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## G = Dic3.D20order 480 = 25·3·5

### 4th non-split extension by Dic3 of D20 acting via D20/C20=C2

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 — D10⋊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=1, b2=d2=a3, bab-1=cac-1=dad-1=a-1, bc=cb, dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 940 in 152 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], C5, S3, C6 [×3], C6, C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×3], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×4], D10 [×6], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30 [×3], C4.4D4, Dic10 [×2], D20 [×2], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C4×Dic3, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15, C60, C6×D5 [×3], D30 [×3], C2×C30, D10⋊C4, D10⋊C4 [×3], C4×C20, C2×Dic10, C2×D20, C23.11D6, C3⋊D20 [×2], C15⋊Q8 [×2], C6×Dic5, C10×Dic3 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C4.D20, D10⋊Dic3, D304C4, C3×D10⋊C4, Dic3×C20, D303C4, C2×C3⋊D20, C2×C15⋊Q8, Dic3.D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C4.4D4, D20 [×2], C22×D5, C4○D12, S3×D4, D42S3, S3×D5, C2×D20, C4○D20 [×2], C23.11D6, C2×S3×D5, C4.D20, D6.D10, S3×D20, Dic5.D6, Dic3.D20

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 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 3 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 20 60 2 2 2 6 6 6 6 20 60 2 2 2 2 2 20 20 2 ··· 2 4 4 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 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 C2 C2 S3 D4 D5 D6 D6 D6 C4○D4 D10 D10 D20 C4○D12 C4○D20 S3×D4 D4⋊2S3 S3×D5 C2×S3×D5 D6.D10 S3×D20 Dic5.D6 kernel Dic3.D20 D10⋊Dic3 D30⋊4C4 C3×D10⋊C4 Dic3×C20 D30⋊3C4 C2×C3⋊D20 C2×C15⋊Q8 D10⋊C4 C5×Dic3 C4×Dic3 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 Dic3 C10 C6 C10 C10 C2×C4 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 4 2 8 4 16 1 1 2 2 4 4 4

Matrix representation of Dic3.D20 in GL4(𝔽61) generated by

 0 60 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 52 43 0 0 52 9 0 0 0 0 1 0 0 0 0 1
,
 23 46 0 0 23 38 0 0 0 0 57 25 0 0 36 34
,
 50 0 0 0 11 11 0 0 0 0 57 25 0 0 36 4
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,1,0,0,0,0,1],[52,52,0,0,43,9,0,0,0,0,1,0,0,0,0,1],[23,23,0,0,46,38,0,0,0,0,57,36,0,0,25,34],[50,11,0,0,0,11,0,0,0,0,57,36,0,0,25,4] >;

Dic3.D20 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_{20}
% in TeX

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

G:=SmallGroup(480,429);
// by ID

G=gap.SmallGroup(480,429);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,58,1356,18822]);
// Polycyclic

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

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