metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.10D6, D30.10D4, C60.21C23, Dic15.42D4, Dic6.10D10, Dic30.6C22, D4⋊D5⋊5S3, D4.S3⋊5D5, C3⋊C8.15D10, (C5×D4).9D6, C6.71(D4×D5), C15⋊17(C4○D8), C5⋊3(D8⋊3S3), C5⋊Dic12⋊4C2, D4.15(S3×D5), (C3×D4).9D10, C10.72(S3×D4), C5⋊2C8.15D6, D15⋊2C8⋊3C2, D20⋊S3⋊1C2, D4⋊2D15⋊2C2, C30.183(C2×D4), C6.D20⋊4C2, C3⋊3(SD16⋊3D5), C20.21(C22×S3), (C3×D20).7C22, (C4×D15).5C22, C12.21(C22×D5), (D4×C15).15C22, C2.24(D10⋊D6), (C5×Dic6).7C22, C4.21(C2×S3×D5), (C3×D4⋊D5)⋊7C2, (C5×D4.S3)⋊7C2, (C5×C3⋊C8).5C22, (C3×C5⋊2C8).5C22, SmallGroup(480,573)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.10D6
G = < a,b,c,d | a20=b2=c6=1, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a15b, dcd-1=a15c-1 >
Subgroups: 716 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C3×D5, D15, C30, C30, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, S3×C8, Dic12, D4.S3, D4.S3, C3×D8, D4⋊2S3, C5×Dic3, Dic15, Dic15, C60, C6×D5, D30, C2×C30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8⋊2D5, D8⋊3S3, C5×C3⋊C8, C3×C5⋊2C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, SD16⋊3D5, D15⋊2C8, C6.D20, C5⋊Dic12, C3×D4⋊D5, C5×D4.S3, D20⋊S3, D4⋊2D15, D20.10D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊3S3, C2×S3×D5, SD16⋊3D5, D10⋊D6, D20.10D6
►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)
(1 174)(2 173)(3 172)(4 171)(5 170)(6 169)(7 168)(8 167)(9 166)(10 165)(11 164)(12 163)(13 162)(14 161)(15 180)(16 179)(17 178)(18 177)(19 176)(20 175)(21 98)(22 97)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 90)(30 89)(31 88)(32 87)(33 86)(34 85)(35 84)(36 83)(37 82)(38 81)(39 100)(40 99)(41 215)(42 214)(43 213)(44 212)(45 211)(46 210)(47 209)(48 208)(49 207)(50 206)(51 205)(52 204)(53 203)(54 202)(55 201)(56 220)(57 219)(58 218)(59 217)(60 216)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)(73 126)(74 125)(75 124)(76 123)(77 122)(78 121)(79 140)(80 139)(101 227)(102 226)(103 225)(104 224)(105 223)(106 222)(107 221)(108 240)(109 239)(110 238)(111 237)(112 236)(113 235)(114 234)(115 233)(116 232)(117 231)(118 230)(119 229)(120 228)(141 188)(142 187)(143 186)(144 185)(145 184)(146 183)(147 182)(148 181)(149 200)(150 199)(151 198)(152 197)(153 196)(154 195)(155 194)(156 193)(157 192)(158 191)(159 190)(160 189)
(1 33 181 74 56 239)(2 24 182 65 57 230)(3 35 183 76 58 221)(4 26 184 67 59 232)(5 37 185 78 60 223)(6 28 186 69 41 234)(7 39 187 80 42 225)(8 30 188 71 43 236)(9 21 189 62 44 227)(10 32 190 73 45 238)(11 23 191 64 46 229)(12 34 192 75 47 240)(13 25 193 66 48 231)(14 36 194 77 49 222)(15 27 195 68 50 233)(16 38 196 79 51 224)(17 29 197 70 52 235)(18 40 198 61 53 226)(19 31 199 72 54 237)(20 22 200 63 55 228)(81 158 140 210 104 164)(82 149 121 201 105 175)(83 160 122 212 106 166)(84 151 123 203 107 177)(85 142 124 214 108 168)(86 153 125 205 109 179)(87 144 126 216 110 170)(88 155 127 207 111 161)(89 146 128 218 112 172)(90 157 129 209 113 163)(91 148 130 220 114 174)(92 159 131 211 115 165)(93 150 132 202 116 176)(94 141 133 213 117 167)(95 152 134 204 118 178)(96 143 135 215 119 169)(97 154 136 206 120 180)(98 145 137 217 101 171)(99 156 138 208 102 162)(100 147 139 219 103 173)
(1 120 6 105 11 110 16 115)(2 101 7 106 12 111 17 116)(3 102 8 107 13 112 18 117)(4 103 9 108 14 113 19 118)(5 104 10 109 15 114 20 119)(21 209 26 214 31 219 36 204)(22 210 27 215 32 220 37 205)(23 211 28 216 33 201 38 206)(24 212 29 217 34 202 39 207)(25 213 30 218 35 203 40 208)(41 82 46 87 51 92 56 97)(42 83 47 88 52 93 57 98)(43 84 48 89 53 94 58 99)(44 85 49 90 54 95 59 100)(45 86 50 91 55 96 60 81)(61 156 66 141 71 146 76 151)(62 157 67 142 72 147 77 152)(63 158 68 143 73 148 78 153)(64 159 69 144 74 149 79 154)(65 160 70 145 75 150 80 155)(121 191 126 196 131 181 136 186)(122 192 127 197 132 182 137 187)(123 193 128 198 133 183 138 188)(124 194 129 199 134 184 139 189)(125 195 130 200 135 185 140 190)(161 230 166 235 171 240 176 225)(162 231 167 236 172 221 177 226)(163 232 168 237 173 222 178 227)(164 233 169 238 174 223 179 228)(165 234 170 239 175 224 180 229)
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), (1,174)(2,173)(3,172)(4,171)(5,170)(6,169)(7,168)(8,167)(9,166)(10,165)(11,164)(12,163)(13,162)(14,161)(15,180)(16,179)(17,178)(18,177)(19,176)(20,175)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,100)(40,99)(41,215)(42,214)(43,213)(44,212)(45,211)(46,210)(47,209)(48,208)(49,207)(50,206)(51,205)(52,204)(53,203)(54,202)(55,201)(56,220)(57,219)(58,218)(59,217)(60,216)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,140)(80,139)(101,227)(102,226)(103,225)(104,224)(105,223)(106,222)(107,221)(108,240)(109,239)(110,238)(111,237)(112,236)(113,235)(114,234)(115,233)(116,232)(117,231)(118,230)(119,229)(120,228)(141,188)(142,187)(143,186)(144,185)(145,184)(146,183)(147,182)(148,181)(149,200)(150,199)(151,198)(152,197)(153,196)(154,195)(155,194)(156,193)(157,192)(158,191)(159,190)(160,189), (1,33,181,74,56,239)(2,24,182,65,57,230)(3,35,183,76,58,221)(4,26,184,67,59,232)(5,37,185,78,60,223)(6,28,186,69,41,234)(7,39,187,80,42,225)(8,30,188,71,43,236)(9,21,189,62,44,227)(10,32,190,73,45,238)(11,23,191,64,46,229)(12,34,192,75,47,240)(13,25,193,66,48,231)(14,36,194,77,49,222)(15,27,195,68,50,233)(16,38,196,79,51,224)(17,29,197,70,52,235)(18,40,198,61,53,226)(19,31,199,72,54,237)(20,22,200,63,55,228)(81,158,140,210,104,164)(82,149,121,201,105,175)(83,160,122,212,106,166)(84,151,123,203,107,177)(85,142,124,214,108,168)(86,153,125,205,109,179)(87,144,126,216,110,170)(88,155,127,207,111,161)(89,146,128,218,112,172)(90,157,129,209,113,163)(91,148,130,220,114,174)(92,159,131,211,115,165)(93,150,132,202,116,176)(94,141,133,213,117,167)(95,152,134,204,118,178)(96,143,135,215,119,169)(97,154,136,206,120,180)(98,145,137,217,101,171)(99,156,138,208,102,162)(100,147,139,219,103,173), (1,120,6,105,11,110,16,115)(2,101,7,106,12,111,17,116)(3,102,8,107,13,112,18,117)(4,103,9,108,14,113,19,118)(5,104,10,109,15,114,20,119)(21,209,26,214,31,219,36,204)(22,210,27,215,32,220,37,205)(23,211,28,216,33,201,38,206)(24,212,29,217,34,202,39,207)(25,213,30,218,35,203,40,208)(41,82,46,87,51,92,56,97)(42,83,47,88,52,93,57,98)(43,84,48,89,53,94,58,99)(44,85,49,90,54,95,59,100)(45,86,50,91,55,96,60,81)(61,156,66,141,71,146,76,151)(62,157,67,142,72,147,77,152)(63,158,68,143,73,148,78,153)(64,159,69,144,74,149,79,154)(65,160,70,145,75,150,80,155)(121,191,126,196,131,181,136,186)(122,192,127,197,132,182,137,187)(123,193,128,198,133,183,138,188)(124,194,129,199,134,184,139,189)(125,195,130,200,135,185,140,190)(161,230,166,235,171,240,176,225)(162,231,167,236,172,221,177,226)(163,232,168,237,173,222,178,227)(164,233,169,238,174,223,179,228)(165,234,170,239,175,224,180,229)>;
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), (1,174)(2,173)(3,172)(4,171)(5,170)(6,169)(7,168)(8,167)(9,166)(10,165)(11,164)(12,163)(13,162)(14,161)(15,180)(16,179)(17,178)(18,177)(19,176)(20,175)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,100)(40,99)(41,215)(42,214)(43,213)(44,212)(45,211)(46,210)(47,209)(48,208)(49,207)(50,206)(51,205)(52,204)(53,203)(54,202)(55,201)(56,220)(57,219)(58,218)(59,217)(60,216)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,140)(80,139)(101,227)(102,226)(103,225)(104,224)(105,223)(106,222)(107,221)(108,240)(109,239)(110,238)(111,237)(112,236)(113,235)(114,234)(115,233)(116,232)(117,231)(118,230)(119,229)(120,228)(141,188)(142,187)(143,186)(144,185)(145,184)(146,183)(147,182)(148,181)(149,200)(150,199)(151,198)(152,197)(153,196)(154,195)(155,194)(156,193)(157,192)(158,191)(159,190)(160,189), (1,33,181,74,56,239)(2,24,182,65,57,230)(3,35,183,76,58,221)(4,26,184,67,59,232)(5,37,185,78,60,223)(6,28,186,69,41,234)(7,39,187,80,42,225)(8,30,188,71,43,236)(9,21,189,62,44,227)(10,32,190,73,45,238)(11,23,191,64,46,229)(12,34,192,75,47,240)(13,25,193,66,48,231)(14,36,194,77,49,222)(15,27,195,68,50,233)(16,38,196,79,51,224)(17,29,197,70,52,235)(18,40,198,61,53,226)(19,31,199,72,54,237)(20,22,200,63,55,228)(81,158,140,210,104,164)(82,149,121,201,105,175)(83,160,122,212,106,166)(84,151,123,203,107,177)(85,142,124,214,108,168)(86,153,125,205,109,179)(87,144,126,216,110,170)(88,155,127,207,111,161)(89,146,128,218,112,172)(90,157,129,209,113,163)(91,148,130,220,114,174)(92,159,131,211,115,165)(93,150,132,202,116,176)(94,141,133,213,117,167)(95,152,134,204,118,178)(96,143,135,215,119,169)(97,154,136,206,120,180)(98,145,137,217,101,171)(99,156,138,208,102,162)(100,147,139,219,103,173), (1,120,6,105,11,110,16,115)(2,101,7,106,12,111,17,116)(3,102,8,107,13,112,18,117)(4,103,9,108,14,113,19,118)(5,104,10,109,15,114,20,119)(21,209,26,214,31,219,36,204)(22,210,27,215,32,220,37,205)(23,211,28,216,33,201,38,206)(24,212,29,217,34,202,39,207)(25,213,30,218,35,203,40,208)(41,82,46,87,51,92,56,97)(42,83,47,88,52,93,57,98)(43,84,48,89,53,94,58,99)(44,85,49,90,54,95,59,100)(45,86,50,91,55,96,60,81)(61,156,66,141,71,146,76,151)(62,157,67,142,72,147,77,152)(63,158,68,143,73,148,78,153)(64,159,69,144,74,149,79,154)(65,160,70,145,75,150,80,155)(121,191,126,196,131,181,136,186)(122,192,127,197,132,182,137,187)(123,193,128,198,133,183,138,188)(124,194,129,199,134,184,139,189)(125,195,130,200,135,185,140,190)(161,230,166,235,171,240,176,225)(162,231,167,236,172,221,177,226)(163,232,168,237,173,222,178,227)(164,233,169,238,174,223,179,228)(165,234,170,239,175,224,180,229) );
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)], [(1,174),(2,173),(3,172),(4,171),(5,170),(6,169),(7,168),(8,167),(9,166),(10,165),(11,164),(12,163),(13,162),(14,161),(15,180),(16,179),(17,178),(18,177),(19,176),(20,175),(21,98),(22,97),(23,96),(24,95),(25,94),(26,93),(27,92),(28,91),(29,90),(30,89),(31,88),(32,87),(33,86),(34,85),(35,84),(36,83),(37,82),(38,81),(39,100),(40,99),(41,215),(42,214),(43,213),(44,212),(45,211),(46,210),(47,209),(48,208),(49,207),(50,206),(51,205),(52,204),(53,203),(54,202),(55,201),(56,220),(57,219),(58,218),(59,217),(60,216),(61,138),(62,137),(63,136),(64,135),(65,134),(66,133),(67,132),(68,131),(69,130),(70,129),(71,128),(72,127),(73,126),(74,125),(75,124),(76,123),(77,122),(78,121),(79,140),(80,139),(101,227),(102,226),(103,225),(104,224),(105,223),(106,222),(107,221),(108,240),(109,239),(110,238),(111,237),(112,236),(113,235),(114,234),(115,233),(116,232),(117,231),(118,230),(119,229),(120,228),(141,188),(142,187),(143,186),(144,185),(145,184),(146,183),(147,182),(148,181),(149,200),(150,199),(151,198),(152,197),(153,196),(154,195),(155,194),(156,193),(157,192),(158,191),(159,190),(160,189)], [(1,33,181,74,56,239),(2,24,182,65,57,230),(3,35,183,76,58,221),(4,26,184,67,59,232),(5,37,185,78,60,223),(6,28,186,69,41,234),(7,39,187,80,42,225),(8,30,188,71,43,236),(9,21,189,62,44,227),(10,32,190,73,45,238),(11,23,191,64,46,229),(12,34,192,75,47,240),(13,25,193,66,48,231),(14,36,194,77,49,222),(15,27,195,68,50,233),(16,38,196,79,51,224),(17,29,197,70,52,235),(18,40,198,61,53,226),(19,31,199,72,54,237),(20,22,200,63,55,228),(81,158,140,210,104,164),(82,149,121,201,105,175),(83,160,122,212,106,166),(84,151,123,203,107,177),(85,142,124,214,108,168),(86,153,125,205,109,179),(87,144,126,216,110,170),(88,155,127,207,111,161),(89,146,128,218,112,172),(90,157,129,209,113,163),(91,148,130,220,114,174),(92,159,131,211,115,165),(93,150,132,202,116,176),(94,141,133,213,117,167),(95,152,134,204,118,178),(96,143,135,215,119,169),(97,154,136,206,120,180),(98,145,137,217,101,171),(99,156,138,208,102,162),(100,147,139,219,103,173)], [(1,120,6,105,11,110,16,115),(2,101,7,106,12,111,17,116),(3,102,8,107,13,112,18,117),(4,103,9,108,14,113,19,118),(5,104,10,109,15,114,20,119),(21,209,26,214,31,219,36,204),(22,210,27,215,32,220,37,205),(23,211,28,216,33,201,38,206),(24,212,29,217,34,202,39,207),(25,213,30,218,35,203,40,208),(41,82,46,87,51,92,56,97),(42,83,47,88,52,93,57,98),(43,84,48,89,53,94,58,99),(44,85,49,90,54,95,59,100),(45,86,50,91,55,96,60,81),(61,156,66,141,71,146,76,151),(62,157,67,142,72,147,77,152),(63,158,68,143,73,148,78,153),(64,159,69,144,74,149,79,154),(65,160,70,145,75,150,80,155),(121,191,126,196,131,181,136,186),(122,192,127,197,132,182,137,187),(123,193,128,198,133,183,138,188),(124,194,129,199,134,184,139,189),(125,195,130,200,135,185,140,190),(161,230,166,235,171,240,176,225),(162,231,167,236,172,221,177,226),(163,232,168,237,173,222,178,227),(164,233,169,238,174,223,179,228),(165,234,170,239,175,224,180,229)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 20 | 30 | 2 | 2 | 12 | 15 | 15 | 60 | 2 | 2 | 2 | 8 | 40 | 6 | 6 | 10 | 10 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 24 | 24 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | C4○D8 | S3×D4 | S3×D5 | D4×D5 | D8⋊3S3 | C2×S3×D5 | SD16⋊3D5 | D10⋊D6 | D20.10D6 |
| kernel | D20.10D6 | D15⋊2C8 | C6.D20 | C5⋊Dic12 | C3×D4⋊D5 | C5×D4.S3 | D20⋊S3 | D4⋊2D15 | D4⋊D5 | Dic15 | D30 | D4.S3 | C5⋊2C8 | D20 | C5×D4 | C3⋊C8 | Dic6 | C3×D4 | C15 | C10 | D4 | 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 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.10D6 ►in GL6(𝔽241)
| 177 | 0 | 0 | 0 | 0 | 0 |
| 110 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 70 | 191 | 0 | 0 | 0 | 0 |
| 151 | 171 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 51 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 69 | 54 | 0 | 0 | 0 | 0 |
| 135 | 172 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 79 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [177,110,0,0,0,0,0,64,0,0,0,0,0,0,51,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[70,151,0,0,0,0,191,171,0,0,0,0,0,0,1,0,0,0,0,0,51,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[69,135,0,0,0,0,54,172,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[30,79,0,0,0,0,0,8,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;
D20.10D6 in GAP, Magma, Sage, TeX
D_{20}._{10}D_6 % in TeX
G:=Group("D20.10D6"); // GroupNames label
G:=SmallGroup(480,573);
// by ID
G=gap.SmallGroup(480,573);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;
// generators/relations