11 | 233 | 0 | 0 |
| 0 | 0 | 127 | 136 |
| 0 | 0 | 9 | 114 |
G:=sub<GL(4,GF(241))| [6,26,0,0,0,201,0,0,0,0,1,0,0,0,0,1],[91,17,0,0,80,150,0,0,0,0,1,0,0,0,0,1],[177,0,0,0,0,177,0,0,0,0,240,1,0,0,240,0],[30,11,0,0,0,233,0,0,0,0,127,9,0,0,136,114] >;
D20.31D6 in GAP, Magma, Sage, TeX
D_{20}._{31}D_6 % in TeX
G:=Group("D20.31D6"); // GroupNames label
G:=SmallGroup(480,387);
// by ID
G=gap.SmallGroup(480,387);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,100,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=a^10,d^2=a^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^5>;
// generators/relations