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G = Dic3⋊D20  order 480 = 25·3·5

1st semidirect product of Dic3 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3⋊2D20, (C6×D5)⋊2D4, (C2×D20)⋊3S3, (C6×D20)⋊15C2, C3⋊4(C4⋊D20), C15⋊2(C4⋊D4), (C5×Dic3)⋊1D4, C2.17(S3×D20), C30.36(C2×D4), (C2×C20).14D6, C6.130(D4×D5), C6.13(C2×D20), C10.13(S3×D4), D10⋊2(C3⋊D4), Dic3⋊C4⋊17D5, D30⋊3C4⋊14C2, C30.56(C4○D4), (C2×C12).223D10, D10⋊Dic3⋊9C2, C5⋊1(C23.14D6), (C2×C30).99C23, (C22×D5).46D6, (C2×C60).254C22, C6.33(Q8⋊2D5), (C2×Dic3).99D10, C2.15(D20⋊S3), C10.12(D4⋊2S3), (C10×Dic3).59C22, (C2×Dic15).79C22, (C22×D15).31C22, (C2×D5×Dic3)⋊5C2, (C2×C3⋊D20)⋊1C2, (C2×C4).38(S3×D5), C2.14(D5×C3⋊D4), C10.32(C2×C3⋊D4), (D5×C2×C6).16C22, C22.168(C2×S3×D5), (C5×Dic3⋊C4)⋊17C2, (C2×C6).111(C22×D5), (C2×C10).111(C22×S3), SmallGroup(480,485)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C30 — Dic3⋊D20
Chief series
C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — Dic3⋊D20
Lower central
C15 — C2×C30 — Dic3⋊D20
Upper central
C1 — C22 — C2×C4

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, cbc-1=dbd=a3b, dcd=c-1 >

Subgroups: 1180 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C3×D5, D15, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, C5×Dic3, Dic15, C60, C6×D5, C6×D5, D30, C2×C30, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, C23.14D6, D5×Dic3, C3⋊D20, C3×D20, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C4⋊D20, D10⋊Dic3, C5×Dic3⋊C4, D30⋊3C4, C2×D5×Dic3, C2×C3⋊D20, C6×D20, Dic3⋊D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, D20, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, Q8⋊2D5, C23.14D6, C2×S3×D5, C4⋊D20, D20⋊S3, S3×D20, D5×C3⋊D4, Dic3⋊D20

Smallest permutation representation of Dic3⋊D20
►On 240 points
Generators in S240
(1 107 234 213 188 48)(2 108 235 214 189 49)(3 109 236 215 190 50)(4 110 237 216 191 51)(5 111 238 217 192 52)(6 112 239 218 193 53)(7 113 240 219 194 54)(8 114 221 220 195 55)(9 115 222 201 196 56)(10 116 223 202 197 57)(11 117 224 203 198 58)(12 118 225 204 199 59)(13 119 226 205 200 60)(14 120 227 206 181 41)(15 101 228 207 182 42)(16 102 229 208 183 43)(17 103 230 209 184 44)(18 104 231 210 185 45)(19 105 232 211 186 46)(20 106 233 212 187 47)(21 93 62 162 149 128)(22 94 63 163 150 129)(23 95 64 164 151 130)(24 96 65 165 152 131)(25 97 66 166 153 132)(26 98 67 167 154 133)(27 99 68 168 155 134)(28 100 69 169 156 135)(29 81 70 170 157 136)(30 82 71 171 158 137)(31 83 72 172 159 138)(32 84 73 173 160 139)(33 85 74 174 141 140)(34 86 75 175 142 121)(35 87 76 176 143 122)(36 88 77 177 144 123)(37 89 78 178 145 124)(38 90 79 179 146 125)(39 91 80 180 147 126)(40 92 61 161 148 127)

(1 142 213 86)(2 87 214 143)(3 144 215 88)(4 89 216 145)(5 146 217 90)(6 91 218 147)(7 148 219 92)(8 93 220 149)(9 150 201 94)(10 95 202 151)(11 152 203 96)(12 97 204 153)(13 154 205 98)(14 99 206 155)(15 156 207 100)(16 81 208 157)(17 158 209 82)(18 83 210 159)(19 160 211 84)(20 85 212 141)(21 195 162 114)(22 115 163 196)(23 197 164 116)(24 117 165 198)(25 199 166 118)(26 119 167 200)(27 181 168 120)(28 101 169 182)(29 183 170 102)(30 103 171 184)(31 185 172 104)(32 105 173 186)(33 187 174 106)(34 107 175 188)(35 189 176 108)(36 109 177 190)(37 191 178 110)(38 111 179 192)(39 193 180 112)(40 113 161 194)(41 68 227 134)(42 135 228 69)(43 70 229 136)(44 137 230 71)(45 72 231 138)(46 139 232 73)(47 74 233 140)(48 121 234 75)(49 76 235 122)(50 123 236 77)(51 78 237 124)(52 125 238 79)(53 80 239 126)(54 127 240 61)(55 62 221 128)(56 129 222 63)(57 64 223 130)(58 131 224 65)(59 66 225 132)(60 133 226 67)

(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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(141 142)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)(161 168)(162 167)(163 166)(164 165)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)(181 194)(182 193)(183 192)(184 191)(185 190)(186 189)(187 188)(195 200)(196 199)(197 198)(201 204)(202 203)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)(221 226)(222 225)(223 224)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)
 
G:=sub<Sym(240)| (1,107,234,213,188,48)(2,108,235,214,189,49)(3,109,236,215,190,50)(4,110,237,216,191,51)(5,111,238,217,192,52)(6,112,239,218,193,53)(7,113,240,219,194,54)(8,114,221,220,195,55)(9,115,222,201,196,56)(10,116,223,202,197,57)(11,117,224,203,198,58)(12,118,225,204,199,59)(13,119,226,205,200,60)(14,120,227,206,181,41)(15,101,228,207,182,42)(16,102,229,208,183,43)(17,103,230,209,184,44)(18,104,231,210,185,45)(19,105,232,211,186,46)(20,106,233,212,187,47)(21,93,62,162,149,128)(22,94,63,163,150,129)(23,95,64,164,151,130)(24,96,65,165,152,131)(25,97,66,166,153,132)(26,98,67,167,154,133)(27,99,68,168,155,134)(28,100,69,169,156,135)(29,81,70,170,157,136)(30,82,71,171,158,137)(31,83,72,172,159,138)(32,84,73,173,160,139)(33,85,74,174,141,140)(34,86,75,175,142,121)(35,87,76,176,143,122)(36,88,77,177,144,123)(37,89,78,178,145,124)(38,90,79,179,146,125)(39,91,80,180,147,126)(40,92,61,161,148,127), (1,142,213,86)(2,87,214,143)(3,144,215,88)(4,89,216,145)(5,146,217,90)(6,91,218,147)(7,148,219,92)(8,93,220,149)(9,150,201,94)(10,95,202,151)(11,152,203,96)(12,97,204,153)(13,154,205,98)(14,99,206,155)(15,156,207,100)(16,81,208,157)(17,158,209,82)(18,83,210,159)(19,160,211,84)(20,85,212,141)(21,195,162,114)(22,115,163,196)(23,197,164,116)(24,117,165,198)(25,199,166,118)(26,119,167,200)(27,181,168,120)(28,101,169,182)(29,183,170,102)(30,103,171,184)(31,185,172,104)(32,105,173,186)(33,187,174,106)(34,107,175,188)(35,189,176,108)(36,109,177,190)(37,191,178,110)(38,111,179,192)(39,193,180,112)(40,113,161,194)(41,68,227,134)(42,135,228,69)(43,70,229,136)(44,137,230,71)(45,72,231,138)(46,139,232,73)(47,74,233,140)(48,121,234,75)(49,76,235,122)(50,123,236,77)(51,78,237,124)(52,125,238,79)(53,80,239,126)(54,127,240,61)(55,62,221,128)(56,129,222,63)(57,64,223,130)(58,131,224,65)(59,66,225,132)(60,133,226,67), (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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,142)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)(161,168)(162,167)(163,166)(164,165)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,200)(196,199)(197,198)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234)>;
 
G:=Group( (1,107,234,213,188,48)(2,108,235,214,189,49)(3,109,236,215,190,50)(4,110,237,216,191,51)(5,111,238,217,192,52)(6,112,239,218,193,53)(7,113,240,219,194,54)(8,114,221,220,195,55)(9,115,222,201,196,56)(10,116,223,202,197,57)(11,117,224,203,198,58)(12,118,225,204,199,59)(13,119,226,205,200,60)(14,120,227,206,181,41)(15,101,228,207,182,42)(16,102,229,208,183,43)(17,103,230,209,184,44)(18,104,231,210,185,45)(19,105,232,211,186,46)(20,106,233,212,187,47)(21,93,62,162,149,128)(22,94,63,163,150,129)(23,95,64,164,151,130)(24,96,65,165,152,131)(25,97,66,166,153,132)(26,98,67,167,154,133)(27,99,68,168,155,134)(28,100,69,169,156,135)(29,81,70,170,157,136)(30,82,71,171,158,137)(31,83,72,172,159,138)(32,84,73,173,160,139)(33,85,74,174,141,140)(34,86,75,175,142,121)(35,87,76,176,143,122)(36,88,77,177,144,123)(37,89,78,178,145,124)(38,90,79,179,146,125)(39,91,80,180,147,126)(40,92,61,161,148,127), (1,142,213,86)(2,87,214,143)(3,144,215,88)(4,89,216,145)(5,146,217,90)(6,91,218,147)(7,148,219,92)(8,93,220,149)(9,150,201,94)(10,95,202,151)(11,152,203,96)(12,97,204,153)(13,154,205,98)(14,99,206,155)(15,156,207,100)(16,81,208,157)(17,158,209,82)(18,83,210,159)(19,160,211,84)(20,85,212,141)(21,195,162,114)(22,115,163,196)(23,197,164,116)(24,117,165,198)(25,199,166,118)(26,119,167,200)(27,181,168,120)(28,101,169,182)(29,183,170,102)(30,103,171,184)(31,185,172,104)(32,105,173,186)(33,187,174,106)(34,107,175,188)(35,189,176,108)(36,109,177,190)(37,191,178,110)(38,111,179,192)(39,193,180,112)(40,113,161,194)(41,68,227,134)(42,135,228,69)(43,70,229,136)(44,137,230,71)(45,72,231,138)(46,139,232,73)(47,74,233,140)(48,121,234,75)(49,76,235,122)(50,123,236,77)(51,78,237,124)(52,125,238,79)(53,80,239,126)(54,127,240,61)(55,62,221,128)(56,129,222,63)(57,64,223,130)(58,131,224,65)(59,66,225,132)(60,133,226,67), (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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,142)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)(161,168)(162,167)(163,166)(164,165)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,200)(196,199)(197,198)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234) );
 
G=PermutationGroup([[(1,107,234,213,188,48),(2,108,235,214,189,49),(3,109,236,215,190,50),(4,110,237,216,191,51),(5,111,238,217,192,52),(6,112,239,218,193,53),(7,113,240,219,194,54),(8,114,221,220,195,55),(9,115,222,201,196,56),(10,116,223,202,197,57),(11,117,224,203,198,58),(12,118,225,204,199,59),(13,119,226,205,200,60),(14,120,227,206,181,41),(15,101,228,207,182,42),(16,102,229,208,183,43),(17,103,230,209,184,44),(18,104,231,210,185,45),(19,105,232,211,186,46),(20,106,233,212,187,47),(21,93,62,162,149,128),(22,94,63,163,150,129),(23,95,64,164,151,130),(24,96,65,165,152,131),(25,97,66,166,153,132),(26,98,67,167,154,133),(27,99,68,168,155,134),(28,100,69,169,156,135),(29,81,70,170,157,136),(30,82,71,171,158,137),(31,83,72,172,159,138),(32,84,73,173,160,139),(33,85,74,174,141,140),(34,86,75,175,142,121),(35,87,76,176,143,122),(36,88,77,177,144,123),(37,89,78,178,145,124),(38,90,79,179,146,125),(39,91,80,180,147,126),(40,92,61,161,148,127)], [(1,142,213,86),(2,87,214,143),(3,144,215,88),(4,89,216,145),(5,146,217,90),(6,91,218,147),(7,148,219,92),(8,93,220,149),(9,150,201,94),(10,95,202,151),(11,152,203,96),(12,97,204,153),(13,154,205,98),(14,99,206,155),(15,156,207,100),(16,81,208,157),(17,158,209,82),(18,83,210,159),(19,160,211,84),(20,85,212,141),(21,195,162,114),(22,115,163,196),(23,197,164,116),(24,117,165,198),(25,199,166,118),(26,119,167,200),(27,181,168,120),(28,101,169,182),(29,183,170,102),(30,103,171,184),(31,185,172,104),(32,105,173,186),(33,187,174,106),(34,107,175,188),(35,189,176,108),(36,109,177,190),(37,191,178,110),(38,111,179,192),(39,193,180,112),(40,113,161,194),(41,68,227,134),(42,135,228,69),(43,70,229,136),(44,137,230,71),(45,72,231,138),(46,139,232,73),(47,74,233,140),(48,121,234,75),(49,76,235,122),(50,123,236,77),(51,78,237,124),(52,125,238,79),(53,80,239,126),(54,127,240,61),(55,62,221,128),(56,129,222,63),(57,64,223,130),(58,131,224,65),(59,66,225,132),(60,133,226,67)], [(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),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58),(61,68),(62,67),(63,66),(64,65),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,120),(114,119),(115,118),(116,117),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131),(141,142),(143,160),(144,159),(145,158),(146,157),(147,156),(148,155),(149,154),(150,153),(151,152),(161,168),(162,167),(163,166),(164,165),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175),(181,194),(182,193),(183,192),(184,191),(185,190),(186,189),(187,188),(195,200),(196,199),(197,198),(201,204),(202,203),(205,220),(206,219),(207,218),(208,217),(209,216),(210,215),(211,214),(212,213),(221,226),(222,225),(223,224),(227,240),(228,239),(229,238),(230,237),(231,236),(232,235),(233,234)]])
 

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F12A12B15A15B20A20B20C20D20E···20L30A···30F60A···60H
order12222222344444455666666610···10121215152020202020···2030···3060···60
size111110102060246612303022222202020202···24444444412···124···44···4

60 irreducible representations

dim111111122222222222444444444
type+++++++++++++++++-+++++
imageC1C2C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10C3⋊D4D20S3×D4D4⋊2S3S3×D5D4×D5Q8⋊2D5C2×S3×D5D20⋊S3S3×D20D5×C3⋊D4
kernelDic3⋊D20D10⋊Dic3C5×Dic3⋊C4D30⋊3C4C2×D5×Dic3C2×C3⋊D20C6×D20C2×D20C5×Dic3C6×D5Dic3⋊C4C2×C20C22×D5C30C2×Dic3C2×C12D10Dic3C10C10C2×C4C6C6C22C2C2C2
# reps111112112221224248112222444

Matrix representation of Dic3⋊D20 ►in GL8(𝔽61)

10000000
01000000
00100000
00010000
000060100
000060000
000000600
000000060
,
10000000
01000000
006000000
000600000
00000100
00001000
000000110
0000001650
,
3910000000
4322000000
004310000
006000000
000060000
000006000
0000002246
0000001239
,
3910000000
3122000000
001430000
000600000
000060000
000006000
0000002246
0000002039

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,16,0,0,0,0,0,0,0,50],[39,43,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,22,12,0,0,0,0,0,0,46,39],[39,31,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,22,20,0,0,0,0,0,0,46,39] >;
 

Dic3⋊D20 in GAP, Magma, Sage, TeX 

{\rm Dic}_3\rtimes D_{20}
 
% in TeX
 
G:=Group("Dic3:D20");
 
// GroupNames label
 
G:=SmallGroup(480,485);
 
// by ID
 
G=gap.SmallGroup(480,485);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,58,1356,18822]);
 
// 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,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^-1>;
 
// generators/relations
 

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