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## G = Dic3⋊4D20order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic3⋊4D20
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — Dic3⋊4D20
 Lower central C15 — C30 — Dic3⋊4D20
 Upper central C1 — C22 — C2×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, 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, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, D30, D30, C2×C30, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C2×C4×D5, C2×D20, Dic34D4, D5×Dic3, D30.C2, C3⋊D20, C6×Dic5, C10×Dic3, 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, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C22×S3, C4×D4, C4×D5, D20, 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
(1 162 24 108 67 148)(2 149 68 109 25 163)(3 164 26 110 69 150)(4 151 70 111 27 165)(5 166 28 112 71 152)(6 153 72 113 29 167)(7 168 30 114 73 154)(8 155 74 115 31 169)(9 170 32 116 75 156)(10 157 76 117 33 171)(11 172 34 118 77 158)(12 159 78 119 35 173)(13 174 36 120 79 160)(14 141 80 101 37 175)(15 176 38 102 61 142)(16 143 62 103 39 177)(17 178 40 104 63 144)(18 145 64 105 21 179)(19 180 22 106 65 146)(20 147 66 107 23 161)(41 214 229 191 91 134)(42 135 92 192 230 215)(43 216 231 193 93 136)(44 137 94 194 232 217)(45 218 233 195 95 138)(46 139 96 196 234 219)(47 220 235 197 97 140)(48 121 98 198 236 201)(49 202 237 199 99 122)(50 123 100 200 238 203)(51 204 239 181 81 124)(52 125 82 182 240 205)(53 206 221 183 83 126)(54 127 84 184 222 207)(55 208 223 185 85 128)(56 129 86 186 224 209)(57 210 225 187 87 130)(58 131 88 188 226 211)(59 212 227 189 89 132)(60 133 90 190 228 213)
(1 198 108 48)(2 199 109 49)(3 200 110 50)(4 181 111 51)(5 182 112 52)(6 183 113 53)(7 184 114 54)(8 185 115 55)(9 186 116 56)(10 187 117 57)(11 188 118 58)(12 189 119 59)(13 190 120 60)(14 191 101 41)(15 192 102 42)(16 193 103 43)(17 194 104 44)(18 195 105 45)(19 196 106 46)(20 197 107 47)(21 138 145 233)(22 139 146 234)(23 140 147 235)(24 121 148 236)(25 122 149 237)(26 123 150 238)(27 124 151 239)(28 125 152 240)(29 126 153 221)(30 127 154 222)(31 128 155 223)(32 129 156 224)(33 130 157 225)(34 131 158 226)(35 132 159 227)(36 133 160 228)(37 134 141 229)(38 135 142 230)(39 136 143 231)(40 137 144 232)(61 215 176 92)(62 216 177 93)(63 217 178 94)(64 218 179 95)(65 219 180 96)(66 220 161 97)(67 201 162 98)(68 202 163 99)(69 203 164 100)(70 204 165 81)(71 205 166 82)(72 206 167 83)(73 207 168 84)(74 208 169 85)(75 209 170 86)(76 210 171 87)(77 211 172 88)(78 212 173 89)(79 213 174 90)(80 214 175 91)
(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 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(95 100)(96 99)(97 98)(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)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 160)(156 159)(157 158)(161 162)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(181 194)(182 193)(183 192)(184 191)(185 190)(186 189)(187 188)(195 200)(196 199)(197 198)(201 220)(202 219)(203 218)(204 217)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)(221 230)(222 229)(223 228)(224 227)(225 226)(231 240)(232 239)(233 238)(234 237)(235 236)

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

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

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 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 C4 S3 D4 D5 D6 D6 D6 C4○D4 D10 D10 C4×S3 C4×D5 D20 C4○D20 S3×D4 D4⋊2S3 S3×D5 C2×S3×D5 C4×S3×D5 S3×D20 Dic5.D6 kernel Dic3⋊4D20 C6.Dic10 C3×D10⋊C4 Dic3×C20 D30⋊3C4 C2×D5×Dic3 C2×D30.C2 C2×C3⋊D20 C3⋊D20 D10⋊C4 C5×Dic3 C4×Dic3 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 D10 Dic3 Dic3 C6 C10 C10 C2×C4 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 1 1 2 4 2 4 8 8 8 1 1 2 2 4 4 4

Matrix representation of Dic34D20 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 60 1 0 0 0 0 60 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 53 0 0 0 0 36 17
,
 0 1 0 0 0 0 60 17 0 0 0 0 0 0 34 25 0 0 0 0 36 57 0 0 0 0 0 0 44 53 0 0 0 0 36 17
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 36 27 0 0 0 0 4 25 0 0 0 0 0 0 44 53 0 0 0 0 36 17

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

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