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## G = C32×Dic7order 252 = 22·32·7

### Direct product of C32 and Dic7

Aliases: C32×Dic7, C216C12, C42.12C6, (C3×C21)⋊5C4, C73(C3×C12), (C3×C6).3D7, C6.4(C3×D7), C2.(C32×D7), C14.3(C3×C6), (C3×C42).3C2, SmallGroup(252,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C32×Dic7
 Chief series C1 — C7 — C14 — C42 — C3×C42 — C32×Dic7
 Lower central C7 — C32×Dic7
 Upper central C1 — C3×C6

Generators and relations for C32×Dic7
G = < a,b,c,d | a3=b3=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of C32×Dic7
Regular action on 252 points
Generators in S252
(1 125 65)(2 126 66)(3 113 67)(4 114 68)(5 115 69)(6 116 70)(7 117 57)(8 118 58)(9 119 59)(10 120 60)(11 121 61)(12 122 62)(13 123 63)(14 124 64)(15 95 80)(16 96 81)(17 97 82)(18 98 83)(19 85 84)(20 86 71)(21 87 72)(22 88 73)(23 89 74)(24 90 75)(25 91 76)(26 92 77)(27 93 78)(28 94 79)(29 112 51)(30 99 52)(31 100 53)(32 101 54)(33 102 55)(34 103 56)(35 104 43)(36 105 44)(37 106 45)(38 107 46)(39 108 47)(40 109 48)(41 110 49)(42 111 50)(127 239 183)(128 240 184)(129 241 185)(130 242 186)(131 243 187)(132 244 188)(133 245 189)(134 246 190)(135 247 191)(136 248 192)(137 249 193)(138 250 194)(139 251 195)(140 252 196)(141 218 204)(142 219 205)(143 220 206)(144 221 207)(145 222 208)(146 223 209)(147 224 210)(148 211 197)(149 212 198)(150 213 199)(151 214 200)(152 215 201)(153 216 202)(154 217 203)(155 232 169)(156 233 170)(157 234 171)(158 235 172)(159 236 173)(160 237 174)(161 238 175)(162 225 176)(163 226 177)(164 227 178)(165 228 179)(166 229 180)(167 230 181)(168 231 182)
(1 29 28)(2 30 15)(3 31 16)(4 32 17)(5 33 18)(6 34 19)(7 35 20)(8 36 21)(9 37 22)(10 38 23)(11 39 24)(12 40 25)(13 41 26)(14 42 27)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)(85 116 103)(86 117 104)(87 118 105)(88 119 106)(89 120 107)(90 121 108)(91 122 109)(92 123 110)(93 124 111)(94 125 112)(95 126 99)(96 113 100)(97 114 101)(98 115 102)(127 155 148)(128 156 149)(129 157 150)(130 158 151)(131 159 152)(132 160 153)(133 161 154)(134 162 141)(135 163 142)(136 164 143)(137 165 144)(138 166 145)(139 167 146)(140 168 147)(169 197 183)(170 198 184)(171 199 185)(172 200 186)(173 201 187)(174 202 188)(175 203 189)(176 204 190)(177 205 191)(178 206 192)(179 207 193)(180 208 194)(181 209 195)(182 210 196)(211 239 232)(212 240 233)(213 241 234)(214 242 235)(215 243 236)(216 244 237)(217 245 238)(218 246 225)(219 247 226)(220 248 227)(221 249 228)(222 250 229)(223 251 230)(224 252 231)
(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)
(1 127 8 134)(2 140 9 133)(3 139 10 132)(4 138 11 131)(5 137 12 130)(6 136 13 129)(7 135 14 128)(15 147 22 154)(16 146 23 153)(17 145 24 152)(18 144 25 151)(19 143 26 150)(20 142 27 149)(21 141 28 148)(29 155 36 162)(30 168 37 161)(31 167 38 160)(32 166 39 159)(33 165 40 158)(34 164 41 157)(35 163 42 156)(43 177 50 170)(44 176 51 169)(45 175 52 182)(46 174 53 181)(47 173 54 180)(48 172 55 179)(49 171 56 178)(57 191 64 184)(58 190 65 183)(59 189 66 196)(60 188 67 195)(61 187 68 194)(62 186 69 193)(63 185 70 192)(71 205 78 198)(72 204 79 197)(73 203 80 210)(74 202 81 209)(75 201 82 208)(76 200 83 207)(77 199 84 206)(85 220 92 213)(86 219 93 212)(87 218 94 211)(88 217 95 224)(89 216 96 223)(90 215 97 222)(91 214 98 221)(99 231 106 238)(100 230 107 237)(101 229 108 236)(102 228 109 235)(103 227 110 234)(104 226 111 233)(105 225 112 232)(113 251 120 244)(114 250 121 243)(115 249 122 242)(116 248 123 241)(117 247 124 240)(118 246 125 239)(119 245 126 252)

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

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

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

90 conjugacy classes

 class 1 2 3A ··· 3H 4A 4B 6A ··· 6H 7A 7B 7C 12A ··· 12P 14A 14B 14C 21A ··· 21X 42A ··· 42X order 1 2 3 ··· 3 4 4 6 ··· 6 7 7 7 12 ··· 12 14 14 14 21 ··· 21 42 ··· 42 size 1 1 1 ··· 1 7 7 1 ··· 1 2 2 2 7 ··· 7 2 2 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 D7 Dic7 C3×D7 C3×Dic7 kernel C32×Dic7 C3×C42 C3×Dic7 C3×C21 C42 C21 C3×C6 C32 C6 C3 # reps 1 1 8 2 8 16 3 3 24 24

Matrix representation of C32×Dic7 in GL4(𝔽337) generated by

 1 0 0 0 0 208 0 0 0 0 1 0 0 0 0 1
,
 208 0 0 0 0 208 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 336 0 0 229 109
,
 1 0 0 0 0 336 0 0 0 0 260 83 0 0 26 77
G:=sub<GL(4,GF(337))| [1,0,0,0,0,208,0,0,0,0,1,0,0,0,0,1],[208,0,0,0,0,208,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,229,0,0,336,109],[1,0,0,0,0,336,0,0,0,0,260,26,0,0,83,77] >;

C32×Dic7 in GAP, Magma, Sage, TeX

C_3^2\times {\rm Dic}_7
% in TeX

G:=Group("C3^2xDic7");
// GroupNames label

G:=SmallGroup(252,20);
// by ID

G=gap.SmallGroup(252,20);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-7,90,5404]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^14=1,d^2=c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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