metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.4D20, (C4×Dic3)⋊8D5, C6.11(C2×D20), C2.16(S3×D20), C30.32(C2×D4), C10.11(S3×D4), D10⋊C4⋊1S3, D30⋊4C4⋊5C2, (C2×C20).258D6, C15⋊3(C4.4D4), C3⋊1(C4.D20), (C2×Dic5).9D6, D30⋊3C4⋊30C2, (C22×D5).3D6, (Dic3×C20)⋊20C2, C6.20(C4○D20), C30.20(C4○D4), (C2×C12).177D10, D10⋊Dic3⋊2C2, (C2×C30).43C23, (C5×Dic3).26D4, C10.23(C4○D12), (C2×C60).402C22, C5⋊1(C23.11D6), C10.39(D4⋊2S3), (C2×Dic3).136D10, (C6×Dic5).25C22, C2.13(D6.D10), C2.11(Dic5.D6), (C2×Dic15).48C22, (C22×D15).21C22, (C10×Dic3).160C22, (C2×C15⋊Q8)⋊4C2, (C2×C4).70(S3×D5), (D5×C2×C6).1C22, (C2×C3⋊D20).2C2, C22.132(C2×S3×D5), (C3×D10⋊C4)⋊30C2, (C2×C6).55(C22×D5), (C2×C10).55(C22×S3), SmallGroup(480,429)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3.D20
G = < a,b,c,d | a6=c20=1, b2=d2=a3, bab-1=cac-1=dad-1=a-1, bc=cb, dbd-1=a3b, dcd-1=a3c-1 >
Subgroups: 940 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4.4D4, Dic10, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, D30, C2×C30, D10⋊C4, D10⋊C4, C4×C20, C2×Dic10, C2×D20, C23.11D6, C3⋊D20, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C4.D20, D10⋊Dic3, D30⋊4C4, C3×D10⋊C4, Dic3×C20, D30⋊3C4, C2×C3⋊D20, C2×C15⋊Q8, Dic3.D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4.4D4, D20, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, C2×D20, C4○D20, C23.11D6, C2×S3×D5, C4.D20, D6.D10, S3×D20, Dic5.D6, Dic3.D20
►On 240 points
Generators in S240
(1 159 197 90 43 229)(2 230 44 91 198 160)(3 141 199 92 45 231)(4 232 46 93 200 142)(5 143 181 94 47 233)(6 234 48 95 182 144)(7 145 183 96 49 235)(8 236 50 97 184 146)(9 147 185 98 51 237)(10 238 52 99 186 148)(11 149 187 100 53 239)(12 240 54 81 188 150)(13 151 189 82 55 221)(14 222 56 83 190 152)(15 153 191 84 57 223)(16 224 58 85 192 154)(17 155 193 86 59 225)(18 226 60 87 194 156)(19 157 195 88 41 227)(20 228 42 89 196 158)(21 66 103 134 208 162)(22 163 209 135 104 67)(23 68 105 136 210 164)(24 165 211 137 106 69)(25 70 107 138 212 166)(26 167 213 139 108 71)(27 72 109 140 214 168)(28 169 215 121 110 73)(29 74 111 122 216 170)(30 171 217 123 112 75)(31 76 113 124 218 172)(32 173 219 125 114 77)(33 78 115 126 220 174)(34 175 201 127 116 79)(35 80 117 128 202 176)(36 177 203 129 118 61)(37 62 119 130 204 178)(38 179 205 131 120 63)(39 64 101 132 206 180)(40 161 207 133 102 65)
(1 115 90 174)(2 116 91 175)(3 117 92 176)(4 118 93 177)(5 119 94 178)(6 120 95 179)(7 101 96 180)(8 102 97 161)(9 103 98 162)(10 104 99 163)(11 105 100 164)(12 106 81 165)(13 107 82 166)(14 108 83 167)(15 109 84 168)(16 110 85 169)(17 111 86 170)(18 112 87 171)(19 113 88 172)(20 114 89 173)(21 237 134 185)(22 238 135 186)(23 239 136 187)(24 240 137 188)(25 221 138 189)(26 222 139 190)(27 223 140 191)(28 224 121 192)(29 225 122 193)(30 226 123 194)(31 227 124 195)(32 228 125 196)(33 229 126 197)(34 230 127 198)(35 231 128 199)(36 232 129 200)(37 233 130 181)(38 234 131 182)(39 235 132 183)(40 236 133 184)(41 218 157 76)(42 219 158 77)(43 220 159 78)(44 201 160 79)(45 202 141 80)(46 203 142 61)(47 204 143 62)(48 205 144 63)(49 206 145 64)(50 207 146 65)(51 208 147 66)(52 209 148 67)(53 210 149 68)(54 211 150 69)(55 212 151 70)(56 213 152 71)(57 214 153 72)(58 215 154 73)(59 216 155 74)(60 217 156 75)
(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)
(1 114 90 173)(2 172 91 113)(3 112 92 171)(4 170 93 111)(5 110 94 169)(6 168 95 109)(7 108 96 167)(8 166 97 107)(9 106 98 165)(10 164 99 105)(11 104 100 163)(12 162 81 103)(13 102 82 161)(14 180 83 101)(15 120 84 179)(16 178 85 119)(17 118 86 177)(18 176 87 117)(19 116 88 175)(20 174 89 115)(21 54 134 150)(22 149 135 53)(23 52 136 148)(24 147 137 51)(25 50 138 146)(26 145 139 49)(27 48 140 144)(28 143 121 47)(29 46 122 142)(30 141 123 45)(31 44 124 160)(32 159 125 43)(33 42 126 158)(34 157 127 41)(35 60 128 156)(36 155 129 59)(37 58 130 154)(38 153 131 57)(39 56 132 152)(40 151 133 55)(61 193 203 225)(62 224 204 192)(63 191 205 223)(64 222 206 190)(65 189 207 221)(66 240 208 188)(67 187 209 239)(68 238 210 186)(69 185 211 237)(70 236 212 184)(71 183 213 235)(72 234 214 182)(73 181 215 233)(74 232 216 200)(75 199 217 231)(76 230 218 198)(77 197 219 229)(78 228 220 196)(79 195 201 227)(80 226 202 194)
G:=sub<Sym(240)| (1,159,197,90,43,229)(2,230,44,91,198,160)(3,141,199,92,45,231)(4,232,46,93,200,142)(5,143,181,94,47,233)(6,234,48,95,182,144)(7,145,183,96,49,235)(8,236,50,97,184,146)(9,147,185,98,51,237)(10,238,52,99,186,148)(11,149,187,100,53,239)(12,240,54,81,188,150)(13,151,189,82,55,221)(14,222,56,83,190,152)(15,153,191,84,57,223)(16,224,58,85,192,154)(17,155,193,86,59,225)(18,226,60,87,194,156)(19,157,195,88,41,227)(20,228,42,89,196,158)(21,66,103,134,208,162)(22,163,209,135,104,67)(23,68,105,136,210,164)(24,165,211,137,106,69)(25,70,107,138,212,166)(26,167,213,139,108,71)(27,72,109,140,214,168)(28,169,215,121,110,73)(29,74,111,122,216,170)(30,171,217,123,112,75)(31,76,113,124,218,172)(32,173,219,125,114,77)(33,78,115,126,220,174)(34,175,201,127,116,79)(35,80,117,128,202,176)(36,177,203,129,118,61)(37,62,119,130,204,178)(38,179,205,131,120,63)(39,64,101,132,206,180)(40,161,207,133,102,65), (1,115,90,174)(2,116,91,175)(3,117,92,176)(4,118,93,177)(5,119,94,178)(6,120,95,179)(7,101,96,180)(8,102,97,161)(9,103,98,162)(10,104,99,163)(11,105,100,164)(12,106,81,165)(13,107,82,166)(14,108,83,167)(15,109,84,168)(16,110,85,169)(17,111,86,170)(18,112,87,171)(19,113,88,172)(20,114,89,173)(21,237,134,185)(22,238,135,186)(23,239,136,187)(24,240,137,188)(25,221,138,189)(26,222,139,190)(27,223,140,191)(28,224,121,192)(29,225,122,193)(30,226,123,194)(31,227,124,195)(32,228,125,196)(33,229,126,197)(34,230,127,198)(35,231,128,199)(36,232,129,200)(37,233,130,181)(38,234,131,182)(39,235,132,183)(40,236,133,184)(41,218,157,76)(42,219,158,77)(43,220,159,78)(44,201,160,79)(45,202,141,80)(46,203,142,61)(47,204,143,62)(48,205,144,63)(49,206,145,64)(50,207,146,65)(51,208,147,66)(52,209,148,67)(53,210,149,68)(54,211,150,69)(55,212,151,70)(56,213,152,71)(57,214,153,72)(58,215,154,73)(59,216,155,74)(60,217,156,75), (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), (1,114,90,173)(2,172,91,113)(3,112,92,171)(4,170,93,111)(5,110,94,169)(6,168,95,109)(7,108,96,167)(8,166,97,107)(9,106,98,165)(10,164,99,105)(11,104,100,163)(12,162,81,103)(13,102,82,161)(14,180,83,101)(15,120,84,179)(16,178,85,119)(17,118,86,177)(18,176,87,117)(19,116,88,175)(20,174,89,115)(21,54,134,150)(22,149,135,53)(23,52,136,148)(24,147,137,51)(25,50,138,146)(26,145,139,49)(27,48,140,144)(28,143,121,47)(29,46,122,142)(30,141,123,45)(31,44,124,160)(32,159,125,43)(33,42,126,158)(34,157,127,41)(35,60,128,156)(36,155,129,59)(37,58,130,154)(38,153,131,57)(39,56,132,152)(40,151,133,55)(61,193,203,225)(62,224,204,192)(63,191,205,223)(64,222,206,190)(65,189,207,221)(66,240,208,188)(67,187,209,239)(68,238,210,186)(69,185,211,237)(70,236,212,184)(71,183,213,235)(72,234,214,182)(73,181,215,233)(74,232,216,200)(75,199,217,231)(76,230,218,198)(77,197,219,229)(78,228,220,196)(79,195,201,227)(80,226,202,194)>;
G:=Group( (1,159,197,90,43,229)(2,230,44,91,198,160)(3,141,199,92,45,231)(4,232,46,93,200,142)(5,143,181,94,47,233)(6,234,48,95,182,144)(7,145,183,96,49,235)(8,236,50,97,184,146)(9,147,185,98,51,237)(10,238,52,99,186,148)(11,149,187,100,53,239)(12,240,54,81,188,150)(13,151,189,82,55,221)(14,222,56,83,190,152)(15,153,191,84,57,223)(16,224,58,85,192,154)(17,155,193,86,59,225)(18,226,60,87,194,156)(19,157,195,88,41,227)(20,228,42,89,196,158)(21,66,103,134,208,162)(22,163,209,135,104,67)(23,68,105,136,210,164)(24,165,211,137,106,69)(25,70,107,138,212,166)(26,167,213,139,108,71)(27,72,109,140,214,168)(28,169,215,121,110,73)(29,74,111,122,216,170)(30,171,217,123,112,75)(31,76,113,124,218,172)(32,173,219,125,114,77)(33,78,115,126,220,174)(34,175,201,127,116,79)(35,80,117,128,202,176)(36,177,203,129,118,61)(37,62,119,130,204,178)(38,179,205,131,120,63)(39,64,101,132,206,180)(40,161,207,133,102,65), (1,115,90,174)(2,116,91,175)(3,117,92,176)(4,118,93,177)(5,119,94,178)(6,120,95,179)(7,101,96,180)(8,102,97,161)(9,103,98,162)(10,104,99,163)(11,105,100,164)(12,106,81,165)(13,107,82,166)(14,108,83,167)(15,109,84,168)(16,110,85,169)(17,111,86,170)(18,112,87,171)(19,113,88,172)(20,114,89,173)(21,237,134,185)(22,238,135,186)(23,239,136,187)(24,240,137,188)(25,221,138,189)(26,222,139,190)(27,223,140,191)(28,224,121,192)(29,225,122,193)(30,226,123,194)(31,227,124,195)(32,228,125,196)(33,229,126,197)(34,230,127,198)(35,231,128,199)(36,232,129,200)(37,233,130,181)(38,234,131,182)(39,235,132,183)(40,236,133,184)(41,218,157,76)(42,219,158,77)(43,220,159,78)(44,201,160,79)(45,202,141,80)(46,203,142,61)(47,204,143,62)(48,205,144,63)(49,206,145,64)(50,207,146,65)(51,208,147,66)(52,209,148,67)(53,210,149,68)(54,211,150,69)(55,212,151,70)(56,213,152,71)(57,214,153,72)(58,215,154,73)(59,216,155,74)(60,217,156,75), (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), (1,114,90,173)(2,172,91,113)(3,112,92,171)(4,170,93,111)(5,110,94,169)(6,168,95,109)(7,108,96,167)(8,166,97,107)(9,106,98,165)(10,164,99,105)(11,104,100,163)(12,162,81,103)(13,102,82,161)(14,180,83,101)(15,120,84,179)(16,178,85,119)(17,118,86,177)(18,176,87,117)(19,116,88,175)(20,174,89,115)(21,54,134,150)(22,149,135,53)(23,52,136,148)(24,147,137,51)(25,50,138,146)(26,145,139,49)(27,48,140,144)(28,143,121,47)(29,46,122,142)(30,141,123,45)(31,44,124,160)(32,159,125,43)(33,42,126,158)(34,157,127,41)(35,60,128,156)(36,155,129,59)(37,58,130,154)(38,153,131,57)(39,56,132,152)(40,151,133,55)(61,193,203,225)(62,224,204,192)(63,191,205,223)(64,222,206,190)(65,189,207,221)(66,240,208,188)(67,187,209,239)(68,238,210,186)(69,185,211,237)(70,236,212,184)(71,183,213,235)(72,234,214,182)(73,181,215,233)(74,232,216,200)(75,199,217,231)(76,230,218,198)(77,197,219,229)(78,228,220,196)(79,195,201,227)(80,226,202,194) );
G=PermutationGroup([[(1,159,197,90,43,229),(2,230,44,91,198,160),(3,141,199,92,45,231),(4,232,46,93,200,142),(5,143,181,94,47,233),(6,234,48,95,182,144),(7,145,183,96,49,235),(8,236,50,97,184,146),(9,147,185,98,51,237),(10,238,52,99,186,148),(11,149,187,100,53,239),(12,240,54,81,188,150),(13,151,189,82,55,221),(14,222,56,83,190,152),(15,153,191,84,57,223),(16,224,58,85,192,154),(17,155,193,86,59,225),(18,226,60,87,194,156),(19,157,195,88,41,227),(20,228,42,89,196,158),(21,66,103,134,208,162),(22,163,209,135,104,67),(23,68,105,136,210,164),(24,165,211,137,106,69),(25,70,107,138,212,166),(26,167,213,139,108,71),(27,72,109,140,214,168),(28,169,215,121,110,73),(29,74,111,122,216,170),(30,171,217,123,112,75),(31,76,113,124,218,172),(32,173,219,125,114,77),(33,78,115,126,220,174),(34,175,201,127,116,79),(35,80,117,128,202,176),(36,177,203,129,118,61),(37,62,119,130,204,178),(38,179,205,131,120,63),(39,64,101,132,206,180),(40,161,207,133,102,65)], [(1,115,90,174),(2,116,91,175),(3,117,92,176),(4,118,93,177),(5,119,94,178),(6,120,95,179),(7,101,96,180),(8,102,97,161),(9,103,98,162),(10,104,99,163),(11,105,100,164),(12,106,81,165),(13,107,82,166),(14,108,83,167),(15,109,84,168),(16,110,85,169),(17,111,86,170),(18,112,87,171),(19,113,88,172),(20,114,89,173),(21,237,134,185),(22,238,135,186),(23,239,136,187),(24,240,137,188),(25,221,138,189),(26,222,139,190),(27,223,140,191),(28,224,121,192),(29,225,122,193),(30,226,123,194),(31,227,124,195),(32,228,125,196),(33,229,126,197),(34,230,127,198),(35,231,128,199),(36,232,129,200),(37,233,130,181),(38,234,131,182),(39,235,132,183),(40,236,133,184),(41,218,157,76),(42,219,158,77),(43,220,159,78),(44,201,160,79),(45,202,141,80),(46,203,142,61),(47,204,143,62),(48,205,144,63),(49,206,145,64),(50,207,146,65),(51,208,147,66),(52,209,148,67),(53,210,149,68),(54,211,150,69),(55,212,151,70),(56,213,152,71),(57,214,153,72),(58,215,154,73),(59,216,155,74),(60,217,156,75)], [(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)], [(1,114,90,173),(2,172,91,113),(3,112,92,171),(4,170,93,111),(5,110,94,169),(6,168,95,109),(7,108,96,167),(8,166,97,107),(9,106,98,165),(10,164,99,105),(11,104,100,163),(12,162,81,103),(13,102,82,161),(14,180,83,101),(15,120,84,179),(16,178,85,119),(17,118,86,177),(18,176,87,117),(19,116,88,175),(20,174,89,115),(21,54,134,150),(22,149,135,53),(23,52,136,148),(24,147,137,51),(25,50,138,146),(26,145,139,49),(27,48,140,144),(28,143,121,47),(29,46,122,142),(30,141,123,45),(31,44,124,160),(32,159,125,43),(33,42,126,158),(34,157,127,41),(35,60,128,156),(36,155,129,59),(37,58,130,154),(38,153,131,57),(39,56,132,152),(40,151,133,55),(61,193,203,225),(62,224,204,192),(63,191,205,223),(64,222,206,190),(65,189,207,221),(66,240,208,188),(67,187,209,239),(68,238,210,186),(69,185,211,237),(70,236,212,184),(71,183,213,235),(72,234,214,182),(73,181,215,233),(74,232,216,200),(75,199,217,231),(76,230,218,198),(77,197,219,229),(78,228,220,196),(79,195,201,227),(80,226,202,194)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 60 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 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 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D20 | C4○D12 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | C2×S3×D5 | D6.D10 | S3×D20 | Dic5.D6 |
| kernel | Dic3.D20 | D10⋊Dic3 | D30⋊4C4 | C3×D10⋊C4 | Dic3×C20 | D30⋊3C4 | C2×C3⋊D20 | C2×C15⋊Q8 | D10⋊C4 | C5×Dic3 | C4×Dic3 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | Dic3 | C10 | C6 | C10 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 8 | 4 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic3.D20 ►in GL4(𝔽61) generated by
| 0 | 60 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 43 | 0 | 0 |
| 52 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 23 | 46 | 0 | 0 |
| 23 | 38 | 0 | 0 |
| 0 | 0 | 57 | 25 |
| 0 | 0 | 36 | 34 |
| 50 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 |
| 0 | 0 | 57 | 25 |
| 0 | 0 | 36 | 4 |
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,1,0,0,0,0,1],[52,52,0,0,43,9,0,0,0,0,1,0,0,0,0,1],[23,23,0,0,46,38,0,0,0,0,57,36,0,0,25,34],[50,11,0,0,0,11,0,0,0,0,57,36,0,0,25,4] >;
Dic3.D20 in GAP, Magma, Sage, TeX
{\rm Dic}_3.D_{20} % in TeX
G:=Group("Dic3.D20"); // GroupNames label
G:=SmallGroup(480,429);
// by ID
G=gap.SmallGroup(480,429);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^20=1,b^2=d^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^-1>;
// generators/relations