direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3×D20, C3⋊5(C4×D20), C12⋊1(C4×D5), C15⋊11(C4×D4), C5⋊2(D4×Dic3), C60⋊14(C2×C4), (C3×D20)⋊9C4, C4⋊1(D5×Dic3), C2.2(S3×D20), (C4×Dic3)⋊5D5, (C5×Dic3)⋊7D4, (C6×D20).5C2, C60⋊5C4⋊25C2, C20⋊7(C2×Dic3), C30.46(C2×D4), C10.15(S3×D4), C6.15(C2×D20), (Dic3×C20)⋊5C2, D10⋊2(C2×Dic3), (C2×D20).11S3, (C2×C20).298D6, C6.55(C4○D20), C30.67(C4○D4), (C2×C12).124D10, (C22×D5).51D6, D10⋊Dic3⋊10C2, C2.4(D20⋊5S3), (C2×C60).117C22, (C2×C30).115C23, C30.127(C22×C4), C10.29(D4⋊2S3), (C2×Dic3).181D10, C10.25(C22×Dic3), (C2×Dic15).93C22, (C10×Dic3).180C22, C6.88(C2×C4×D5), (C6×D5)⋊3(C2×C4), (C2×D5×Dic3)⋊8C2, C2.13(C2×D5×Dic3), C22.55(C2×S3×D5), (C2×C4).108(S3×D5), (D5×C2×C6).21C22, (C2×C6).127(C22×D5), (C2×C10).127(C22×S3), SmallGroup(480,501)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3×D20
G = < a,b,c,d | a6=c20=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 892 in 188 conjugacy classes, 72 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, C5×Dic3, C5×Dic3, Dic15, C60, C6×D5, C6×D5, C2×C30, C4⋊Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×D20, D4×Dic3, D5×Dic3, C3×D20, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C4×D20, D10⋊Dic3, Dic3×C20, C60⋊5C4, C2×D5×Dic3, C6×D20, Dic3×D20
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, Dic3, D6, C22×C4, C2×D4, C4○D4, D10, C2×Dic3, C22×S3, C4×D4, C4×D5, D20, C22×D5, S3×D4, D4⋊2S3, C22×Dic3, S3×D5, C2×C4×D5, C2×D20, C4○D20, D4×Dic3, D5×Dic3, C2×S3×D5, C4×D20, D20⋊5S3, S3×D20, C2×D5×Dic3, Dic3×D20
►On 240 points
Generators in S240
(1 219 35 122 116 172)(2 220 36 123 117 173)(3 201 37 124 118 174)(4 202 38 125 119 175)(5 203 39 126 120 176)(6 204 40 127 101 177)(7 205 21 128 102 178)(8 206 22 129 103 179)(9 207 23 130 104 180)(10 208 24 131 105 161)(11 209 25 132 106 162)(12 210 26 133 107 163)(13 211 27 134 108 164)(14 212 28 135 109 165)(15 213 29 136 110 166)(16 214 30 137 111 167)(17 215 31 138 112 168)(18 216 32 139 113 169)(19 217 33 140 114 170)(20 218 34 121 115 171)(41 92 79 157 240 181)(42 93 80 158 221 182)(43 94 61 159 222 183)(44 95 62 160 223 184)(45 96 63 141 224 185)(46 97 64 142 225 186)(47 98 65 143 226 187)(48 99 66 144 227 188)(49 100 67 145 228 189)(50 81 68 146 229 190)(51 82 69 147 230 191)(52 83 70 148 231 192)(53 84 71 149 232 193)(54 85 72 150 233 194)(55 86 73 151 234 195)(56 87 74 152 235 196)(57 88 75 153 236 197)(58 89 76 154 237 198)(59 90 77 155 238 199)(60 91 78 156 239 200)
(1 57 122 153)(2 58 123 154)(3 59 124 155)(4 60 125 156)(5 41 126 157)(6 42 127 158)(7 43 128 159)(8 44 129 160)(9 45 130 141)(10 46 131 142)(11 47 132 143)(12 48 133 144)(13 49 134 145)(14 50 135 146)(15 51 136 147)(16 52 137 148)(17 53 138 149)(18 54 139 150)(19 55 140 151)(20 56 121 152)(21 222 178 94)(22 223 179 95)(23 224 180 96)(24 225 161 97)(25 226 162 98)(26 227 163 99)(27 228 164 100)(28 229 165 81)(29 230 166 82)(30 231 167 83)(31 232 168 84)(32 233 169 85)(33 234 170 86)(34 235 171 87)(35 236 172 88)(36 237 173 89)(37 238 174 90)(38 239 175 91)(39 240 176 92)(40 221 177 93)(61 205 183 102)(62 206 184 103)(63 207 185 104)(64 208 186 105)(65 209 187 106)(66 210 188 107)(67 211 189 108)(68 212 190 109)(69 213 191 110)(70 214 192 111)(71 215 193 112)(72 216 194 113)(73 217 195 114)(74 218 196 115)(75 219 197 116)(76 220 198 117)(77 201 199 118)(78 202 200 119)(79 203 181 120)(80 204 182 101)
(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)
(1 136)(2 135)(3 134)(4 133)(5 132)(6 131)(7 130)(8 129)(9 128)(10 127)(11 126)(12 125)(13 124)(14 123)(15 122)(16 121)(17 140)(18 139)(19 138)(20 137)(21 180)(22 179)(23 178)(24 177)(25 176)(26 175)(27 174)(28 173)(29 172)(30 171)(31 170)(32 169)(33 168)(34 167)(35 166)(36 165)(37 164)(38 163)(39 162)(40 161)(41 143)(42 142)(43 141)(44 160)(45 159)(46 158)(47 157)(48 156)(49 155)(50 154)(51 153)(52 152)(53 151)(54 150)(55 149)(56 148)(57 147)(58 146)(59 145)(60 144)(61 185)(62 184)(63 183)(64 182)(65 181)(66 200)(67 199)(68 198)(69 197)(70 196)(71 195)(72 194)(73 193)(74 192)(75 191)(76 190)(77 189)(78 188)(79 187)(80 186)(81 237)(82 236)(83 235)(84 234)(85 233)(86 232)(87 231)(88 230)(89 229)(90 228)(91 227)(92 226)(93 225)(94 224)(95 223)(96 222)(97 221)(98 240)(99 239)(100 238)(101 208)(102 207)(103 206)(104 205)(105 204)(106 203)(107 202)(108 201)(109 220)(110 219)(111 218)(112 217)(113 216)(114 215)(115 214)(116 213)(117 212)(118 211)(119 210)(120 209)
G:=sub<Sym(240)| (1,219,35,122,116,172)(2,220,36,123,117,173)(3,201,37,124,118,174)(4,202,38,125,119,175)(5,203,39,126,120,176)(6,204,40,127,101,177)(7,205,21,128,102,178)(8,206,22,129,103,179)(9,207,23,130,104,180)(10,208,24,131,105,161)(11,209,25,132,106,162)(12,210,26,133,107,163)(13,211,27,134,108,164)(14,212,28,135,109,165)(15,213,29,136,110,166)(16,214,30,137,111,167)(17,215,31,138,112,168)(18,216,32,139,113,169)(19,217,33,140,114,170)(20,218,34,121,115,171)(41,92,79,157,240,181)(42,93,80,158,221,182)(43,94,61,159,222,183)(44,95,62,160,223,184)(45,96,63,141,224,185)(46,97,64,142,225,186)(47,98,65,143,226,187)(48,99,66,144,227,188)(49,100,67,145,228,189)(50,81,68,146,229,190)(51,82,69,147,230,191)(52,83,70,148,231,192)(53,84,71,149,232,193)(54,85,72,150,233,194)(55,86,73,151,234,195)(56,87,74,152,235,196)(57,88,75,153,236,197)(58,89,76,154,237,198)(59,90,77,155,238,199)(60,91,78,156,239,200), (1,57,122,153)(2,58,123,154)(3,59,124,155)(4,60,125,156)(5,41,126,157)(6,42,127,158)(7,43,128,159)(8,44,129,160)(9,45,130,141)(10,46,131,142)(11,47,132,143)(12,48,133,144)(13,49,134,145)(14,50,135,146)(15,51,136,147)(16,52,137,148)(17,53,138,149)(18,54,139,150)(19,55,140,151)(20,56,121,152)(21,222,178,94)(22,223,179,95)(23,224,180,96)(24,225,161,97)(25,226,162,98)(26,227,163,99)(27,228,164,100)(28,229,165,81)(29,230,166,82)(30,231,167,83)(31,232,168,84)(32,233,169,85)(33,234,170,86)(34,235,171,87)(35,236,172,88)(36,237,173,89)(37,238,174,90)(38,239,175,91)(39,240,176,92)(40,221,177,93)(61,205,183,102)(62,206,184,103)(63,207,185,104)(64,208,186,105)(65,209,187,106)(66,210,188,107)(67,211,189,108)(68,212,190,109)(69,213,191,110)(70,214,192,111)(71,215,193,112)(72,216,194,113)(73,217,195,114)(74,218,196,115)(75,219,197,116)(76,220,198,117)(77,201,199,118)(78,202,200,119)(79,203,181,120)(80,204,182,101), (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), (1,136)(2,135)(3,134)(4,133)(5,132)(6,131)(7,130)(8,129)(9,128)(10,127)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,140)(18,139)(19,138)(20,137)(21,180)(22,179)(23,178)(24,177)(25,176)(26,175)(27,174)(28,173)(29,172)(30,171)(31,170)(32,169)(33,168)(34,167)(35,166)(36,165)(37,164)(38,163)(39,162)(40,161)(41,143)(42,142)(43,141)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(61,185)(62,184)(63,183)(64,182)(65,181)(66,200)(67,199)(68,198)(69,197)(70,196)(71,195)(72,194)(73,193)(74,192)(75,191)(76,190)(77,189)(78,188)(79,187)(80,186)(81,237)(82,236)(83,235)(84,234)(85,233)(86,232)(87,231)(88,230)(89,229)(90,228)(91,227)(92,226)(93,225)(94,224)(95,223)(96,222)(97,221)(98,240)(99,239)(100,238)(101,208)(102,207)(103,206)(104,205)(105,204)(106,203)(107,202)(108,201)(109,220)(110,219)(111,218)(112,217)(113,216)(114,215)(115,214)(116,213)(117,212)(118,211)(119,210)(120,209)>;
G:=Group( (1,219,35,122,116,172)(2,220,36,123,117,173)(3,201,37,124,118,174)(4,202,38,125,119,175)(5,203,39,126,120,176)(6,204,40,127,101,177)(7,205,21,128,102,178)(8,206,22,129,103,179)(9,207,23,130,104,180)(10,208,24,131,105,161)(11,209,25,132,106,162)(12,210,26,133,107,163)(13,211,27,134,108,164)(14,212,28,135,109,165)(15,213,29,136,110,166)(16,214,30,137,111,167)(17,215,31,138,112,168)(18,216,32,139,113,169)(19,217,33,140,114,170)(20,218,34,121,115,171)(41,92,79,157,240,181)(42,93,80,158,221,182)(43,94,61,159,222,183)(44,95,62,160,223,184)(45,96,63,141,224,185)(46,97,64,142,225,186)(47,98,65,143,226,187)(48,99,66,144,227,188)(49,100,67,145,228,189)(50,81,68,146,229,190)(51,82,69,147,230,191)(52,83,70,148,231,192)(53,84,71,149,232,193)(54,85,72,150,233,194)(55,86,73,151,234,195)(56,87,74,152,235,196)(57,88,75,153,236,197)(58,89,76,154,237,198)(59,90,77,155,238,199)(60,91,78,156,239,200), (1,57,122,153)(2,58,123,154)(3,59,124,155)(4,60,125,156)(5,41,126,157)(6,42,127,158)(7,43,128,159)(8,44,129,160)(9,45,130,141)(10,46,131,142)(11,47,132,143)(12,48,133,144)(13,49,134,145)(14,50,135,146)(15,51,136,147)(16,52,137,148)(17,53,138,149)(18,54,139,150)(19,55,140,151)(20,56,121,152)(21,222,178,94)(22,223,179,95)(23,224,180,96)(24,225,161,97)(25,226,162,98)(26,227,163,99)(27,228,164,100)(28,229,165,81)(29,230,166,82)(30,231,167,83)(31,232,168,84)(32,233,169,85)(33,234,170,86)(34,235,171,87)(35,236,172,88)(36,237,173,89)(37,238,174,90)(38,239,175,91)(39,240,176,92)(40,221,177,93)(61,205,183,102)(62,206,184,103)(63,207,185,104)(64,208,186,105)(65,209,187,106)(66,210,188,107)(67,211,189,108)(68,212,190,109)(69,213,191,110)(70,214,192,111)(71,215,193,112)(72,216,194,113)(73,217,195,114)(74,218,196,115)(75,219,197,116)(76,220,198,117)(77,201,199,118)(78,202,200,119)(79,203,181,120)(80,204,182,101), (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), (1,136)(2,135)(3,134)(4,133)(5,132)(6,131)(7,130)(8,129)(9,128)(10,127)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,140)(18,139)(19,138)(20,137)(21,180)(22,179)(23,178)(24,177)(25,176)(26,175)(27,174)(28,173)(29,172)(30,171)(31,170)(32,169)(33,168)(34,167)(35,166)(36,165)(37,164)(38,163)(39,162)(40,161)(41,143)(42,142)(43,141)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(61,185)(62,184)(63,183)(64,182)(65,181)(66,200)(67,199)(68,198)(69,197)(70,196)(71,195)(72,194)(73,193)(74,192)(75,191)(76,190)(77,189)(78,188)(79,187)(80,186)(81,237)(82,236)(83,235)(84,234)(85,233)(86,232)(87,231)(88,230)(89,229)(90,228)(91,227)(92,226)(93,225)(94,224)(95,223)(96,222)(97,221)(98,240)(99,239)(100,238)(101,208)(102,207)(103,206)(104,205)(105,204)(106,203)(107,202)(108,201)(109,220)(110,219)(111,218)(112,217)(113,216)(114,215)(115,214)(116,213)(117,212)(118,211)(119,210)(120,209) );
G=PermutationGroup([[(1,219,35,122,116,172),(2,220,36,123,117,173),(3,201,37,124,118,174),(4,202,38,125,119,175),(5,203,39,126,120,176),(6,204,40,127,101,177),(7,205,21,128,102,178),(8,206,22,129,103,179),(9,207,23,130,104,180),(10,208,24,131,105,161),(11,209,25,132,106,162),(12,210,26,133,107,163),(13,211,27,134,108,164),(14,212,28,135,109,165),(15,213,29,136,110,166),(16,214,30,137,111,167),(17,215,31,138,112,168),(18,216,32,139,113,169),(19,217,33,140,114,170),(20,218,34,121,115,171),(41,92,79,157,240,181),(42,93,80,158,221,182),(43,94,61,159,222,183),(44,95,62,160,223,184),(45,96,63,141,224,185),(46,97,64,142,225,186),(47,98,65,143,226,187),(48,99,66,144,227,188),(49,100,67,145,228,189),(50,81,68,146,229,190),(51,82,69,147,230,191),(52,83,70,148,231,192),(53,84,71,149,232,193),(54,85,72,150,233,194),(55,86,73,151,234,195),(56,87,74,152,235,196),(57,88,75,153,236,197),(58,89,76,154,237,198),(59,90,77,155,238,199),(60,91,78,156,239,200)], [(1,57,122,153),(2,58,123,154),(3,59,124,155),(4,60,125,156),(5,41,126,157),(6,42,127,158),(7,43,128,159),(8,44,129,160),(9,45,130,141),(10,46,131,142),(11,47,132,143),(12,48,133,144),(13,49,134,145),(14,50,135,146),(15,51,136,147),(16,52,137,148),(17,53,138,149),(18,54,139,150),(19,55,140,151),(20,56,121,152),(21,222,178,94),(22,223,179,95),(23,224,180,96),(24,225,161,97),(25,226,162,98),(26,227,163,99),(27,228,164,100),(28,229,165,81),(29,230,166,82),(30,231,167,83),(31,232,168,84),(32,233,169,85),(33,234,170,86),(34,235,171,87),(35,236,172,88),(36,237,173,89),(37,238,174,90),(38,239,175,91),(39,240,176,92),(40,221,177,93),(61,205,183,102),(62,206,184,103),(63,207,185,104),(64,208,186,105),(65,209,187,106),(66,210,188,107),(67,211,189,108),(68,212,190,109),(69,213,191,110),(70,214,192,111),(71,215,193,112),(72,216,194,113),(73,217,195,114),(74,218,196,115),(75,219,197,116),(76,220,198,117),(77,201,199,118),(78,202,200,119),(79,203,181,120),(80,204,182,101)], [(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)], [(1,136),(2,135),(3,134),(4,133),(5,132),(6,131),(7,130),(8,129),(9,128),(10,127),(11,126),(12,125),(13,124),(14,123),(15,122),(16,121),(17,140),(18,139),(19,138),(20,137),(21,180),(22,179),(23,178),(24,177),(25,176),(26,175),(27,174),(28,173),(29,172),(30,171),(31,170),(32,169),(33,168),(34,167),(35,166),(36,165),(37,164),(38,163),(39,162),(40,161),(41,143),(42,142),(43,141),(44,160),(45,159),(46,158),(47,157),(48,156),(49,155),(50,154),(51,153),(52,152),(53,151),(54,150),(55,149),(56,148),(57,147),(58,146),(59,145),(60,144),(61,185),(62,184),(63,183),(64,182),(65,181),(66,200),(67,199),(68,198),(69,197),(70,196),(71,195),(72,194),(73,193),(74,192),(75,191),(76,190),(77,189),(78,188),(79,187),(80,186),(81,237),(82,236),(83,235),(84,234),(85,233),(86,232),(87,231),(88,230),(89,229),(90,228),(91,227),(92,226),(93,225),(94,224),(95,223),(96,222),(97,221),(98,240),(99,239),(100,238),(101,208),(102,207),(103,206),(104,205),(105,204),(106,203),(107,202),(108,201),(109,220),(110,219),(111,218),(112,217),(113,216),(114,215),(115,214),(116,213),(117,212),(118,211),(119,210),(120,209)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 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 | S3 | D4 | D5 | Dic3 | D6 | D6 | C4○D4 | D10 | D10 | D20 | C4×D5 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D5×Dic3 | C2×S3×D5 | D20⋊5S3 | S3×D20 |
| kernel | Dic3×D20 | D10⋊Dic3 | Dic3×C20 | C60⋊5C4 | C2×D5×Dic3 | C6×D20 | C3×D20 | C2×D20 | C5×Dic3 | C4×Dic3 | D20 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | Dic3 | C12 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 2 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic3×D20 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 36 |
| 0 | 0 | 0 | 0 | 26 | 10 |
| 52 | 5 | 0 | 0 | 0 | 0 |
| 8 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,26,0,0,0,0,36,10],[52,8,0,0,0,0,5,9,0,0,0,0,0,0,0,60,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,28,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
Dic3×D20 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times D_{20} % in TeX
G:=Group("Dic3xD20"); // GroupNames label
G:=SmallGroup(480,501);
// by ID
G=gap.SmallGroup(480,501);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^20=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations