metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.Dic3, (C3×Q8).F5, C3⋊3(Q8.F5), Q8.2(C3⋊F5), C15⋊19(C8○D4), C60.38(C2×C4), (C3×D20).1C4, (C4×D5).41D6, 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), (C3×Dic5).66C23, Dic5.52(C22×S3), 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
Subgroups: 492 in 124 conjugacy classes, 57 normal (18 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×3], C5, C6, C6 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, C12 [×3], C12, C2×C6 [×3], C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, C20 [×3], D10 [×3], C3⋊C8 [×4], C2×C12 [×3], C3×D4 [×3], C3×Q8, C3×D5 [×3], C30, C8○D4, C5⋊C8 [×4], C4×D5 [×3], D20 [×3], C5×Q8, C2×C3⋊C8 [×3], C4.Dic3 [×3], C3×C4○D4, C3×Dic5, C60 [×3], C6×D5 [×3], D5⋊C8 [×3], C4.F5 [×3], Q8⋊2D5, D4.Dic3, C15⋊C8, C15⋊C8 [×3], D5×C12 [×3], C3×D20 [×3], Q8×C15, Q8.F5, C60.C4 [×3], C12.F5 [×3], C3×Q8⋊2D5, D20.Dic3
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, F5, C2×Dic3 [×6], C22×S3, C8○D4, C2×F5 [×3], C22×Dic3, C3⋊F5, C22×F5, D4.Dic3, C2×C3⋊F5 [×3], Q8.F5, C22×C3⋊F5, D20.Dic3
Generators and relations
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 >
►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 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 88)(82 87)(83 86)(84 85)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 122)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 160)(156 159)(157 158)(161 172)(162 171)(163 170)(164 169)(165 168)(166 167)(173 180)(174 179)(175 178)(176 177)(181 190)(182 189)(183 188)(184 187)(185 186)(191 200)(192 199)(193 198)(194 197)(195 196)(201 208)(202 207)(203 206)(204 205)(209 220)(210 219)(211 218)(212 217)(213 216)(214 215)(221 240)(222 239)(223 238)(224 237)(225 236)(226 235)(227 234)(228 233)(229 232)(230 231)
(1 186 122 158 85 33 11 196 132 148 95 23)(2 195 123 147 86 22 12 185 133 157 96 32)(3 184 124 156 87 31 13 194 134 146 97 21)(4 193 125 145 88 40 14 183 135 155 98 30)(5 182 126 154 89 29 15 192 136 144 99 39)(6 191 127 143 90 38 16 181 137 153 100 28)(7 200 128 152 91 27 17 190 138 142 81 37)(8 189 129 141 92 36 18 199 139 151 82 26)(9 198 130 150 93 25 19 188 140 160 83 35)(10 187 131 159 94 34 20 197 121 149 84 24)(41 213 179 111 78 239 51 203 169 101 68 229)(42 202 180 120 79 228 52 212 170 110 69 238)(43 211 161 109 80 237 53 201 171 119 70 227)(44 220 162 118 61 226 54 210 172 108 71 236)(45 209 163 107 62 235 55 219 173 117 72 225)(46 218 164 116 63 224 56 208 174 106 73 234)(47 207 165 105 64 233 57 217 175 115 74 223)(48 216 166 114 65 222 58 206 176 104 75 232)(49 205 167 103 66 231 59 215 177 113 76 221)(50 214 168 112 67 240 60 204 178 102 77 230)
(1 76 148 215 11 66 158 205)(2 69 157 212 12 79 147 202)(3 62 146 209 13 72 156 219)(4 75 155 206 14 65 145 216)(5 68 144 203 15 78 154 213)(6 61 153 220 16 71 143 210)(7 74 142 217 17 64 152 207)(8 67 151 214 18 77 141 204)(9 80 160 211 19 70 150 201)(10 73 149 208 20 63 159 218)(21 225 134 173 31 235 124 163)(22 238 123 170 32 228 133 180)(23 231 132 167 33 221 122 177)(24 224 121 164 34 234 131 174)(25 237 130 161 35 227 140 171)(26 230 139 178 36 240 129 168)(27 223 128 175 37 233 138 165)(28 236 137 172 38 226 127 162)(29 229 126 169 39 239 136 179)(30 222 135 166 40 232 125 176)(41 192 101 89 51 182 111 99)(42 185 110 86 52 195 120 96)(43 198 119 83 53 188 109 93)(44 191 108 100 54 181 118 90)(45 184 117 97 55 194 107 87)(46 197 106 94 56 187 116 84)(47 190 115 91 57 200 105 81)(48 183 104 88 58 193 114 98)(49 196 113 85 59 186 103 95)(50 189 102 82 60 199 112 92)
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)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,88)(82,87)(83,86)(84,85)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,122)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,186,122,158,85,33,11,196,132,148,95,23)(2,195,123,147,86,22,12,185,133,157,96,32)(3,184,124,156,87,31,13,194,134,146,97,21)(4,193,125,145,88,40,14,183,135,155,98,30)(5,182,126,154,89,29,15,192,136,144,99,39)(6,191,127,143,90,38,16,181,137,153,100,28)(7,200,128,152,91,27,17,190,138,142,81,37)(8,189,129,141,92,36,18,199,139,151,82,26)(9,198,130,150,93,25,19,188,140,160,83,35)(10,187,131,159,94,34,20,197,121,149,84,24)(41,213,179,111,78,239,51,203,169,101,68,229)(42,202,180,120,79,228,52,212,170,110,69,238)(43,211,161,109,80,237,53,201,171,119,70,227)(44,220,162,118,61,226,54,210,172,108,71,236)(45,209,163,107,62,235,55,219,173,117,72,225)(46,218,164,116,63,224,56,208,174,106,73,234)(47,207,165,105,64,233,57,217,175,115,74,223)(48,216,166,114,65,222,58,206,176,104,75,232)(49,205,167,103,66,231,59,215,177,113,76,221)(50,214,168,112,67,240,60,204,178,102,77,230), (1,76,148,215,11,66,158,205)(2,69,157,212,12,79,147,202)(3,62,146,209,13,72,156,219)(4,75,155,206,14,65,145,216)(5,68,144,203,15,78,154,213)(6,61,153,220,16,71,143,210)(7,74,142,217,17,64,152,207)(8,67,151,214,18,77,141,204)(9,80,160,211,19,70,150,201)(10,73,149,208,20,63,159,218)(21,225,134,173,31,235,124,163)(22,238,123,170,32,228,133,180)(23,231,132,167,33,221,122,177)(24,224,121,164,34,234,131,174)(25,237,130,161,35,227,140,171)(26,230,139,178,36,240,129,168)(27,223,128,175,37,233,138,165)(28,236,137,172,38,226,127,162)(29,229,126,169,39,239,136,179)(30,222,135,166,40,232,125,176)(41,192,101,89,51,182,111,99)(42,185,110,86,52,195,120,96)(43,198,119,83,53,188,109,93)(44,191,108,100,54,181,118,90)(45,184,117,97,55,194,107,87)(46,197,106,94,56,187,116,84)(47,190,115,91,57,200,105,81)(48,183,104,88,58,193,114,98)(49,196,113,85,59,186,103,95)(50,189,102,82,60,199,112,92)>;
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)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,88)(82,87)(83,86)(84,85)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(121,122)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)(155,160)(156,159)(157,158)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)(173,180)(174,179)(175,178)(176,177)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,186,122,158,85,33,11,196,132,148,95,23)(2,195,123,147,86,22,12,185,133,157,96,32)(3,184,124,156,87,31,13,194,134,146,97,21)(4,193,125,145,88,40,14,183,135,155,98,30)(5,182,126,154,89,29,15,192,136,144,99,39)(6,191,127,143,90,38,16,181,137,153,100,28)(7,200,128,152,91,27,17,190,138,142,81,37)(8,189,129,141,92,36,18,199,139,151,82,26)(9,198,130,150,93,25,19,188,140,160,83,35)(10,187,131,159,94,34,20,197,121,149,84,24)(41,213,179,111,78,239,51,203,169,101,68,229)(42,202,180,120,79,228,52,212,170,110,69,238)(43,211,161,109,80,237,53,201,171,119,70,227)(44,220,162,118,61,226,54,210,172,108,71,236)(45,209,163,107,62,235,55,219,173,117,72,225)(46,218,164,116,63,224,56,208,174,106,73,234)(47,207,165,105,64,233,57,217,175,115,74,223)(48,216,166,114,65,222,58,206,176,104,75,232)(49,205,167,103,66,231,59,215,177,113,76,221)(50,214,168,112,67,240,60,204,178,102,77,230), (1,76,148,215,11,66,158,205)(2,69,157,212,12,79,147,202)(3,62,146,209,13,72,156,219)(4,75,155,206,14,65,145,216)(5,68,144,203,15,78,154,213)(6,61,153,220,16,71,143,210)(7,74,142,217,17,64,152,207)(8,67,151,214,18,77,141,204)(9,80,160,211,19,70,150,201)(10,73,149,208,20,63,159,218)(21,225,134,173,31,235,124,163)(22,238,123,170,32,228,133,180)(23,231,132,167,33,221,122,177)(24,224,121,164,34,234,131,174)(25,237,130,161,35,227,140,171)(26,230,139,178,36,240,129,168)(27,223,128,175,37,233,138,165)(28,236,137,172,38,226,127,162)(29,229,126,169,39,239,136,179)(30,222,135,166,40,232,125,176)(41,192,101,89,51,182,111,99)(42,185,110,86,52,195,120,96)(43,198,119,83,53,188,109,93)(44,191,108,100,54,181,118,90)(45,184,117,97,55,194,107,87)(46,197,106,94,56,187,116,84)(47,190,115,91,57,200,105,81)(48,183,104,88,58,193,114,98)(49,196,113,85,59,186,103,95)(50,189,102,82,60,199,112,92) );
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),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,60),(58,59),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,88),(82,87),(83,86),(84,85),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95),(101,104),(102,103),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113),(121,122),(123,140),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133),(131,132),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148),(155,160),(156,159),(157,158),(161,172),(162,171),(163,170),(164,169),(165,168),(166,167),(173,180),(174,179),(175,178),(176,177),(181,190),(182,189),(183,188),(184,187),(185,186),(191,200),(192,199),(193,198),(194,197),(195,196),(201,208),(202,207),(203,206),(204,205),(209,220),(210,219),(211,218),(212,217),(213,216),(214,215),(221,240),(222,239),(223,238),(224,237),(225,236),(226,235),(227,234),(228,233),(229,232),(230,231)], [(1,186,122,158,85,33,11,196,132,148,95,23),(2,195,123,147,86,22,12,185,133,157,96,32),(3,184,124,156,87,31,13,194,134,146,97,21),(4,193,125,145,88,40,14,183,135,155,98,30),(5,182,126,154,89,29,15,192,136,144,99,39),(6,191,127,143,90,38,16,181,137,153,100,28),(7,200,128,152,91,27,17,190,138,142,81,37),(8,189,129,141,92,36,18,199,139,151,82,26),(9,198,130,150,93,25,19,188,140,160,83,35),(10,187,131,159,94,34,20,197,121,149,84,24),(41,213,179,111,78,239,51,203,169,101,68,229),(42,202,180,120,79,228,52,212,170,110,69,238),(43,211,161,109,80,237,53,201,171,119,70,227),(44,220,162,118,61,226,54,210,172,108,71,236),(45,209,163,107,62,235,55,219,173,117,72,225),(46,218,164,116,63,224,56,208,174,106,73,234),(47,207,165,105,64,233,57,217,175,115,74,223),(48,216,166,114,65,222,58,206,176,104,75,232),(49,205,167,103,66,231,59,215,177,113,76,221),(50,214,168,112,67,240,60,204,178,102,77,230)], [(1,76,148,215,11,66,158,205),(2,69,157,212,12,79,147,202),(3,62,146,209,13,72,156,219),(4,75,155,206,14,65,145,216),(5,68,144,203,15,78,154,213),(6,61,153,220,16,71,143,210),(7,74,142,217,17,64,152,207),(8,67,151,214,18,77,141,204),(9,80,160,211,19,70,150,201),(10,73,149,208,20,63,159,218),(21,225,134,173,31,235,124,163),(22,238,123,170,32,228,133,180),(23,231,132,167,33,221,122,177),(24,224,121,164,34,234,131,174),(25,237,130,161,35,227,140,171),(26,230,139,178,36,240,129,168),(27,223,128,175,37,233,138,165),(28,236,137,172,38,226,127,162),(29,229,126,169,39,239,136,179),(30,222,135,166,40,232,125,176),(41,192,101,89,51,182,111,99),(42,185,110,86,52,195,120,96),(43,198,119,83,53,188,109,93),(44,191,108,100,54,181,118,90),(45,184,117,97,55,194,107,87),(46,197,106,94,56,187,116,84),(47,190,115,91,57,200,105,81),(48,183,104,88,58,193,114,98),(49,196,113,85,59,186,103,95),(50,189,102,82,60,199,112,92)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
D_{20}.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
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