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G = Dic34D20order 480 = 25·3·5

1st semidirect product of Dic3 and D20 acting through Inn(Dic3)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic34D20, C32(C4×D20), C155(C4×D4), D308(C2×C4), D103(C4×S3), C3⋊D203C4, C2.1(S3×D20), Dic33(C4×D5), (C5×Dic3)⋊6D4, C6.12(C2×D20), C10.12(S3×D4), C30.34(C2×D4), (C4×Dic3)⋊14D5, (C2×C20).263D6, D10⋊C420S3, D303C431C2, (Dic3×C20)⋊24C2, C6.69(C4○D20), (C2×C12).189D10, C52(Dic34D4), C30.47(C22×C4), (C2×C30).85C23, C6.Dic1013C2, (C2×Dic5).99D6, (C22×D5).43D6, C30.111(C4○D4), (C2×C60).407C22, C10.43(D42S3), (C2×Dic3).177D10, C2.3(Dic5.D6), (C6×Dic5).50C22, (C2×Dic15).71C22, (C22×D15).27C22, (C10×Dic3).175C22, C2.18(C4×S3×D5), C6.15(C2×C4×D5), (C6×D5)⋊1(C2×C4), C10.47(S3×C2×C4), (C2×D5×Dic3)⋊1C2, (C2×C4).75(S3×D5), C22.40(C2×S3×D5), (C2×D30.C2)⋊2C2, (C2×C3⋊D20).5C2, (D5×C2×C6).12C22, (C5×Dic3)⋊14(C2×C4), (C3×D10⋊C4)⋊31C2, (C2×C6).97(C22×D5), (C2×C10).97(C22×S3), SmallGroup(480,471)

Series: Derived Chief Lower central Upper central

C1C30 — Dic34D20
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — Dic34D20
C15C30 — Dic34D20
C1C22C2×C4

Generators and relations for Dic34D20
 G = < a,b,c,d | a6=c20=d2=1, b2=a3, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1036 in 188 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×4], Dic3, C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×5], D10 [×2], D10 [×6], C2×C10, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C2×C12, C22×S3, C22×C6, C3×D5 [×2], D15 [×2], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×4], C3×Dic5, Dic15, C60, C6×D5 [×2], C6×D5 [×2], D30 [×2], D30 [×2], C2×C30, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C2×C4×D5 [×2], C2×D20, Dic34D4, D5×Dic3 [×2], D30.C2 [×2], C3⋊D20 [×4], C6×Dic5, C10×Dic3 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C4×D20, C6.Dic10, C3×D10⋊C4, Dic3×C20, D303C4, C2×D5×Dic3, C2×D30.C2, C2×C3⋊D20, Dic34D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C22×S3, C4×D4, C4×D5 [×2], D20 [×2], C22×D5, S3×C2×C4, S3×D4, D42S3, S3×D5, C2×C4×D5, C2×D20, C4○D20, Dic34D4, C2×S3×D5, C4×D20, C4×S3×D5, S3×D20, Dic5.D6, Dic34D20

Smallest permutation representation of Dic34D20
On 240 points
Generators in S240
(1 226 171 118 193 208)(2 209 194 119 172 227)(3 228 173 120 195 210)(4 211 196 101 174 229)(5 230 175 102 197 212)(6 213 198 103 176 231)(7 232 177 104 199 214)(8 215 200 105 178 233)(9 234 179 106 181 216)(10 217 182 107 180 235)(11 236 161 108 183 218)(12 219 184 109 162 237)(13 238 163 110 185 220)(14 201 186 111 164 239)(15 240 165 112 187 202)(16 203 188 113 166 221)(17 222 167 114 189 204)(18 205 190 115 168 223)(19 224 169 116 191 206)(20 207 192 117 170 225)(21 140 70 49 146 83)(22 84 147 50 71 121)(23 122 72 51 148 85)(24 86 149 52 73 123)(25 124 74 53 150 87)(26 88 151 54 75 125)(27 126 76 55 152 89)(28 90 153 56 77 127)(29 128 78 57 154 91)(30 92 155 58 79 129)(31 130 80 59 156 93)(32 94 157 60 61 131)(33 132 62 41 158 95)(34 96 159 42 63 133)(35 134 64 43 160 97)(36 98 141 44 65 135)(37 136 66 45 142 99)(38 100 143 46 67 137)(39 138 68 47 144 81)(40 82 145 48 69 139)
(1 32 118 60)(2 33 119 41)(3 34 120 42)(4 35 101 43)(5 36 102 44)(6 37 103 45)(7 38 104 46)(8 39 105 47)(9 40 106 48)(10 21 107 49)(11 22 108 50)(12 23 109 51)(13 24 110 52)(14 25 111 53)(15 26 112 54)(16 27 113 55)(17 28 114 56)(18 29 115 57)(19 30 116 58)(20 31 117 59)(61 208 94 171)(62 209 95 172)(63 210 96 173)(64 211 97 174)(65 212 98 175)(66 213 99 176)(67 214 100 177)(68 215 81 178)(69 216 82 179)(70 217 83 180)(71 218 84 161)(72 219 85 162)(73 220 86 163)(74 201 87 164)(75 202 88 165)(76 203 89 166)(77 204 90 167)(78 205 91 168)(79 206 92 169)(80 207 93 170)(121 183 147 236)(122 184 148 237)(123 185 149 238)(124 186 150 239)(125 187 151 240)(126 188 152 221)(127 189 153 222)(128 190 154 223)(129 191 155 224)(130 192 156 225)(131 193 157 226)(132 194 158 227)(133 195 159 228)(134 196 160 229)(135 197 141 230)(136 198 142 231)(137 199 143 232)(138 200 144 233)(139 181 145 234)(140 182 146 235)
(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 22)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(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)(181 184)(182 183)(185 200)(186 199)(187 198)(188 197)(189 196)(190 195)(191 194)(192 193)(201 214)(202 213)(203 212)(204 211)(205 210)(206 209)(207 208)(215 220)(216 219)(217 218)(221 230)(222 229)(223 228)(224 227)(225 226)(231 240)(232 239)(233 238)(234 237)(235 236)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A···20H20I···20X30A···30F60A···60H
order122222223444444444444556666610···1012121212151520···2020···2030···3060···60
size111110103030222333366101030302222220202···2442020442···26···64···44···4

78 irreducible representations

dim11111111122222222222224444444
type++++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6D6C4○D4D10D10C4×S3C4×D5D20C4○D20S3×D4D42S3S3×D5C2×S3×D5C4×S3×D5S3×D20Dic5.D6
kernelDic34D20C6.Dic10C3×D10⋊C4Dic3×C20D303C4C2×D5×Dic3C2×D30.C2C2×C3⋊D20C3⋊D20D10⋊C4C5×Dic3C4×Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12D10Dic3Dic3C6C10C10C2×C4C22C2C2C2
# reps11111111812211124248881122444

Matrix representation of Dic34D20 in GL6(𝔽61)

6000000
0600000
001000
000100
0000601
0000600
,
5000000
0500000
0060000
0006000
00004453
00003617
,
010000
60170000
00342500
00365700
00004453
00003617
,
010000
100000
00362700
0042500
00004453
00003617

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,36,0,0,0,0,53,17],[0,60,0,0,0,0,1,17,0,0,0,0,0,0,34,36,0,0,0,0,25,57,0,0,0,0,0,0,44,36,0,0,0,0,53,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,4,0,0,0,0,27,25,0,0,0,0,0,0,44,36,0,0,0,0,53,17] >;

Dic34D20 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_4D_{20}
% in TeX

G:=Group("Dic3:4D20");
// GroupNames label

G:=SmallGroup(480,471);
// by ID

G=gap.SmallGroup(480,471);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^20=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽