metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.2Dic3, C60.177C23, Dic10.2Dic3, C3⋊C8.25D10, C15⋊12(C8○D4), C4○D20.3S3, (C3×D20).4C4, (C4×D5).59D6, (C2×C20).78D6, C12.20(C4×D5), C4.Dic3⋊5D5, C4.9(D5×Dic3), C60.129(C2×C4), (C2×C12).79D10, C5⋊4(D4.Dic3), C5⋊D4.2Dic3, C3⋊5(D20.2C4), C30.99(C22×C4), D10.4(C2×Dic3), (C3×Dic10).4C4, C20.32(C2×Dic3), C20.32D6⋊14C2, C22.2(D5×Dic3), (C2×C60).213C22, C20.174(C22×S3), C15⋊3C8.48C22, Dic5.4(C2×Dic3), (D5×C12).59C22, C12.174(C22×D5), C10.17(C22×Dic3), (D5×C3⋊C8)⋊13C2, C6.80(C2×C4×D5), C4.147(C2×S3×D5), (C2×C6).8(C4×D5), C2.6(C2×D5×Dic3), (C6×D5).6(C2×C4), (C2×C15⋊3C8)⋊18C2, (C3×C5⋊D4).4C4, (C2×C30).96(C2×C4), (C3×C4○D20).6C2, (C2×C4).193(S3×D5), (C5×C3⋊C8).25C22, (C5×C4.Dic3)⋊11C2, (C3×Dic5).6(C2×C4), (C2×C10).25(C2×Dic3), SmallGroup(480,360)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.2Dic3
G = < a,b,c,d | a20=b2=1, c6=a10, d2=a10c3, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c5 >
Subgroups: 428 in 124 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C8○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C3⋊C8, C4.Dic3, C4.Dic3, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C8×D5, C8⋊D5, C2×C5⋊2C8, C5×M4(2), C4○D20, D4.Dic3, C5×C3⋊C8, C15⋊3C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D20.2C4, D5×C3⋊C8, C20.32D6, C5×C4.Dic3, C2×C15⋊3C8, C3×C4○D20, D20.2Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, D10, C2×Dic3, C22×S3, C8○D4, C4×D5, C22×D5, C22×Dic3, S3×D5, C2×C4×D5, D4.Dic3, D5×Dic3, C2×S3×D5, D20.2C4, C2×D5×Dic3, D20.2Dic3
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 80)(76 79)(77 78)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 160)(156 159)(157 158)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 180)(176 179)(177 178)(181 198)(182 197)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(199 200)(201 212)(202 211)(203 210)(204 209)(205 208)(206 207)(213 220)(214 219)(215 218)(216 217)(221 240)(222 239)(223 238)(224 237)(225 236)(226 235)(227 234)(228 233)(229 232)(230 231)
(1 221 178 158 82 207 11 231 168 148 92 217)(2 222 179 159 83 208 12 232 169 149 93 218)(3 223 180 160 84 209 13 233 170 150 94 219)(4 224 161 141 85 210 14 234 171 151 95 220)(5 225 162 142 86 211 15 235 172 152 96 201)(6 226 163 143 87 212 16 236 173 153 97 202)(7 227 164 144 88 213 17 237 174 154 98 203)(8 228 165 145 89 214 18 238 175 155 99 204)(9 229 166 146 90 215 19 239 176 156 100 205)(10 230 167 147 91 216 20 240 177 157 81 206)(21 191 69 60 132 103 31 181 79 50 122 113)(22 192 70 41 133 104 32 182 80 51 123 114)(23 193 71 42 134 105 33 183 61 52 124 115)(24 194 72 43 135 106 34 184 62 53 125 116)(25 195 73 44 136 107 35 185 63 54 126 117)(26 196 74 45 137 108 36 186 64 55 127 118)(27 197 75 46 138 109 37 187 65 56 128 119)(28 198 76 47 139 110 38 188 66 57 129 120)(29 199 77 48 140 111 39 189 67 58 130 101)(30 200 78 49 121 112 40 190 68 59 131 102)
(1 68 148 112 11 78 158 102)(2 79 149 103 12 69 159 113)(3 70 150 114 13 80 160 104)(4 61 151 105 14 71 141 115)(5 72 152 116 15 62 142 106)(6 63 153 107 16 73 143 117)(7 74 154 118 17 64 144 108)(8 65 155 109 18 75 145 119)(9 76 156 120 19 66 146 110)(10 67 157 111 20 77 147 101)(21 208 50 179 31 218 60 169)(22 219 51 170 32 209 41 180)(23 210 52 161 33 220 42 171)(24 201 53 172 34 211 43 162)(25 212 54 163 35 202 44 173)(26 203 55 174 36 213 45 164)(27 214 56 165 37 204 46 175)(28 205 57 176 38 215 47 166)(29 216 58 167 39 206 48 177)(30 207 59 178 40 217 49 168)(81 130 240 189 91 140 230 199)(82 121 221 200 92 131 231 190)(83 132 222 191 93 122 232 181)(84 123 223 182 94 133 233 192)(85 134 224 193 95 124 234 183)(86 125 225 184 96 135 235 194)(87 136 226 195 97 126 236 185)(88 127 227 186 98 137 237 196)(89 138 228 197 99 128 238 187)(90 129 229 188 100 139 239 198)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,80)(76,79)(77,78)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(199,200)(201,212)(202,211)(203,210)(204,209)(205,208)(206,207)(213,220)(214,219)(215,218)(216,217)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,221,178,158,82,207,11,231,168,148,92,217)(2,222,179,159,83,208,12,232,169,149,93,218)(3,223,180,160,84,209,13,233,170,150,94,219)(4,224,161,141,85,210,14,234,171,151,95,220)(5,225,162,142,86,211,15,235,172,152,96,201)(6,226,163,143,87,212,16,236,173,153,97,202)(7,227,164,144,88,213,17,237,174,154,98,203)(8,228,165,145,89,214,18,238,175,155,99,204)(9,229,166,146,90,215,19,239,176,156,100,205)(10,230,167,147,91,216,20,240,177,157,81,206)(21,191,69,60,132,103,31,181,79,50,122,113)(22,192,70,41,133,104,32,182,80,51,123,114)(23,193,71,42,134,105,33,183,61,52,124,115)(24,194,72,43,135,106,34,184,62,53,125,116)(25,195,73,44,136,107,35,185,63,54,126,117)(26,196,74,45,137,108,36,186,64,55,127,118)(27,197,75,46,138,109,37,187,65,56,128,119)(28,198,76,47,139,110,38,188,66,57,129,120)(29,199,77,48,140,111,39,189,67,58,130,101)(30,200,78,49,121,112,40,190,68,59,131,102), (1,68,148,112,11,78,158,102)(2,79,149,103,12,69,159,113)(3,70,150,114,13,80,160,104)(4,61,151,105,14,71,141,115)(5,72,152,116,15,62,142,106)(6,63,153,107,16,73,143,117)(7,74,154,118,17,64,144,108)(8,65,155,109,18,75,145,119)(9,76,156,120,19,66,146,110)(10,67,157,111,20,77,147,101)(21,208,50,179,31,218,60,169)(22,219,51,170,32,209,41,180)(23,210,52,161,33,220,42,171)(24,201,53,172,34,211,43,162)(25,212,54,163,35,202,44,173)(26,203,55,174,36,213,45,164)(27,214,56,165,37,204,46,175)(28,205,57,176,38,215,47,166)(29,216,58,167,39,206,48,177)(30,207,59,178,40,217,49,168)(81,130,240,189,91,140,230,199)(82,121,221,200,92,131,231,190)(83,132,222,191,93,122,232,181)(84,123,223,182,94,133,233,192)(85,134,224,193,95,124,234,183)(86,125,225,184,96,135,235,194)(87,136,226,195,97,126,236,185)(88,127,227,186,98,137,237,196)(89,138,228,197,99,128,238,187)(90,129,229,188,100,139,239,198)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,80)(76,79)(77,78)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(199,200)(201,212)(202,211)(203,210)(204,209)(205,208)(206,207)(213,220)(214,219)(215,218)(216,217)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,221,178,158,82,207,11,231,168,148,92,217)(2,222,179,159,83,208,12,232,169,149,93,218)(3,223,180,160,84,209,13,233,170,150,94,219)(4,224,161,141,85,210,14,234,171,151,95,220)(5,225,162,142,86,211,15,235,172,152,96,201)(6,226,163,143,87,212,16,236,173,153,97,202)(7,227,164,144,88,213,17,237,174,154,98,203)(8,228,165,145,89,214,18,238,175,155,99,204)(9,229,166,146,90,215,19,239,176,156,100,205)(10,230,167,147,91,216,20,240,177,157,81,206)(21,191,69,60,132,103,31,181,79,50,122,113)(22,192,70,41,133,104,32,182,80,51,123,114)(23,193,71,42,134,105,33,183,61,52,124,115)(24,194,72,43,135,106,34,184,62,53,125,116)(25,195,73,44,136,107,35,185,63,54,126,117)(26,196,74,45,137,108,36,186,64,55,127,118)(27,197,75,46,138,109,37,187,65,56,128,119)(28,198,76,47,139,110,38,188,66,57,129,120)(29,199,77,48,140,111,39,189,67,58,130,101)(30,200,78,49,121,112,40,190,68,59,131,102), (1,68,148,112,11,78,158,102)(2,79,149,103,12,69,159,113)(3,70,150,114,13,80,160,104)(4,61,151,105,14,71,141,115)(5,72,152,116,15,62,142,106)(6,63,153,107,16,73,143,117)(7,74,154,118,17,64,144,108)(8,65,155,109,18,75,145,119)(9,76,156,120,19,66,146,110)(10,67,157,111,20,77,147,101)(21,208,50,179,31,218,60,169)(22,219,51,170,32,209,41,180)(23,210,52,161,33,220,42,171)(24,201,53,172,34,211,43,162)(25,212,54,163,35,202,44,173)(26,203,55,174,36,213,45,164)(27,214,56,165,37,204,46,175)(28,205,57,176,38,215,47,166)(29,216,58,167,39,206,48,177)(30,207,59,178,40,217,49,168)(81,130,240,189,91,140,230,199)(82,121,221,200,92,131,231,190)(83,132,222,191,93,122,232,181)(84,123,223,182,94,133,233,192)(85,134,224,193,95,124,234,183)(86,125,225,184,96,135,235,194)(87,136,226,195,97,126,236,185)(88,127,227,186,98,137,237,196)(89,138,228,197,99,128,238,187)(90,129,229,188,100,139,239,198) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,60),(58,59),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(75,80),(76,79),(77,78),(81,82),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148),(155,160),(156,159),(157,158),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168),(175,180),(176,179),(177,178),(181,198),(182,197),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(199,200),(201,212),(202,211),(203,210),(204,209),(205,208),(206,207),(213,220),(214,219),(215,218),(216,217),(221,240),(222,239),(223,238),(224,237),(225,236),(226,235),(227,234),(228,233),(229,232),(230,231)], [(1,221,178,158,82,207,11,231,168,148,92,217),(2,222,179,159,83,208,12,232,169,149,93,218),(3,223,180,160,84,209,13,233,170,150,94,219),(4,224,161,141,85,210,14,234,171,151,95,220),(5,225,162,142,86,211,15,235,172,152,96,201),(6,226,163,143,87,212,16,236,173,153,97,202),(7,227,164,144,88,213,17,237,174,154,98,203),(8,228,165,145,89,214,18,238,175,155,99,204),(9,229,166,146,90,215,19,239,176,156,100,205),(10,230,167,147,91,216,20,240,177,157,81,206),(21,191,69,60,132,103,31,181,79,50,122,113),(22,192,70,41,133,104,32,182,80,51,123,114),(23,193,71,42,134,105,33,183,61,52,124,115),(24,194,72,43,135,106,34,184,62,53,125,116),(25,195,73,44,136,107,35,185,63,54,126,117),(26,196,74,45,137,108,36,186,64,55,127,118),(27,197,75,46,138,109,37,187,65,56,128,119),(28,198,76,47,139,110,38,188,66,57,129,120),(29,199,77,48,140,111,39,189,67,58,130,101),(30,200,78,49,121,112,40,190,68,59,131,102)], [(1,68,148,112,11,78,158,102),(2,79,149,103,12,69,159,113),(3,70,150,114,13,80,160,104),(4,61,151,105,14,71,141,115),(5,72,152,116,15,62,142,106),(6,63,153,107,16,73,143,117),(7,74,154,118,17,64,144,108),(8,65,155,109,18,75,145,119),(9,76,156,120,19,66,146,110),(10,67,157,111,20,77,147,101),(21,208,50,179,31,218,60,169),(22,219,51,170,32,209,41,180),(23,210,52,161,33,220,42,171),(24,201,53,172,34,211,43,162),(25,212,54,163,35,202,44,173),(26,203,55,174,36,213,45,164),(27,214,56,165,37,204,46,175),(28,205,57,176,38,215,47,166),(29,216,58,167,39,206,48,177),(30,207,59,178,40,217,49,168),(81,130,240,189,91,140,230,199),(82,121,221,200,92,131,231,190),(83,132,222,191,93,122,232,181),(84,123,223,182,94,133,233,192),(85,134,224,193,95,124,234,183),(86,125,225,184,96,135,235,194),(87,136,226,195,97,126,236,185),(88,127,227,186,98,137,237,196),(89,138,228,197,99,128,238,187),(90,129,229,188,100,139,239,198)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 10 | 10 | 2 | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 2 | 4 | 20 | 20 | 6 | 6 | 6 | 6 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4 | C4 | C4 | S3 | D5 | Dic3 | D6 | Dic3 | Dic3 | D6 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | S3×D5 | D4.Dic3 | D5×Dic3 | C2×S3×D5 | D5×Dic3 | D20.2C4 | D20.2Dic3 |
| kernel | D20.2Dic3 | D5×C3⋊C8 | C20.32D6 | C5×C4.Dic3 | C2×C15⋊3C8 | C3×C4○D20 | C3×Dic10 | C3×D20 | C3×C5⋊D4 | C4○D20 | C4.Dic3 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | C3⋊C8 | C2×C12 | C15 | C12 | C2×C6 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D20.2Dic3 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 235 |
| 0 | 0 | 0 | 0 | 181 | 230 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 190 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 235 |
| 0 | 0 | 0 | 0 | 20 | 230 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 0 | 0 | 177 |
| 26 | 69 | 0 | 0 | 0 | 0 |
| 95 | 215 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 211 | 0 |
| 0 | 0 | 0 | 0 | 131 | 30 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,190,1,0,0,0,0,240,0,0,0,0,0,0,0,11,181,0,0,0,0,235,230],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,190,1,0,0,0,0,0,0,11,20,0,0,0,0,235,230],[240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[26,95,0,0,0,0,69,215,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,211,131,0,0,0,0,0,30] >;
D20.2Dic3 in GAP, Magma, Sage, TeX
D_{20}._2{\rm Dic}_3 % in TeX
G:=Group("D20.2Dic3"); // GroupNames label
G:=SmallGroup(480,360);
// by ID
G=gap.SmallGroup(480,360);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,219,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^6=a^10,d^2=a^10*c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^5>;
// generators/relations