metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.3Dic3, C60.176C23, Dic10.3Dic3, C3⋊C8.34D10, C15⋊11(C8○D4), C60.99(C2×C4), C4○D20.6S3, (C3×D20).3C4, (C4×D5).58D6, C12.19(C4×D5), C60.7C4⋊9C2, C4.4(D5×Dic3), (C2×C12).78D10, (C2×C20).311D6, C5⋊3(D4.Dic3), C5⋊D4.3Dic3, C3⋊5(D20.3C4), C30.98(C22×C4), (C2×C60).39C22, (C3×Dic10).3C4, D10.3(C2×Dic3), C20.40(C2×Dic3), C20.32D6⋊13C2, C22.1(D5×Dic3), C20.173(C22×S3), C15⋊3C8.29C22, Dic5.3(C2×Dic3), (D5×C12).58C22, C12.173(C22×D5), C10.16(C22×Dic3), (C2×C3⋊C8)⋊3D5, (C10×C3⋊C8)⋊1C2, (D5×C3⋊C8)⋊12C2, C6.79(C2×C4×D5), C4.146(C2×S3×D5), (C2×C6).7(C4×D5), C2.5(C2×D5×Dic3), (C6×D5).5(C2×C4), (C2×C4).91(S3×D5), (C3×C5⋊D4).3C4, (C2×C30).95(C2×C4), (C3×C4○D20).3C2, (C5×C3⋊C8).39C22, (C3×Dic5).5(C2×C4), (C2×C10).35(C2×Dic3), SmallGroup(480,359)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.3Dic3
G = < a,b,c,d | a20=b2=1, c6=a10, d2=a10c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 428 in 124 conjugacy classes, 60 normal (38 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, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C3×Dic5, C60, C6×D5, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, D4.Dic3, C5×C3⋊C8, C15⋊3C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D20.3C4, D5×C3⋊C8, C20.32D6, C10×C3⋊C8, C60.7C4, C3×C4○D20, D20.3Dic3
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, D20.3C4, C2×D5×Dic3, D20.3Dic3
►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 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(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)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 180)(176 179)(177 178)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 204)(202 203)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)(221 222)(223 240)(224 239)(225 238)(226 237)(227 236)(228 235)(229 234)(230 233)(231 232)
(1 203 222 51 112 26 11 213 232 41 102 36)(2 204 223 52 113 27 12 214 233 42 103 37)(3 205 224 53 114 28 13 215 234 43 104 38)(4 206 225 54 115 29 14 216 235 44 105 39)(5 207 226 55 116 30 15 217 236 45 106 40)(6 208 227 56 117 31 16 218 237 46 107 21)(7 209 228 57 118 32 17 219 238 47 108 22)(8 210 229 58 119 33 18 220 239 48 109 23)(9 211 230 59 120 34 19 201 240 49 110 24)(10 212 231 60 101 35 20 202 221 50 111 25)(61 148 82 124 166 189 71 158 92 134 176 199)(62 149 83 125 167 190 72 159 93 135 177 200)(63 150 84 126 168 191 73 160 94 136 178 181)(64 151 85 127 169 192 74 141 95 137 179 182)(65 152 86 128 170 193 75 142 96 138 180 183)(66 153 87 129 171 194 76 143 97 139 161 184)(67 154 88 130 172 195 77 144 98 140 162 185)(68 155 89 131 173 196 78 145 99 121 163 186)(69 156 90 132 174 197 79 146 100 122 164 187)(70 157 91 133 175 198 80 147 81 123 165 188)
(1 84 41 181 11 94 51 191)(2 85 42 182 12 95 52 192)(3 86 43 183 13 96 53 193)(4 87 44 184 14 97 54 194)(5 88 45 185 15 98 55 195)(6 89 46 186 16 99 56 196)(7 90 47 187 17 100 57 197)(8 91 48 188 18 81 58 198)(9 92 49 189 19 82 59 199)(10 93 50 190 20 83 60 200)(21 121 237 78 31 131 227 68)(22 122 238 79 32 132 228 69)(23 123 239 80 33 133 229 70)(24 124 240 61 34 134 230 71)(25 125 221 62 35 135 231 72)(26 126 222 63 36 136 232 73)(27 127 223 64 37 137 233 74)(28 128 224 65 38 138 234 75)(29 129 225 66 39 139 235 76)(30 130 226 67 40 140 236 77)(101 167 212 149 111 177 202 159)(102 168 213 150 112 178 203 160)(103 169 214 151 113 179 204 141)(104 170 215 152 114 180 205 142)(105 171 216 153 115 161 206 143)(106 172 217 154 116 162 207 144)(107 173 218 155 117 163 208 145)(108 174 219 156 118 164 209 146)(109 175 220 157 119 165 210 147)(110 176 201 158 120 166 211 148)
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)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(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)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232), (1,203,222,51,112,26,11,213,232,41,102,36)(2,204,223,52,113,27,12,214,233,42,103,37)(3,205,224,53,114,28,13,215,234,43,104,38)(4,206,225,54,115,29,14,216,235,44,105,39)(5,207,226,55,116,30,15,217,236,45,106,40)(6,208,227,56,117,31,16,218,237,46,107,21)(7,209,228,57,118,32,17,219,238,47,108,22)(8,210,229,58,119,33,18,220,239,48,109,23)(9,211,230,59,120,34,19,201,240,49,110,24)(10,212,231,60,101,35,20,202,221,50,111,25)(61,148,82,124,166,189,71,158,92,134,176,199)(62,149,83,125,167,190,72,159,93,135,177,200)(63,150,84,126,168,191,73,160,94,136,178,181)(64,151,85,127,169,192,74,141,95,137,179,182)(65,152,86,128,170,193,75,142,96,138,180,183)(66,153,87,129,171,194,76,143,97,139,161,184)(67,154,88,130,172,195,77,144,98,140,162,185)(68,155,89,131,173,196,78,145,99,121,163,186)(69,156,90,132,174,197,79,146,100,122,164,187)(70,157,91,133,175,198,80,147,81,123,165,188), (1,84,41,181,11,94,51,191)(2,85,42,182,12,95,52,192)(3,86,43,183,13,96,53,193)(4,87,44,184,14,97,54,194)(5,88,45,185,15,98,55,195)(6,89,46,186,16,99,56,196)(7,90,47,187,17,100,57,197)(8,91,48,188,18,81,58,198)(9,92,49,189,19,82,59,199)(10,93,50,190,20,83,60,200)(21,121,237,78,31,131,227,68)(22,122,238,79,32,132,228,69)(23,123,239,80,33,133,229,70)(24,124,240,61,34,134,230,71)(25,125,221,62,35,135,231,72)(26,126,222,63,36,136,232,73)(27,127,223,64,37,137,233,74)(28,128,224,65,38,138,234,75)(29,129,225,66,39,139,235,76)(30,130,226,67,40,140,236,77)(101,167,212,149,111,177,202,159)(102,168,213,150,112,178,203,160)(103,169,214,151,113,179,204,141)(104,170,215,152,114,180,205,142)(105,171,216,153,115,161,206,143)(106,172,217,154,116,162,207,144)(107,173,218,155,117,163,208,145)(108,174,219,156,118,164,209,146)(109,175,220,157,119,165,210,147)(110,176,201,158,120,166,211,148)>;
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)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(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)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)(159,160)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,180)(176,179)(177,178)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232), (1,203,222,51,112,26,11,213,232,41,102,36)(2,204,223,52,113,27,12,214,233,42,103,37)(3,205,224,53,114,28,13,215,234,43,104,38)(4,206,225,54,115,29,14,216,235,44,105,39)(5,207,226,55,116,30,15,217,236,45,106,40)(6,208,227,56,117,31,16,218,237,46,107,21)(7,209,228,57,118,32,17,219,238,47,108,22)(8,210,229,58,119,33,18,220,239,48,109,23)(9,211,230,59,120,34,19,201,240,49,110,24)(10,212,231,60,101,35,20,202,221,50,111,25)(61,148,82,124,166,189,71,158,92,134,176,199)(62,149,83,125,167,190,72,159,93,135,177,200)(63,150,84,126,168,191,73,160,94,136,178,181)(64,151,85,127,169,192,74,141,95,137,179,182)(65,152,86,128,170,193,75,142,96,138,180,183)(66,153,87,129,171,194,76,143,97,139,161,184)(67,154,88,130,172,195,77,144,98,140,162,185)(68,155,89,131,173,196,78,145,99,121,163,186)(69,156,90,132,174,197,79,146,100,122,164,187)(70,157,91,133,175,198,80,147,81,123,165,188), (1,84,41,181,11,94,51,191)(2,85,42,182,12,95,52,192)(3,86,43,183,13,96,53,193)(4,87,44,184,14,97,54,194)(5,88,45,185,15,98,55,195)(6,89,46,186,16,99,56,196)(7,90,47,187,17,100,57,197)(8,91,48,188,18,81,58,198)(9,92,49,189,19,82,59,199)(10,93,50,190,20,83,60,200)(21,121,237,78,31,131,227,68)(22,122,238,79,32,132,228,69)(23,123,239,80,33,133,229,70)(24,124,240,61,34,134,230,71)(25,125,221,62,35,135,231,72)(26,126,222,63,36,136,232,73)(27,127,223,64,37,137,233,74)(28,128,224,65,38,138,234,75)(29,129,225,66,39,139,235,76)(30,130,226,67,40,140,236,77)(101,167,212,149,111,177,202,159)(102,168,213,150,112,178,203,160)(103,169,214,151,113,179,204,141)(104,170,215,152,114,180,205,142)(105,171,216,153,115,161,206,143)(106,172,217,154,116,162,207,144)(107,173,218,155,117,163,208,145)(108,174,219,156,118,164,209,146)(109,175,220,157,119,165,210,147)(110,176,201,158,120,166,211,148) );
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),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,64),(62,63),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(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),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,158),(142,157),(143,156),(144,155),(145,154),(146,153),(147,152),(148,151),(149,150),(159,160),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168),(175,180),(176,179),(177,178),(181,200),(182,199),(183,198),(184,197),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(201,204),(202,203),(205,220),(206,219),(207,218),(208,217),(209,216),(210,215),(211,214),(212,213),(221,222),(223,240),(224,239),(225,238),(226,237),(227,236),(228,235),(229,234),(230,233),(231,232)], [(1,203,222,51,112,26,11,213,232,41,102,36),(2,204,223,52,113,27,12,214,233,42,103,37),(3,205,224,53,114,28,13,215,234,43,104,38),(4,206,225,54,115,29,14,216,235,44,105,39),(5,207,226,55,116,30,15,217,236,45,106,40),(6,208,227,56,117,31,16,218,237,46,107,21),(7,209,228,57,118,32,17,219,238,47,108,22),(8,210,229,58,119,33,18,220,239,48,109,23),(9,211,230,59,120,34,19,201,240,49,110,24),(10,212,231,60,101,35,20,202,221,50,111,25),(61,148,82,124,166,189,71,158,92,134,176,199),(62,149,83,125,167,190,72,159,93,135,177,200),(63,150,84,126,168,191,73,160,94,136,178,181),(64,151,85,127,169,192,74,141,95,137,179,182),(65,152,86,128,170,193,75,142,96,138,180,183),(66,153,87,129,171,194,76,143,97,139,161,184),(67,154,88,130,172,195,77,144,98,140,162,185),(68,155,89,131,173,196,78,145,99,121,163,186),(69,156,90,132,174,197,79,146,100,122,164,187),(70,157,91,133,175,198,80,147,81,123,165,188)], [(1,84,41,181,11,94,51,191),(2,85,42,182,12,95,52,192),(3,86,43,183,13,96,53,193),(4,87,44,184,14,97,54,194),(5,88,45,185,15,98,55,195),(6,89,46,186,16,99,56,196),(7,90,47,187,17,100,57,197),(8,91,48,188,18,81,58,198),(9,92,49,189,19,82,59,199),(10,93,50,190,20,83,60,200),(21,121,237,78,31,131,227,68),(22,122,238,79,32,132,228,69),(23,123,239,80,33,133,229,70),(24,124,240,61,34,134,230,71),(25,125,221,62,35,135,231,72),(26,126,222,63,36,136,232,73),(27,127,223,64,37,137,233,74),(28,128,224,65,38,138,234,75),(29,129,225,66,39,139,235,76),(30,130,226,67,40,140,236,77),(101,167,212,149,111,177,202,159),(102,168,213,150,112,178,203,160),(103,169,214,151,113,179,204,141),(104,170,215,152,114,180,205,142),(105,171,216,153,115,161,206,143),(106,172,217,154,116,162,207,144),(107,173,218,155,117,163,208,145),(108,174,219,156,118,164,209,146),(109,175,220,157,119,165,210,147),(110,176,201,158,120,166,211,148)]])
78 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 | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 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 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 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 | 3 | 3 | 3 | 3 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 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 |
| 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 | D20.3C4 | S3×D5 | D4.Dic3 | D5×Dic3 | C2×S3×D5 | D5×Dic3 | D20.3Dic3 |
| kernel | D20.3Dic3 | D5×C3⋊C8 | C20.32D6 | C10×C3⋊C8 | C60.7C4 | C3×C4○D20 | C3×Dic10 | C3×D20 | C3×C5⋊D4 | C4○D20 | C2×C3⋊C8 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | C3⋊C8 | C2×C12 | C15 | C12 | C2×C6 | C3 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of D20.3Dic3 ►in GL4(𝔽241) generated by
| 156 | 200 | 0 | 0 |
| 41 | 119 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 156 | 200 | 0 | 0 |
| 41 | 85 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 177 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 144 | 239 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 150 | 151 |
| 0 | 0 | 92 | 91 |
G:=sub<GL(4,GF(241))| [156,41,0,0,200,119,0,0,0,0,1,0,0,0,0,1],[156,41,0,0,200,85,0,0,0,0,1,0,0,0,0,1],[177,0,0,0,0,177,0,0,0,0,1,144,0,0,5,239],[30,0,0,0,0,30,0,0,0,0,150,92,0,0,151,91] >;
D20.3Dic3 in GAP, Magma, Sage, TeX
D_{20}._3{\rm Dic}_3 % in TeX
G:=Group("D20.3Dic3"); // GroupNames label
G:=SmallGroup(480,359);
// by ID
G=gap.SmallGroup(480,359);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,80,1356,18822]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations