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G = Dic3⋊D20order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic32D20, (C6×D5)⋊2D4, (C2×D20)⋊3S3, (C6×D20)⋊15C2, C34(C4⋊D20), C152(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), D102(C3⋊D4), Dic3⋊C417D5, D303C414C2, C30.56(C4○D4), (C2×C12).223D10, D10⋊Dic39C2, C51(C23.14D6), (C2×C30).99C23, (C22×D5).46D6, (C2×C60).254C22, C6.33(Q82D5), (C2×Dic3).99D10, C2.15(D20⋊S3), C10.12(D42S3), (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

C1C2×C30 — Dic3⋊D20
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — Dic3⋊D20
C15C2×C30 — Dic3⋊D20
C1C22C2×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 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3, C6 [×3], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×4], C10 [×3], Dic3 [×2], Dic3 [×2], C12, D6 [×3], C2×C6, C2×C6 [×7], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×4], D10 [×2], D10 [×8], C2×C10, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C3×D5 [×3], D15, C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5 [×2], C22×D5, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], C5×Dic3, Dic15, C60, C6×D5 [×2], C6×D5 [×5], D30 [×3], C2×C30, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20 [×2], C23.14D6, D5×Dic3 [×2], C3⋊D20 [×4], C3×D20 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, D5×C2×C6 [×2], C22×D15, C4⋊D20, D10⋊Dic3, C5×Dic3⋊C4, D303C4, C2×D5×Dic3, C2×C3⋊D20 [×2], C6×D20, Dic3⋊D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊D4, D20 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, Q82D5, 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 81 181 129 76 217)(2 82 182 130 77 218)(3 83 183 131 78 219)(4 84 184 132 79 220)(5 85 185 133 80 201)(6 86 186 134 61 202)(7 87 187 135 62 203)(8 88 188 136 63 204)(9 89 189 137 64 205)(10 90 190 138 65 206)(11 91 191 139 66 207)(12 92 192 140 67 208)(13 93 193 121 68 209)(14 94 194 122 69 210)(15 95 195 123 70 211)(16 96 196 124 71 212)(17 97 197 125 72 213)(18 98 198 126 73 214)(19 99 199 127 74 215)(20 100 200 128 75 216)(21 175 52 113 141 221)(22 176 53 114 142 222)(23 177 54 115 143 223)(24 178 55 116 144 224)(25 179 56 117 145 225)(26 180 57 118 146 226)(27 161 58 119 147 227)(28 162 59 120 148 228)(29 163 60 101 149 229)(30 164 41 102 150 230)(31 165 42 103 151 231)(32 166 43 104 152 232)(33 167 44 105 153 233)(34 168 45 106 154 234)(35 169 46 107 155 235)(36 170 47 108 156 236)(37 171 48 109 157 237)(38 172 49 110 158 238)(39 173 50 111 159 239)(40 174 51 112 160 240)
(1 147 129 161)(2 162 130 148)(3 149 131 163)(4 164 132 150)(5 151 133 165)(6 166 134 152)(7 153 135 167)(8 168 136 154)(9 155 137 169)(10 170 138 156)(11 157 139 171)(12 172 140 158)(13 159 121 173)(14 174 122 160)(15 141 123 175)(16 176 124 142)(17 143 125 177)(18 178 126 144)(19 145 127 179)(20 180 128 146)(21 95 113 70)(22 71 114 96)(23 97 115 72)(24 73 116 98)(25 99 117 74)(26 75 118 100)(27 81 119 76)(28 77 120 82)(29 83 101 78)(30 79 102 84)(31 85 103 80)(32 61 104 86)(33 87 105 62)(34 63 106 88)(35 89 107 64)(36 65 108 90)(37 91 109 66)(38 67 110 92)(39 93 111 68)(40 69 112 94)(41 184 230 220)(42 201 231 185)(43 186 232 202)(44 203 233 187)(45 188 234 204)(46 205 235 189)(47 190 236 206)(48 207 237 191)(49 192 238 208)(50 209 239 193)(51 194 240 210)(52 211 221 195)(53 196 222 212)(54 213 223 197)(55 198 224 214)(56 215 225 199)(57 200 226 216)(58 217 227 181)(59 182 228 218)(60 219 229 183)
(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 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 140)(138 139)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 160)(154 159)(155 158)(156 157)(161 180)(162 179)(163 178)(164 177)(165 176)(166 175)(167 174)(168 173)(169 172)(170 171)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 212)(202 211)(203 210)(204 209)(205 208)(206 207)(213 220)(214 219)(215 218)(216 217)(221 232)(222 231)(223 230)(224 229)(225 228)(226 227)(233 240)(234 239)(235 238)(236 237)

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

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

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

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×D4D42S3S3×D5D4×D5Q82D5C2×S3×D5D20⋊S3S3×D20D5×C3⋊D4
kernelDic3⋊D20D10⋊Dic3C5×Dic3⋊C4D303C4C2×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

׿
×
𝔽