metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.62D4, D20.37D6, C60.154C23, Dic6.37D10, Dic10.37D6, C4○D20.4S3, (C2×Dic6)⋊8D5, (C2×C30).50D4, C30.82(C2×D4), (C2×C20).93D6, C60.7C4⋊4C2, (C10×Dic6)⋊1C2, C15⋊Q16⋊13C2, (C2×C12).93D10, C5⋊4(Q8.14D6), C15⋊7(C8.C22), C30.D4⋊14C2, C4.17(C15⋊D4), C3⋊4(C20.C23), C12.83(C5⋊D4), C20.27(C3⋊D4), C20.93(C22×S3), (C2×C60).26C22, C12.91(C22×D5), C15⋊3C8.25C22, (C3×D20).43C22, C22.10(C15⋊D4), (C5×Dic6).44C22, (C3×Dic10).44C22, C4.127(C2×S3×D5), (C2×C4).12(S3×D5), C6.76(C2×C5⋊D4), (C3×C4○D20).1C2, C10.77(C2×C3⋊D4), C2.10(C2×C15⋊D4), (C2×C6).13(C5⋊D4), (C2×C10).54(C3⋊D4), SmallGroup(480,383)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.37D6
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=a10c5 >
Subgroups: 492 in 120 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C8.C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, C5×Dic3, C3×Dic5, C60, C6×D5, C2×C30, C4.Dic5, Q8⋊D5, C5⋊Q16, C4○D20, Q8×C10, Q8.14D6, C15⋊3C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, C20.C23, C30.D4, C15⋊Q16, C60.7C4, C3×C4○D20, C10×Dic6, D20.37D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8.C22, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, Q8.14D6, C15⋊D4, C2×S3×D5, C20.C23, C2×C15⋊D4, D20.37D6
►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)
(1 165)(2 164)(3 163)(4 162)(5 161)(6 180)(7 179)(8 178)(9 177)(10 176)(11 175)(12 174)(13 173)(14 172)(15 171)(16 170)(17 169)(18 168)(19 167)(20 166)(21 122)(22 121)(23 140)(24 139)(25 138)(26 137)(27 136)(28 135)(29 134)(30 133)(31 132)(32 131)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 205)(42 204)(43 203)(44 202)(45 201)(46 220)(47 219)(48 218)(49 217)(50 216)(51 215)(52 214)(53 213)(54 212)(55 211)(56 210)(57 209)(58 208)(59 207)(60 206)(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)(81 222)(82 221)(83 240)(84 239)(85 238)(86 237)(87 236)(88 235)(89 234)(90 233)(91 232)(92 231)(93 230)(94 229)(95 228)(96 227)(97 226)(98 225)(99 224)(100 223)(101 158)(102 157)(103 156)(104 155)(105 154)(106 153)(107 152)(108 151)(109 150)(110 149)(111 148)(112 147)(113 146)(114 145)(115 144)(116 143)(117 142)(118 141)(119 160)(120 159)
(1 215 122 6 220 127 11 205 132 16 210 137)(2 216 123 7 201 128 12 206 133 17 211 138)(3 217 124 8 202 129 13 207 134 18 212 139)(4 218 125 9 203 130 14 208 135 19 213 140)(5 219 126 10 204 131 15 209 136 20 214 121)(21 180 46 36 175 41 31 170 56 26 165 51)(22 161 47 37 176 42 32 171 57 27 166 52)(23 162 48 38 177 43 33 172 58 28 167 53)(24 163 49 39 178 44 34 173 59 29 168 54)(25 164 50 40 179 45 35 174 60 30 169 55)(61 152 84 76 147 99 71 142 94 66 157 89)(62 153 85 77 148 100 72 143 95 67 158 90)(63 154 86 78 149 81 73 144 96 68 159 91)(64 155 87 79 150 82 74 145 97 69 160 92)(65 156 88 80 151 83 75 146 98 70 141 93)(101 233 183 106 238 188 111 223 193 116 228 198)(102 234 184 107 239 189 112 224 194 117 229 199)(103 235 185 108 240 190 113 225 195 118 230 200)(104 236 186 109 221 191 114 226 196 119 231 181)(105 237 187 110 222 192 115 227 197 120 232 182)
(1 186 11 196)(2 197 12 187)(3 188 13 198)(4 199 14 189)(5 190 15 200)(6 181 16 191)(7 192 17 182)(8 183 18 193)(9 194 19 184)(10 185 20 195)(21 160 31 150)(22 151 32 141)(23 142 33 152)(24 153 34 143)(25 144 35 154)(26 155 36 145)(27 146 37 156)(28 157 38 147)(29 148 39 158)(30 159 40 149)(41 82 51 92)(42 93 52 83)(43 84 53 94)(44 95 54 85)(45 86 55 96)(46 97 56 87)(47 88 57 98)(48 99 58 89)(49 90 59 100)(50 81 60 91)(61 162 71 172)(62 173 72 163)(63 164 73 174)(64 175 74 165)(65 166 75 176)(66 177 76 167)(67 168 77 178)(68 179 78 169)(69 170 79 180)(70 161 80 171)(101 139 111 129)(102 130 112 140)(103 121 113 131)(104 132 114 122)(105 123 115 133)(106 134 116 124)(107 125 117 135)(108 136 118 126)(109 127 119 137)(110 138 120 128)(201 222 211 232)(202 233 212 223)(203 224 213 234)(204 235 214 225)(205 226 215 236)(206 237 216 227)(207 228 217 238)(208 239 218 229)(209 230 219 240)(210 221 220 231)
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), (1,165)(2,164)(3,163)(4,162)(5,161)(6,180)(7,179)(8,178)(9,177)(10,176)(11,175)(12,174)(13,173)(14,172)(15,171)(16,170)(17,169)(18,168)(19,167)(20,166)(21,122)(22,121)(23,140)(24,139)(25,138)(26,137)(27,136)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,205)(42,204)(43,203)(44,202)(45,201)(46,220)(47,219)(48,218)(49,217)(50,216)(51,215)(52,214)(53,213)(54,212)(55,211)(56,210)(57,209)(58,208)(59,207)(60,206)(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)(81,222)(82,221)(83,240)(84,239)(85,238)(86,237)(87,236)(88,235)(89,234)(90,233)(91,232)(92,231)(93,230)(94,229)(95,228)(96,227)(97,226)(98,225)(99,224)(100,223)(101,158)(102,157)(103,156)(104,155)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,160)(120,159), (1,215,122,6,220,127,11,205,132,16,210,137)(2,216,123,7,201,128,12,206,133,17,211,138)(3,217,124,8,202,129,13,207,134,18,212,139)(4,218,125,9,203,130,14,208,135,19,213,140)(5,219,126,10,204,131,15,209,136,20,214,121)(21,180,46,36,175,41,31,170,56,26,165,51)(22,161,47,37,176,42,32,171,57,27,166,52)(23,162,48,38,177,43,33,172,58,28,167,53)(24,163,49,39,178,44,34,173,59,29,168,54)(25,164,50,40,179,45,35,174,60,30,169,55)(61,152,84,76,147,99,71,142,94,66,157,89)(62,153,85,77,148,100,72,143,95,67,158,90)(63,154,86,78,149,81,73,144,96,68,159,91)(64,155,87,79,150,82,74,145,97,69,160,92)(65,156,88,80,151,83,75,146,98,70,141,93)(101,233,183,106,238,188,111,223,193,116,228,198)(102,234,184,107,239,189,112,224,194,117,229,199)(103,235,185,108,240,190,113,225,195,118,230,200)(104,236,186,109,221,191,114,226,196,119,231,181)(105,237,187,110,222,192,115,227,197,120,232,182), (1,186,11,196)(2,197,12,187)(3,188,13,198)(4,199,14,189)(5,190,15,200)(6,181,16,191)(7,192,17,182)(8,183,18,193)(9,194,19,184)(10,185,20,195)(21,160,31,150)(22,151,32,141)(23,142,33,152)(24,153,34,143)(25,144,35,154)(26,155,36,145)(27,146,37,156)(28,157,38,147)(29,148,39,158)(30,159,40,149)(41,82,51,92)(42,93,52,83)(43,84,53,94)(44,95,54,85)(45,86,55,96)(46,97,56,87)(47,88,57,98)(48,99,58,89)(49,90,59,100)(50,81,60,91)(61,162,71,172)(62,173,72,163)(63,164,73,174)(64,175,74,165)(65,166,75,176)(66,177,76,167)(67,168,77,178)(68,179,78,169)(69,170,79,180)(70,161,80,171)(101,139,111,129)(102,130,112,140)(103,121,113,131)(104,132,114,122)(105,123,115,133)(106,134,116,124)(107,125,117,135)(108,136,118,126)(109,127,119,137)(110,138,120,128)(201,222,211,232)(202,233,212,223)(203,224,213,234)(204,235,214,225)(205,226,215,236)(206,237,216,227)(207,228,217,238)(208,239,218,229)(209,230,219,240)(210,221,220,231)>;
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), (1,165)(2,164)(3,163)(4,162)(5,161)(6,180)(7,179)(8,178)(9,177)(10,176)(11,175)(12,174)(13,173)(14,172)(15,171)(16,170)(17,169)(18,168)(19,167)(20,166)(21,122)(22,121)(23,140)(24,139)(25,138)(26,137)(27,136)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,205)(42,204)(43,203)(44,202)(45,201)(46,220)(47,219)(48,218)(49,217)(50,216)(51,215)(52,214)(53,213)(54,212)(55,211)(56,210)(57,209)(58,208)(59,207)(60,206)(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)(81,222)(82,221)(83,240)(84,239)(85,238)(86,237)(87,236)(88,235)(89,234)(90,233)(91,232)(92,231)(93,230)(94,229)(95,228)(96,227)(97,226)(98,225)(99,224)(100,223)(101,158)(102,157)(103,156)(104,155)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,160)(120,159), (1,215,122,6,220,127,11,205,132,16,210,137)(2,216,123,7,201,128,12,206,133,17,211,138)(3,217,124,8,202,129,13,207,134,18,212,139)(4,218,125,9,203,130,14,208,135,19,213,140)(5,219,126,10,204,131,15,209,136,20,214,121)(21,180,46,36,175,41,31,170,56,26,165,51)(22,161,47,37,176,42,32,171,57,27,166,52)(23,162,48,38,177,43,33,172,58,28,167,53)(24,163,49,39,178,44,34,173,59,29,168,54)(25,164,50,40,179,45,35,174,60,30,169,55)(61,152,84,76,147,99,71,142,94,66,157,89)(62,153,85,77,148,100,72,143,95,67,158,90)(63,154,86,78,149,81,73,144,96,68,159,91)(64,155,87,79,150,82,74,145,97,69,160,92)(65,156,88,80,151,83,75,146,98,70,141,93)(101,233,183,106,238,188,111,223,193,116,228,198)(102,234,184,107,239,189,112,224,194,117,229,199)(103,235,185,108,240,190,113,225,195,118,230,200)(104,236,186,109,221,191,114,226,196,119,231,181)(105,237,187,110,222,192,115,227,197,120,232,182), (1,186,11,196)(2,197,12,187)(3,188,13,198)(4,199,14,189)(5,190,15,200)(6,181,16,191)(7,192,17,182)(8,183,18,193)(9,194,19,184)(10,185,20,195)(21,160,31,150)(22,151,32,141)(23,142,33,152)(24,153,34,143)(25,144,35,154)(26,155,36,145)(27,146,37,156)(28,157,38,147)(29,148,39,158)(30,159,40,149)(41,82,51,92)(42,93,52,83)(43,84,53,94)(44,95,54,85)(45,86,55,96)(46,97,56,87)(47,88,57,98)(48,99,58,89)(49,90,59,100)(50,81,60,91)(61,162,71,172)(62,173,72,163)(63,164,73,174)(64,175,74,165)(65,166,75,176)(66,177,76,167)(67,168,77,178)(68,179,78,169)(69,170,79,180)(70,161,80,171)(101,139,111,129)(102,130,112,140)(103,121,113,131)(104,132,114,122)(105,123,115,133)(106,134,116,124)(107,125,117,135)(108,136,118,126)(109,127,119,137)(110,138,120,128)(201,222,211,232)(202,233,212,223)(203,224,213,234)(204,235,214,225)(205,226,215,236)(206,237,216,227)(207,228,217,238)(208,239,218,229)(209,230,219,240)(210,221,220,231) );
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)], [(1,165),(2,164),(3,163),(4,162),(5,161),(6,180),(7,179),(8,178),(9,177),(10,176),(11,175),(12,174),(13,173),(14,172),(15,171),(16,170),(17,169),(18,168),(19,167),(20,166),(21,122),(22,121),(23,140),(24,139),(25,138),(26,137),(27,136),(28,135),(29,134),(30,133),(31,132),(32,131),(33,130),(34,129),(35,128),(36,127),(37,126),(38,125),(39,124),(40,123),(41,205),(42,204),(43,203),(44,202),(45,201),(46,220),(47,219),(48,218),(49,217),(50,216),(51,215),(52,214),(53,213),(54,212),(55,211),(56,210),(57,209),(58,208),(59,207),(60,206),(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),(81,222),(82,221),(83,240),(84,239),(85,238),(86,237),(87,236),(88,235),(89,234),(90,233),(91,232),(92,231),(93,230),(94,229),(95,228),(96,227),(97,226),(98,225),(99,224),(100,223),(101,158),(102,157),(103,156),(104,155),(105,154),(106,153),(107,152),(108,151),(109,150),(110,149),(111,148),(112,147),(113,146),(114,145),(115,144),(116,143),(117,142),(118,141),(119,160),(120,159)], [(1,215,122,6,220,127,11,205,132,16,210,137),(2,216,123,7,201,128,12,206,133,17,211,138),(3,217,124,8,202,129,13,207,134,18,212,139),(4,218,125,9,203,130,14,208,135,19,213,140),(5,219,126,10,204,131,15,209,136,20,214,121),(21,180,46,36,175,41,31,170,56,26,165,51),(22,161,47,37,176,42,32,171,57,27,166,52),(23,162,48,38,177,43,33,172,58,28,167,53),(24,163,49,39,178,44,34,173,59,29,168,54),(25,164,50,40,179,45,35,174,60,30,169,55),(61,152,84,76,147,99,71,142,94,66,157,89),(62,153,85,77,148,100,72,143,95,67,158,90),(63,154,86,78,149,81,73,144,96,68,159,91),(64,155,87,79,150,82,74,145,97,69,160,92),(65,156,88,80,151,83,75,146,98,70,141,93),(101,233,183,106,238,188,111,223,193,116,228,198),(102,234,184,107,239,189,112,224,194,117,229,199),(103,235,185,108,240,190,113,225,195,118,230,200),(104,236,186,109,221,191,114,226,196,119,231,181),(105,237,187,110,222,192,115,227,197,120,232,182)], [(1,186,11,196),(2,197,12,187),(3,188,13,198),(4,199,14,189),(5,190,15,200),(6,181,16,191),(7,192,17,182),(8,183,18,193),(9,194,19,184),(10,185,20,195),(21,160,31,150),(22,151,32,141),(23,142,33,152),(24,153,34,143),(25,144,35,154),(26,155,36,145),(27,146,37,156),(28,157,38,147),(29,148,39,158),(30,159,40,149),(41,82,51,92),(42,93,52,83),(43,84,53,94),(44,95,54,85),(45,86,55,96),(46,97,56,87),(47,88,57,98),(48,99,58,89),(49,90,59,100),(50,81,60,91),(61,162,71,172),(62,173,72,163),(63,164,73,174),(64,175,74,165),(65,166,75,176),(66,177,76,167),(67,168,77,178),(68,179,78,169),(69,170,79,180),(70,161,80,171),(101,139,111,129),(102,130,112,140),(103,121,113,131),(104,132,114,122),(105,123,115,133),(106,134,116,124),(107,125,117,135),(108,136,118,126),(109,127,119,137),(110,138,120,128),(201,222,211,232),(202,233,212,223),(203,224,213,234),(204,235,214,225),(205,226,215,236),(206,237,216,227),(207,228,217,238),(208,239,218,229),(209,230,219,240),(210,221,220,231)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 2 | 2 | 2 | 12 | 12 | 20 | 2 | 2 | 2 | 4 | 20 | 20 | 60 | 60 | 2 | ··· | 2 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D5 | Q8.14D6 | C15⋊D4 | C2×S3×D5 | C15⋊D4 | C20.C23 | D20.37D6 |
| kernel | D20.37D6 | C30.D4 | C15⋊Q16 | C60.7C4 | C3×C4○D20 | C10×Dic6 | C4○D20 | C60 | C2×C30 | C2×Dic6 | Dic10 | D20 | C2×C20 | Dic6 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D20.37D6 ►in GL4(𝔽241) generated by
| 55 | 22 | 0 | 0 |
| 138 | 186 | 0 | 0 |
| 47 | 124 | 92 | 184 |
| 91 | 124 | 57 | 149 |
| 35 | 0 | 7 | 14 |
| 44 | 0 | 206 | 206 |
| 194 | 117 | 149 | 57 |
| 91 | 62 | 149 | 57 |
| 0 | 37 | 0 | 0 |
| 13 | 56 | 0 | 0 |
| 0 | 99 | 99 | 142 |
| 13 | 99 | 99 | 198 |
| 217 | 181 | 0 | 0 |
| 102 | 24 | 0 | 0 |
| 159 | 100 | 91 | 125 |
| 186 | 100 | 34 | 150 |
G:=sub<GL(4,GF(241))| [55,138,47,91,22,186,124,124,0,0,92,57,0,0,184,149],[35,44,194,91,0,0,117,62,7,206,149,149,14,206,57,57],[0,13,0,13,37,56,99,99,0,0,99,99,0,0,142,198],[217,102,159,186,181,24,100,100,0,0,91,34,0,0,125,150] >;
D20.37D6 in GAP, Magma, Sage, TeX
D_{20}._{37}D_6 % in TeX
G:=Group("D20.37D6"); // GroupNames label
G:=SmallGroup(480,383);
// by ID
G=gap.SmallGroup(480,383);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,219,100,675,346,80,1356,18822]);
// 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=a^10*c^5>;
// generators/relations