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## G = C7×C3⋊Dic3order 252 = 22·32·7

### Direct product of C7 and C3⋊Dic3

Aliases: C7×C3⋊Dic3, C42.7S3, C323C28, C213Dic3, (C3×C21)⋊7C4, C3⋊(C7×Dic3), C6.3(S3×C7), (C3×C6).2C14, (C3×C42).5C2, C14.2(C3⋊S3), C2.(C7×C3⋊S3), SmallGroup(252,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C7×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C42 — C7×C3⋊Dic3
 Lower central C32 — C7×C3⋊Dic3
 Upper central C1 — C14

Generators and relations for C7×C3⋊Dic3
G = < a,b,c,d | a7=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Smallest permutation representation of C7×C3⋊Dic3
Regular action on 252 points
Generators in S252
(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)(239 240 241 242 243 244 245)(246 247 248 249 250 251 252)
(1 23 79)(2 24 80)(3 25 81)(4 26 82)(5 27 83)(6 28 84)(7 22 78)(8 154 70)(9 148 64)(10 149 65)(11 150 66)(12 151 67)(13 152 68)(14 153 69)(15 246 225)(16 247 226)(17 248 227)(18 249 228)(19 250 229)(20 251 230)(21 252 231)(29 133 63)(30 127 57)(31 128 58)(32 129 59)(33 130 60)(34 131 61)(35 132 62)(36 201 210)(37 202 204)(38 203 205)(39 197 206)(40 198 207)(41 199 208)(42 200 209)(43 99 192)(44 100 193)(45 101 194)(46 102 195)(47 103 196)(48 104 190)(49 105 191)(50 162 135)(51 163 136)(52 164 137)(53 165 138)(54 166 139)(55 167 140)(56 168 134)(71 212 155)(72 213 156)(73 214 157)(74 215 158)(75 216 159)(76 217 160)(77 211 161)(85 233 177)(86 234 178)(87 235 179)(88 236 180)(89 237 181)(90 238 182)(91 232 176)(92 120 169)(93 121 170)(94 122 171)(95 123 172)(96 124 173)(97 125 174)(98 126 175)(106 145 244)(107 146 245)(108 147 239)(109 141 240)(110 142 241)(111 143 242)(112 144 243)(113 223 186)(114 224 187)(115 218 188)(116 219 189)(117 220 183)(118 221 184)(119 222 185)
(1 104 219 95 69 31)(2 105 220 96 70 32)(3 99 221 97 64 33)(4 100 222 98 65 34)(5 101 223 92 66 35)(6 102 224 93 67 29)(7 103 218 94 68 30)(8 129 24 191 183 124)(9 130 25 192 184 125)(10 131 26 193 185 126)(11 132 27 194 186 120)(12 133 28 195 187 121)(13 127 22 196 188 122)(14 128 23 190 189 123)(15 235 243 162 206 75)(16 236 244 163 207 76)(17 237 245 164 208 77)(18 238 239 165 209 71)(19 232 240 166 210 72)(20 233 241 167 204 73)(21 234 242 168 205 74)(36 213 250 176 109 139)(37 214 251 177 110 140)(38 215 252 178 111 134)(39 216 246 179 112 135)(40 217 247 180 106 136)(41 211 248 181 107 137)(42 212 249 182 108 138)(43 118 174 148 60 81)(44 119 175 149 61 82)(45 113 169 150 62 83)(46 114 170 151 63 84)(47 115 171 152 57 78)(48 116 172 153 58 79)(49 117 173 154 59 80)(50 197 159 225 87 144)(51 198 160 226 88 145)(52 199 161 227 89 146)(53 200 155 228 90 147)(54 201 156 229 91 141)(55 202 157 230 85 142)(56 203 158 231 86 143)
(1 228 95 53)(2 229 96 54)(3 230 97 55)(4 231 98 56)(5 225 92 50)(6 226 93 51)(7 227 94 52)(8 109 191 213)(9 110 192 214)(10 111 193 215)(11 112 194 216)(12 106 195 217)(13 107 196 211)(14 108 190 212)(15 169 162 83)(16 170 163 84)(17 171 164 78)(18 172 165 79)(19 173 166 80)(20 174 167 81)(21 175 168 82)(22 248 122 137)(23 249 123 138)(24 250 124 139)(25 251 125 140)(26 252 126 134)(27 246 120 135)(28 247 121 136)(29 88 224 198)(30 89 218 199)(31 90 219 200)(32 91 220 201)(33 85 221 202)(34 86 222 203)(35 87 223 197)(36 129 176 183)(37 130 177 184)(38 131 178 185)(39 132 179 186)(40 133 180 187)(41 127 181 188)(42 128 182 189)(43 73 148 241)(44 74 149 242)(45 75 150 243)(46 76 151 244)(47 77 152 245)(48 71 153 239)(49 72 154 240)(57 237 115 208)(58 238 116 209)(59 232 117 210)(60 233 118 204)(61 234 119 205)(62 235 113 206)(63 236 114 207)(64 142 99 157)(65 143 100 158)(66 144 101 159)(67 145 102 160)(68 146 103 161)(69 147 104 155)(70 141 105 156)

G:=sub<Sym(252)| (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)(239,240,241,242,243,244,245)(246,247,248,249,250,251,252), (1,23,79)(2,24,80)(3,25,81)(4,26,82)(5,27,83)(6,28,84)(7,22,78)(8,154,70)(9,148,64)(10,149,65)(11,150,66)(12,151,67)(13,152,68)(14,153,69)(15,246,225)(16,247,226)(17,248,227)(18,249,228)(19,250,229)(20,251,230)(21,252,231)(29,133,63)(30,127,57)(31,128,58)(32,129,59)(33,130,60)(34,131,61)(35,132,62)(36,201,210)(37,202,204)(38,203,205)(39,197,206)(40,198,207)(41,199,208)(42,200,209)(43,99,192)(44,100,193)(45,101,194)(46,102,195)(47,103,196)(48,104,190)(49,105,191)(50,162,135)(51,163,136)(52,164,137)(53,165,138)(54,166,139)(55,167,140)(56,168,134)(71,212,155)(72,213,156)(73,214,157)(74,215,158)(75,216,159)(76,217,160)(77,211,161)(85,233,177)(86,234,178)(87,235,179)(88,236,180)(89,237,181)(90,238,182)(91,232,176)(92,120,169)(93,121,170)(94,122,171)(95,123,172)(96,124,173)(97,125,174)(98,126,175)(106,145,244)(107,146,245)(108,147,239)(109,141,240)(110,142,241)(111,143,242)(112,144,243)(113,223,186)(114,224,187)(115,218,188)(116,219,189)(117,220,183)(118,221,184)(119,222,185), (1,104,219,95,69,31)(2,105,220,96,70,32)(3,99,221,97,64,33)(4,100,222,98,65,34)(5,101,223,92,66,35)(6,102,224,93,67,29)(7,103,218,94,68,30)(8,129,24,191,183,124)(9,130,25,192,184,125)(10,131,26,193,185,126)(11,132,27,194,186,120)(12,133,28,195,187,121)(13,127,22,196,188,122)(14,128,23,190,189,123)(15,235,243,162,206,75)(16,236,244,163,207,76)(17,237,245,164,208,77)(18,238,239,165,209,71)(19,232,240,166,210,72)(20,233,241,167,204,73)(21,234,242,168,205,74)(36,213,250,176,109,139)(37,214,251,177,110,140)(38,215,252,178,111,134)(39,216,246,179,112,135)(40,217,247,180,106,136)(41,211,248,181,107,137)(42,212,249,182,108,138)(43,118,174,148,60,81)(44,119,175,149,61,82)(45,113,169,150,62,83)(46,114,170,151,63,84)(47,115,171,152,57,78)(48,116,172,153,58,79)(49,117,173,154,59,80)(50,197,159,225,87,144)(51,198,160,226,88,145)(52,199,161,227,89,146)(53,200,155,228,90,147)(54,201,156,229,91,141)(55,202,157,230,85,142)(56,203,158,231,86,143), (1,228,95,53)(2,229,96,54)(3,230,97,55)(4,231,98,56)(5,225,92,50)(6,226,93,51)(7,227,94,52)(8,109,191,213)(9,110,192,214)(10,111,193,215)(11,112,194,216)(12,106,195,217)(13,107,196,211)(14,108,190,212)(15,169,162,83)(16,170,163,84)(17,171,164,78)(18,172,165,79)(19,173,166,80)(20,174,167,81)(21,175,168,82)(22,248,122,137)(23,249,123,138)(24,250,124,139)(25,251,125,140)(26,252,126,134)(27,246,120,135)(28,247,121,136)(29,88,224,198)(30,89,218,199)(31,90,219,200)(32,91,220,201)(33,85,221,202)(34,86,222,203)(35,87,223,197)(36,129,176,183)(37,130,177,184)(38,131,178,185)(39,132,179,186)(40,133,180,187)(41,127,181,188)(42,128,182,189)(43,73,148,241)(44,74,149,242)(45,75,150,243)(46,76,151,244)(47,77,152,245)(48,71,153,239)(49,72,154,240)(57,237,115,208)(58,238,116,209)(59,232,117,210)(60,233,118,204)(61,234,119,205)(62,235,113,206)(63,236,114,207)(64,142,99,157)(65,143,100,158)(66,144,101,159)(67,145,102,160)(68,146,103,161)(69,147,104,155)(70,141,105,156)>;

G:=Group( (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)(239,240,241,242,243,244,245)(246,247,248,249,250,251,252), (1,23,79)(2,24,80)(3,25,81)(4,26,82)(5,27,83)(6,28,84)(7,22,78)(8,154,70)(9,148,64)(10,149,65)(11,150,66)(12,151,67)(13,152,68)(14,153,69)(15,246,225)(16,247,226)(17,248,227)(18,249,228)(19,250,229)(20,251,230)(21,252,231)(29,133,63)(30,127,57)(31,128,58)(32,129,59)(33,130,60)(34,131,61)(35,132,62)(36,201,210)(37,202,204)(38,203,205)(39,197,206)(40,198,207)(41,199,208)(42,200,209)(43,99,192)(44,100,193)(45,101,194)(46,102,195)(47,103,196)(48,104,190)(49,105,191)(50,162,135)(51,163,136)(52,164,137)(53,165,138)(54,166,139)(55,167,140)(56,168,134)(71,212,155)(72,213,156)(73,214,157)(74,215,158)(75,216,159)(76,217,160)(77,211,161)(85,233,177)(86,234,178)(87,235,179)(88,236,180)(89,237,181)(90,238,182)(91,232,176)(92,120,169)(93,121,170)(94,122,171)(95,123,172)(96,124,173)(97,125,174)(98,126,175)(106,145,244)(107,146,245)(108,147,239)(109,141,240)(110,142,241)(111,143,242)(112,144,243)(113,223,186)(114,224,187)(115,218,188)(116,219,189)(117,220,183)(118,221,184)(119,222,185), (1,104,219,95,69,31)(2,105,220,96,70,32)(3,99,221,97,64,33)(4,100,222,98,65,34)(5,101,223,92,66,35)(6,102,224,93,67,29)(7,103,218,94,68,30)(8,129,24,191,183,124)(9,130,25,192,184,125)(10,131,26,193,185,126)(11,132,27,194,186,120)(12,133,28,195,187,121)(13,127,22,196,188,122)(14,128,23,190,189,123)(15,235,243,162,206,75)(16,236,244,163,207,76)(17,237,245,164,208,77)(18,238,239,165,209,71)(19,232,240,166,210,72)(20,233,241,167,204,73)(21,234,242,168,205,74)(36,213,250,176,109,139)(37,214,251,177,110,140)(38,215,252,178,111,134)(39,216,246,179,112,135)(40,217,247,180,106,136)(41,211,248,181,107,137)(42,212,249,182,108,138)(43,118,174,148,60,81)(44,119,175,149,61,82)(45,113,169,150,62,83)(46,114,170,151,63,84)(47,115,171,152,57,78)(48,116,172,153,58,79)(49,117,173,154,59,80)(50,197,159,225,87,144)(51,198,160,226,88,145)(52,199,161,227,89,146)(53,200,155,228,90,147)(54,201,156,229,91,141)(55,202,157,230,85,142)(56,203,158,231,86,143), (1,228,95,53)(2,229,96,54)(3,230,97,55)(4,231,98,56)(5,225,92,50)(6,226,93,51)(7,227,94,52)(8,109,191,213)(9,110,192,214)(10,111,193,215)(11,112,194,216)(12,106,195,217)(13,107,196,211)(14,108,190,212)(15,169,162,83)(16,170,163,84)(17,171,164,78)(18,172,165,79)(19,173,166,80)(20,174,167,81)(21,175,168,82)(22,248,122,137)(23,249,123,138)(24,250,124,139)(25,251,125,140)(26,252,126,134)(27,246,120,135)(28,247,121,136)(29,88,224,198)(30,89,218,199)(31,90,219,200)(32,91,220,201)(33,85,221,202)(34,86,222,203)(35,87,223,197)(36,129,176,183)(37,130,177,184)(38,131,178,185)(39,132,179,186)(40,133,180,187)(41,127,181,188)(42,128,182,189)(43,73,148,241)(44,74,149,242)(45,75,150,243)(46,76,151,244)(47,77,152,245)(48,71,153,239)(49,72,154,240)(57,237,115,208)(58,238,116,209)(59,232,117,210)(60,233,118,204)(61,234,119,205)(62,235,113,206)(63,236,114,207)(64,142,99,157)(65,143,100,158)(66,144,101,159)(67,145,102,160)(68,146,103,161)(69,147,104,155)(70,141,105,156) );

G=PermutationGroup([[(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),(239,240,241,242,243,244,245),(246,247,248,249,250,251,252)], [(1,23,79),(2,24,80),(3,25,81),(4,26,82),(5,27,83),(6,28,84),(7,22,78),(8,154,70),(9,148,64),(10,149,65),(11,150,66),(12,151,67),(13,152,68),(14,153,69),(15,246,225),(16,247,226),(17,248,227),(18,249,228),(19,250,229),(20,251,230),(21,252,231),(29,133,63),(30,127,57),(31,128,58),(32,129,59),(33,130,60),(34,131,61),(35,132,62),(36,201,210),(37,202,204),(38,203,205),(39,197,206),(40,198,207),(41,199,208),(42,200,209),(43,99,192),(44,100,193),(45,101,194),(46,102,195),(47,103,196),(48,104,190),(49,105,191),(50,162,135),(51,163,136),(52,164,137),(53,165,138),(54,166,139),(55,167,140),(56,168,134),(71,212,155),(72,213,156),(73,214,157),(74,215,158),(75,216,159),(76,217,160),(77,211,161),(85,233,177),(86,234,178),(87,235,179),(88,236,180),(89,237,181),(90,238,182),(91,232,176),(92,120,169),(93,121,170),(94,122,171),(95,123,172),(96,124,173),(97,125,174),(98,126,175),(106,145,244),(107,146,245),(108,147,239),(109,141,240),(110,142,241),(111,143,242),(112,144,243),(113,223,186),(114,224,187),(115,218,188),(116,219,189),(117,220,183),(118,221,184),(119,222,185)], [(1,104,219,95,69,31),(2,105,220,96,70,32),(3,99,221,97,64,33),(4,100,222,98,65,34),(5,101,223,92,66,35),(6,102,224,93,67,29),(7,103,218,94,68,30),(8,129,24,191,183,124),(9,130,25,192,184,125),(10,131,26,193,185,126),(11,132,27,194,186,120),(12,133,28,195,187,121),(13,127,22,196,188,122),(14,128,23,190,189,123),(15,235,243,162,206,75),(16,236,244,163,207,76),(17,237,245,164,208,77),(18,238,239,165,209,71),(19,232,240,166,210,72),(20,233,241,167,204,73),(21,234,242,168,205,74),(36,213,250,176,109,139),(37,214,251,177,110,140),(38,215,252,178,111,134),(39,216,246,179,112,135),(40,217,247,180,106,136),(41,211,248,181,107,137),(42,212,249,182,108,138),(43,118,174,148,60,81),(44,119,175,149,61,82),(45,113,169,150,62,83),(46,114,170,151,63,84),(47,115,171,152,57,78),(48,116,172,153,58,79),(49,117,173,154,59,80),(50,197,159,225,87,144),(51,198,160,226,88,145),(52,199,161,227,89,146),(53,200,155,228,90,147),(54,201,156,229,91,141),(55,202,157,230,85,142),(56,203,158,231,86,143)], [(1,228,95,53),(2,229,96,54),(3,230,97,55),(4,231,98,56),(5,225,92,50),(6,226,93,51),(7,227,94,52),(8,109,191,213),(9,110,192,214),(10,111,193,215),(11,112,194,216),(12,106,195,217),(13,107,196,211),(14,108,190,212),(15,169,162,83),(16,170,163,84),(17,171,164,78),(18,172,165,79),(19,173,166,80),(20,174,167,81),(21,175,168,82),(22,248,122,137),(23,249,123,138),(24,250,124,139),(25,251,125,140),(26,252,126,134),(27,246,120,135),(28,247,121,136),(29,88,224,198),(30,89,218,199),(31,90,219,200),(32,91,220,201),(33,85,221,202),(34,86,222,203),(35,87,223,197),(36,129,176,183),(37,130,177,184),(38,131,178,185),(39,132,179,186),(40,133,180,187),(41,127,181,188),(42,128,182,189),(43,73,148,241),(44,74,149,242),(45,75,150,243),(46,76,151,244),(47,77,152,245),(48,71,153,239),(49,72,154,240),(57,237,115,208),(58,238,116,209),(59,232,117,210),(60,233,118,204),(61,234,119,205),(62,235,113,206),(63,236,114,207),(64,142,99,157),(65,143,100,158),(66,144,101,159),(67,145,102,160),(68,146,103,161),(69,147,104,155),(70,141,105,156)]])

84 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 7A ··· 7F 14A ··· 14F 21A ··· 21X 28A ··· 28L 42A ··· 42X order 1 2 3 3 3 3 4 4 6 6 6 6 7 ··· 7 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 size 1 1 2 2 2 2 9 9 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C7 C14 C28 S3 Dic3 S3×C7 C7×Dic3 kernel C7×C3⋊Dic3 C3×C42 C3×C21 C3⋊Dic3 C3×C6 C32 C42 C21 C6 C3 # reps 1 1 2 6 6 12 4 4 24 24

Matrix representation of C7×C3⋊Dic3 in GL4(𝔽337) generated by

 52 0 0 0 0 52 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 336 1 0 0 336 0
,
 1 336 0 0 1 0 0 0 0 0 1 336 0 0 1 0
,
 81 121 0 0 202 256 0 0 0 0 263 210 0 0 136 74
G:=sub<GL(4,GF(337))| [52,0,0,0,0,52,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,336,336,0,0,1,0],[1,1,0,0,336,0,0,0,0,0,1,1,0,0,336,0],[81,202,0,0,121,256,0,0,0,0,263,136,0,0,210,74] >;

C7×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_7\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C7xC3:Dic3");
// GroupNames label

G:=SmallGroup(252,23);
// by ID

G=gap.SmallGroup(252,23);
# by ID

G:=PCGroup([5,-2,-7,-2,-3,-3,70,1123,4204]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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