metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.Dic3, (C3×Q8).F5, Q8.2(C3⋊F5), C3⋊3(Q8.F5), C15⋊19(C8○D4), C60.38(C2×C4), (C4×D5).41D6, (C3×D20).1C4, C12.20(C2×F5), (Q8×C15).1C4, C60.C4⋊8C2, C5⋊2(D4.Dic3), Q8⋊2D5.5S3, (C5×Q8).4Dic3, C12.F5⋊10C2, C20.6(C2×Dic3), C6.40(C22×F5), C30.78(C22×C4), C15⋊C8.8C22, D10.2(C2×Dic3), (D5×C12).74C22, C10.9(C22×Dic3), Dic5.52(C22×S3), (C3×Dic5).66C23, C4.6(C2×C3⋊F5), C2.10(C22×C3⋊F5), (C6×D5).27(C2×C4), (C3×Q8⋊2D5).3C2, SmallGroup(480,1068)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.Dic3
G = < a,b,c,d | a20=b2=1, c6=a10, d2=a10c3, bab=a-1, cac-1=a9, dad-1=a17, cbc-1=a8b, dbd-1=a16b, dcd-1=c5 >
Subgroups: 492 in 124 conjugacy classes, 57 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, C20, D10, C3⋊C8, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C8○D4, C5⋊C8, C4×D5, D20, C5×Q8, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C3×Dic5, C60, C6×D5, D5⋊C8, C4.F5, Q8⋊2D5, D4.Dic3, C15⋊C8, C15⋊C8, D5×C12, C3×D20, Q8×C15, Q8.F5, C60.C4, C12.F5, C3×Q8⋊2D5, D20.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, F5, C2×Dic3, C22×S3, C8○D4, C2×F5, C22×Dic3, C3⋊F5, C22×F5, D4.Dic3, C2×C3⋊F5, Q8.F5, C22×C3⋊F5, D20.Dic3
►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 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)(141 150)(142 149)(143 148)(144 147)(145 146)(151 160)(152 159)(153 158)(154 157)(155 156)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 180)(176 179)(177 178)(181 188)(182 187)(183 186)(184 185)(189 200)(190 199)(191 198)(192 197)(193 196)(194 195)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 210)(208 209)(217 220)(218 219)(221 228)(222 227)(223 226)(224 225)(229 240)(230 239)(231 238)(232 237)(233 236)(234 235)
(1 57 102 135 219 64 11 47 112 125 209 74)(2 46 103 124 220 73 12 56 113 134 210 63)(3 55 104 133 201 62 13 45 114 123 211 72)(4 44 105 122 202 71 14 54 115 132 212 61)(5 53 106 131 203 80 15 43 116 121 213 70)(6 42 107 140 204 69 16 52 117 130 214 79)(7 51 108 129 205 78 17 41 118 139 215 68)(8 60 109 138 206 67 18 50 119 128 216 77)(9 49 110 127 207 76 19 59 120 137 217 66)(10 58 111 136 208 65 20 48 101 126 218 75)(21 184 157 177 236 93 31 194 147 167 226 83)(22 193 158 166 237 82 32 183 148 176 227 92)(23 182 159 175 238 91 33 192 149 165 228 81)(24 191 160 164 239 100 34 181 150 174 229 90)(25 200 141 173 240 89 35 190 151 163 230 99)(26 189 142 162 221 98 36 199 152 172 231 88)(27 198 143 171 222 87 37 188 153 161 232 97)(28 187 144 180 223 96 38 197 154 170 233 86)(29 196 145 169 224 85 39 186 155 179 234 95)(30 185 146 178 225 94 40 195 156 168 235 84)
(1 156 125 94 11 146 135 84)(2 149 134 91 12 159 124 81)(3 142 123 88 13 152 133 98)(4 155 132 85 14 145 122 95)(5 148 121 82 15 158 131 92)(6 141 130 99 16 151 140 89)(7 154 139 96 17 144 129 86)(8 147 128 93 18 157 138 83)(9 160 137 90 19 150 127 100)(10 153 126 87 20 143 136 97)(21 67 167 109 31 77 177 119)(22 80 176 106 32 70 166 116)(23 73 165 103 33 63 175 113)(24 66 174 120 34 76 164 110)(25 79 163 117 35 69 173 107)(26 72 172 114 36 62 162 104)(27 65 161 111 37 75 171 101)(28 78 170 108 38 68 180 118)(29 71 179 105 39 61 169 115)(30 64 168 102 40 74 178 112)(41 197 205 223 51 187 215 233)(42 190 214 240 52 200 204 230)(43 183 203 237 53 193 213 227)(44 196 212 234 54 186 202 224)(45 189 201 231 55 199 211 221)(46 182 210 228 56 192 220 238)(47 195 219 225 57 185 209 235)(48 188 208 222 58 198 218 232)(49 181 217 239 59 191 207 229)(50 194 206 236 60 184 216 226)
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)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(141,150)(142,149)(143,148)(144,147)(145,146)(151,160)(152,159)(153,158)(154,157)(155,156)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,188)(182,187)(183,186)(184,185)(189,200)(190,199)(191,198)(192,197)(193,196)(194,195)(201,216)(202,215)(203,214)(204,213)(205,212)(206,211)(207,210)(208,209)(217,220)(218,219)(221,228)(222,227)(223,226)(224,225)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235), (1,57,102,135,219,64,11,47,112,125,209,74)(2,46,103,124,220,73,12,56,113,134,210,63)(3,55,104,133,201,62,13,45,114,123,211,72)(4,44,105,122,202,71,14,54,115,132,212,61)(5,53,106,131,203,80,15,43,116,121,213,70)(6,42,107,140,204,69,16,52,117,130,214,79)(7,51,108,129,205,78,17,41,118,139,215,68)(8,60,109,138,206,67,18,50,119,128,216,77)(9,49,110,127,207,76,19,59,120,137,217,66)(10,58,111,136,208,65,20,48,101,126,218,75)(21,184,157,177,236,93,31,194,147,167,226,83)(22,193,158,166,237,82,32,183,148,176,227,92)(23,182,159,175,238,91,33,192,149,165,228,81)(24,191,160,164,239,100,34,181,150,174,229,90)(25,200,141,173,240,89,35,190,151,163,230,99)(26,189,142,162,221,98,36,199,152,172,231,88)(27,198,143,171,222,87,37,188,153,161,232,97)(28,187,144,180,223,96,38,197,154,170,233,86)(29,196,145,169,224,85,39,186,155,179,234,95)(30,185,146,178,225,94,40,195,156,168,235,84), (1,156,125,94,11,146,135,84)(2,149,134,91,12,159,124,81)(3,142,123,88,13,152,133,98)(4,155,132,85,14,145,122,95)(5,148,121,82,15,158,131,92)(6,141,130,99,16,151,140,89)(7,154,139,96,17,144,129,86)(8,147,128,93,18,157,138,83)(9,160,137,90,19,150,127,100)(10,153,126,87,20,143,136,97)(21,67,167,109,31,77,177,119)(22,80,176,106,32,70,166,116)(23,73,165,103,33,63,175,113)(24,66,174,120,34,76,164,110)(25,79,163,117,35,69,173,107)(26,72,172,114,36,62,162,104)(27,65,161,111,37,75,171,101)(28,78,170,108,38,68,180,118)(29,71,179,105,39,61,169,115)(30,64,168,102,40,74,178,112)(41,197,205,223,51,187,215,233)(42,190,214,240,52,200,204,230)(43,183,203,237,53,193,213,227)(44,196,212,234,54,186,202,224)(45,189,201,231,55,199,211,221)(46,182,210,228,56,192,220,238)(47,195,219,225,57,185,209,235)(48,188,208,222,58,198,218,232)(49,181,217,239,59,191,207,229)(50,194,206,236,60,184,216,226)>;
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)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(141,150)(142,149)(143,148)(144,147)(145,146)(151,160)(152,159)(153,158)(154,157)(155,156)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,188)(182,187)(183,186)(184,185)(189,200)(190,199)(191,198)(192,197)(193,196)(194,195)(201,216)(202,215)(203,214)(204,213)(205,212)(206,211)(207,210)(208,209)(217,220)(218,219)(221,228)(222,227)(223,226)(224,225)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235), (1,57,102,135,219,64,11,47,112,125,209,74)(2,46,103,124,220,73,12,56,113,134,210,63)(3,55,104,133,201,62,13,45,114,123,211,72)(4,44,105,122,202,71,14,54,115,132,212,61)(5,53,106,131,203,80,15,43,116,121,213,70)(6,42,107,140,204,69,16,52,117,130,214,79)(7,51,108,129,205,78,17,41,118,139,215,68)(8,60,109,138,206,67,18,50,119,128,216,77)(9,49,110,127,207,76,19,59,120,137,217,66)(10,58,111,136,208,65,20,48,101,126,218,75)(21,184,157,177,236,93,31,194,147,167,226,83)(22,193,158,166,237,82,32,183,148,176,227,92)(23,182,159,175,238,91,33,192,149,165,228,81)(24,191,160,164,239,100,34,181,150,174,229,90)(25,200,141,173,240,89,35,190,151,163,230,99)(26,189,142,162,221,98,36,199,152,172,231,88)(27,198,143,171,222,87,37,188,153,161,232,97)(28,187,144,180,223,96,38,197,154,170,233,86)(29,196,145,169,224,85,39,186,155,179,234,95)(30,185,146,178,225,94,40,195,156,168,235,84), (1,156,125,94,11,146,135,84)(2,149,134,91,12,159,124,81)(3,142,123,88,13,152,133,98)(4,155,132,85,14,145,122,95)(5,148,121,82,15,158,131,92)(6,141,130,99,16,151,140,89)(7,154,139,96,17,144,129,86)(8,147,128,93,18,157,138,83)(9,160,137,90,19,150,127,100)(10,153,126,87,20,143,136,97)(21,67,167,109,31,77,177,119)(22,80,176,106,32,70,166,116)(23,73,165,103,33,63,175,113)(24,66,174,120,34,76,164,110)(25,79,163,117,35,69,173,107)(26,72,172,114,36,62,162,104)(27,65,161,111,37,75,171,101)(28,78,170,108,38,68,180,118)(29,71,179,105,39,61,169,115)(30,64,168,102,40,74,178,112)(41,197,205,223,51,187,215,233)(42,190,214,240,52,200,204,230)(43,183,203,237,53,193,213,227)(44,196,212,234,54,186,202,224)(45,189,201,231,55,199,211,221)(46,182,210,228,56,192,220,238)(47,195,219,225,57,185,209,235)(48,188,208,222,58,198,218,232)(49,181,217,239,59,191,207,229)(50,194,206,236,60,184,216,226) );
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),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(121,128),(122,127),(123,126),(124,125),(129,140),(130,139),(131,138),(132,137),(133,136),(134,135),(141,150),(142,149),(143,148),(144,147),(145,146),(151,160),(152,159),(153,158),(154,157),(155,156),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168),(175,180),(176,179),(177,178),(181,188),(182,187),(183,186),(184,185),(189,200),(190,199),(191,198),(192,197),(193,196),(194,195),(201,216),(202,215),(203,214),(204,213),(205,212),(206,211),(207,210),(208,209),(217,220),(218,219),(221,228),(222,227),(223,226),(224,225),(229,240),(230,239),(231,238),(232,237),(233,236),(234,235)], [(1,57,102,135,219,64,11,47,112,125,209,74),(2,46,103,124,220,73,12,56,113,134,210,63),(3,55,104,133,201,62,13,45,114,123,211,72),(4,44,105,122,202,71,14,54,115,132,212,61),(5,53,106,131,203,80,15,43,116,121,213,70),(6,42,107,140,204,69,16,52,117,130,214,79),(7,51,108,129,205,78,17,41,118,139,215,68),(8,60,109,138,206,67,18,50,119,128,216,77),(9,49,110,127,207,76,19,59,120,137,217,66),(10,58,111,136,208,65,20,48,101,126,218,75),(21,184,157,177,236,93,31,194,147,167,226,83),(22,193,158,166,237,82,32,183,148,176,227,92),(23,182,159,175,238,91,33,192,149,165,228,81),(24,191,160,164,239,100,34,181,150,174,229,90),(25,200,141,173,240,89,35,190,151,163,230,99),(26,189,142,162,221,98,36,199,152,172,231,88),(27,198,143,171,222,87,37,188,153,161,232,97),(28,187,144,180,223,96,38,197,154,170,233,86),(29,196,145,169,224,85,39,186,155,179,234,95),(30,185,146,178,225,94,40,195,156,168,235,84)], [(1,156,125,94,11,146,135,84),(2,149,134,91,12,159,124,81),(3,142,123,88,13,152,133,98),(4,155,132,85,14,145,122,95),(5,148,121,82,15,158,131,92),(6,141,130,99,16,151,140,89),(7,154,139,96,17,144,129,86),(8,147,128,93,18,157,138,83),(9,160,137,90,19,150,127,100),(10,153,126,87,20,143,136,97),(21,67,167,109,31,77,177,119),(22,80,176,106,32,70,166,116),(23,73,165,103,33,63,175,113),(24,66,174,120,34,76,164,110),(25,79,163,117,35,69,173,107),(26,72,172,114,36,62,162,104),(27,65,161,111,37,75,171,101),(28,78,170,108,38,68,180,118),(29,71,179,105,39,61,169,115),(30,64,168,102,40,74,178,112),(41,197,205,223,51,187,215,233),(42,190,214,240,52,200,204,230),(43,183,203,237,53,193,213,227),(44,196,212,234,54,186,202,224),(45,189,201,231,55,199,211,221),(46,182,210,228,56,192,220,238),(47,195,219,225,57,185,209,235),(48,188,208,222,58,198,218,232),(49,181,217,239,59,191,207,229),(50,194,206,236,60,184,216,226)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10 | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 5 | 5 | 4 | 2 | 20 | 20 | 20 | 15 | 15 | 15 | 15 | 30 | ··· | 30 | 4 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 8 | ··· | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | - | - | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | Dic3 | Dic3 | C8○D4 | F5 | C2×F5 | C3⋊F5 | D4.Dic3 | C2×C3⋊F5 | Q8.F5 | D20.Dic3 |
| kernel | D20.Dic3 | C60.C4 | C12.F5 | C3×Q8⋊2D5 | C3×D20 | Q8×C15 | Q8⋊2D5 | C4×D5 | D20 | C5×Q8 | C15 | C3×Q8 | C12 | Q8 | C5 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 1 | 3 | 3 | 1 | 4 | 1 | 3 | 2 | 2 | 6 | 1 | 2 |
Matrix representation of D20.Dic3 ►in GL6(𝔽241)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 240 | 0 |
| 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 115 | 126 | 229 | 0 |
| 0 | 0 | 0 | 114 | 229 | 12 |
| 0 | 0 | 229 | 114 | 0 | 127 |
| 0 | 0 | 229 | 126 | 115 | 12 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 211 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 224 | 17 | 51 |
| 0 | 0 | 0 | 224 | 207 | 34 |
| 0 | 0 | 224 | 0 | 17 | 34 |
| 0 | 0 | 224 | 190 | 0 | 51 |
G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,0,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,240,240,240,240,0,0,0,0,0,1],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,115,0,229,229,0,0,126,114,114,126,0,0,229,229,0,115,0,0,0,12,127,12],[211,0,0,0,0,0,0,211,0,0,0,0,0,0,190,0,224,224,0,0,224,224,0,190,0,0,17,207,17,0,0,0,51,34,34,51] >;
D20.Dic3 in GAP, Magma, Sage, TeX
D_{20}.{\rm Dic}_3 % in TeX
G:=Group("D20.Dic3"); // GroupNames label
G:=SmallGroup(480,1068);
// by ID
G=gap.SmallGroup(480,1068);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,219,100,80,2693,14118,2379]);
// 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,c*a*c^-1=a^9,d*a*d^-1=a^17,c*b*c^-1=a^8*b,d*b*d^-1=a^16*b,d*c*d^-1=c^5>;
// generators/relations