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## G = C8⋊Dic7⋊C2order 448 = 26·7

### 13rd semidirect product of C8⋊Dic7 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C8⋊Dic7⋊C2
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C4×D7 — C4⋊C4⋊7D7 — C8⋊Dic7⋊C2
 Lower central C7 — C14 — C2×C28 — C8⋊Dic7⋊C2
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for C8⋊Dic7⋊C2
G = < a,b,c,d | a8=b14=d2=1, c2=b7, ab=ba, cac-1=a3, dad=a-1b7, cbc-1=b-1, bd=db, dcd=b7c >

Subgroups: 564 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C23.19D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×C7⋊D4, D4×C14, C28.Q8, C8⋊Dic7, D14⋊C8, D4⋊Dic7, C7×D4⋊C4, C4⋊C47D7, C282D4, C8⋊Dic7⋊C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8⋊C22, C22×D7, C23.19D4, C4○D28, D4×D7, D42D7, D14.D4, D8⋊D7, SD163D7, C8⋊Dic7⋊C2

Smallest permutation representation of C8⋊Dic7⋊C2
On 224 points
Generators in S224
(1 157 169 113 105 42 74 130)(2 158 170 114 106 29 75 131)(3 159 171 115 107 30 76 132)(4 160 172 116 108 31 77 133)(5 161 173 117 109 32 78 134)(6 162 174 118 110 33 79 135)(7 163 175 119 111 34 80 136)(8 164 176 120 112 35 81 137)(9 165 177 121 99 36 82 138)(10 166 178 122 100 37 83 139)(11 167 179 123 101 38 84 140)(12 168 180 124 102 39 71 127)(13 155 181 125 103 40 72 128)(14 156 182 126 104 41 73 129)(15 206 194 141 94 65 219 50)(16 207 195 142 95 66 220 51)(17 208 196 143 96 67 221 52)(18 209 183 144 97 68 222 53)(19 210 184 145 98 69 223 54)(20 197 185 146 85 70 224 55)(21 198 186 147 86 57 211 56)(22 199 187 148 87 58 212 43)(23 200 188 149 88 59 213 44)(24 201 189 150 89 60 214 45)(25 202 190 151 90 61 215 46)(26 203 191 152 91 62 216 47)(27 204 192 153 92 63 217 48)(28 205 193 154 93 64 218 49)
(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)
(1 19 8 26)(2 18 9 25)(3 17 10 24)(4 16 11 23)(5 15 12 22)(6 28 13 21)(7 27 14 20)(29 53 36 46)(30 52 37 45)(31 51 38 44)(32 50 39 43)(33 49 40 56)(34 48 41 55)(35 47 42 54)(57 135 64 128)(58 134 65 127)(59 133 66 140)(60 132 67 139)(61 131 68 138)(62 130 69 137)(63 129 70 136)(71 187 78 194)(72 186 79 193)(73 185 80 192)(74 184 81 191)(75 183 82 190)(76 196 83 189)(77 195 84 188)(85 111 92 104)(86 110 93 103)(87 109 94 102)(88 108 95 101)(89 107 96 100)(90 106 97 99)(91 105 98 112)(113 210 120 203)(114 209 121 202)(115 208 122 201)(116 207 123 200)(117 206 124 199)(118 205 125 198)(119 204 126 197)(141 168 148 161)(142 167 149 160)(143 166 150 159)(144 165 151 158)(145 164 152 157)(146 163 153 156)(147 162 154 155)(169 223 176 216)(170 222 177 215)(171 221 178 214)(172 220 179 213)(173 219 180 212)(174 218 181 211)(175 217 182 224)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 15)(12 16)(13 17)(14 18)(29 153)(30 154)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 167)(44 168)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 161)(52 162)(53 163)(54 164)(55 165)(56 166)(57 122)(58 123)(59 124)(60 125)(61 126)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 121)(71 195)(72 196)(73 183)(74 184)(75 185)(76 186)(77 187)(78 188)(79 189)(80 190)(81 191)(82 192)(83 193)(84 194)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(127 200)(128 201)(129 202)(130 203)(131 204)(132 205)(133 206)(134 207)(135 208)(136 209)(137 210)(138 197)(139 198)(140 199)(169 223)(170 224)(171 211)(172 212)(173 213)(174 214)(175 215)(176 216)(177 217)(178 218)(179 219)(180 220)(181 221)(182 222)

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 8 28 2 2 4 4 14 14 28 28 56 2 2 2 4 4 28 28 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 D14 C4○D8 C4○D28 C8⋊C22 D4⋊2D7 D4×D7 D8⋊D7 SD16⋊3D7 kernel C8⋊Dic7⋊C2 C28.Q8 C8⋊Dic7 D14⋊C8 D4⋊Dic7 C7×D4⋊C4 C4⋊C4⋊7D7 C28⋊2D4 C2×Dic7 C22×D7 D4⋊C4 C28 C4⋊C4 C2×C8 C2×D4 C14 C4 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 3 4 3 3 3 4 12 1 3 3 6 6

Matrix representation of C8⋊Dic7⋊C2 in GL4(𝔽113) generated by

 69 0 0 0 80 18 0 0 0 0 17 46 0 0 33 96
,
 112 0 0 0 0 112 0 0 0 0 10 1 0 0 13 24
,
 103 77 0 0 75 10 0 0 0 0 106 31 0 0 64 7
,
 10 36 0 0 82 103 0 0 0 0 84 101 0 0 70 29
G:=sub<GL(4,GF(113))| [69,80,0,0,0,18,0,0,0,0,17,33,0,0,46,96],[112,0,0,0,0,112,0,0,0,0,10,13,0,0,1,24],[103,75,0,0,77,10,0,0,0,0,106,64,0,0,31,7],[10,82,0,0,36,103,0,0,0,0,84,70,0,0,101,29] >;

C8⋊Dic7⋊C2 in GAP, Magma, Sage, TeX

C_8\rtimes {\rm Dic}_7\rtimes C_2
% in TeX

G:=Group("C8:Dic7:C2");
// GroupNames label

G:=SmallGroup(448,313);
// by ID

G=gap.SmallGroup(448,313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,64,254,219,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^14=d^2=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^3,d*a*d=a^-1*b^7,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^7*c>;
// generators/relations

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