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## G = C2×D14⋊C8order 448 = 26·7

### Direct product of C2 and D14⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×D14⋊C8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — D7×C22×C4 — C2×D14⋊C8
 Lower central C7 — C14 — C2×D14⋊C8
 Upper central C1 — C22×C4 — C22×C8

Generators and relations for C2×D14⋊C8
G = < a,b,c,d | a2=b14=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b7c >

Subgroups: 932 in 202 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C23, D7, C14, C14, C2×C8, C2×C8, C22×C4, C22×C4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22⋊C8, C22×C8, C22×C8, C23×C4, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C22⋊C8, C2×C7⋊C8, C2×C7⋊C8, C2×C56, C2×C56, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, D14⋊C8, C22×C7⋊C8, C22×C56, D7×C22×C4, C2×D14⋊C8
Quotients:

Smallest permutation representation of C2×D14⋊C8
On 224 points
Generators in S224
(1 110)(2 111)(3 112)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 69)(16 70)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 96)(30 97)(31 98)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 209)(44 210)(45 197)(46 198)(47 199)(48 200)(49 201)(50 202)(51 203)(52 204)(53 205)(54 206)(55 207)(56 208)(71 122)(72 123)(73 124)(74 125)(75 126)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(127 173)(128 174)(129 175)(130 176)(131 177)(132 178)(133 179)(134 180)(135 181)(136 182)(137 169)(138 170)(139 171)(140 172)(141 211)(142 212)(143 213)(144 214)(145 215)(146 216)(147 217)(148 218)(149 219)(150 220)(151 221)(152 222)(153 223)(154 224)(155 191)(156 192)(157 193)(158 194)(159 195)(160 196)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)
(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)
(1 115)(2 114)(3 113)(4 126)(5 125)(6 124)(7 123)(8 122)(9 121)(10 120)(11 119)(12 118)(13 117)(14 116)(15 223)(16 222)(17 221)(18 220)(19 219)(20 218)(21 217)(22 216)(23 215)(24 214)(25 213)(26 212)(27 211)(28 224)(29 130)(30 129)(31 128)(32 127)(33 140)(34 139)(35 138)(36 137)(37 136)(38 135)(39 134)(40 133)(41 132)(42 131)(43 190)(44 189)(45 188)(46 187)(47 186)(48 185)(49 184)(50 183)(51 196)(52 195)(53 194)(54 193)(55 192)(56 191)(57 151)(58 150)(59 149)(60 148)(61 147)(62 146)(63 145)(64 144)(65 143)(66 142)(67 141)(68 154)(69 153)(70 152)(71 103)(72 102)(73 101)(74 100)(75 99)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)(85 173)(86 172)(87 171)(88 170)(89 169)(90 182)(91 181)(92 180)(93 179)(94 178)(95 177)(96 176)(97 175)(98 174)(155 208)(156 207)(157 206)(158 205)(159 204)(160 203)(161 202)(162 201)(163 200)(164 199)(165 198)(166 197)(167 210)(168 209)
(1 16 37 205 72 153 176 195)(2 17 38 206 73 154 177 196)(3 18 39 207 74 141 178 183)(4 19 40 208 75 142 179 184)(5 20 41 209 76 143 180 185)(6 21 42 210 77 144 181 186)(7 22 29 197 78 145 182 187)(8 23 30 198 79 146 169 188)(9 24 31 199 80 147 170 189)(10 25 32 200 81 148 171 190)(11 26 33 201 82 149 172 191)(12 27 34 202 83 150 173 192)(13 28 35 203 84 151 174 193)(14 15 36 204 71 152 175 194)(43 113 213 134 163 100 60 94)(44 114 214 135 164 101 61 95)(45 115 215 136 165 102 62 96)(46 116 216 137 166 103 63 97)(47 117 217 138 167 104 64 98)(48 118 218 139 168 105 65 85)(49 119 219 140 155 106 66 86)(50 120 220 127 156 107 67 87)(51 121 221 128 157 108 68 88)(52 122 222 129 158 109 69 89)(53 123 223 130 159 110 70 90)(54 124 224 131 160 111 57 91)(55 125 211 132 161 112 58 92)(56 126 212 133 162 99 59 93)

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

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

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

136 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14U 28A ··· 28X 56A ··· 56AV order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 7 7 7 8 ··· 8 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 ··· 1 14 14 14 14 1 ··· 1 14 14 14 14 2 2 2 2 ··· 2 14 ··· 14 2 ··· 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D4 D7 M4(2) D14 D14 C4×D7 D28 C7⋊D4 C4×D7 C8×D7 C8⋊D7 kernel C2×D14⋊C8 D14⋊C8 C22×C7⋊C8 C22×C56 D7×C22×C4 C2×C4×D7 C22×Dic7 C23×D7 C22×D7 C2×C28 C22×C8 C2×C14 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 4 2 2 16 4 3 4 6 3 6 12 12 6 24 24

Matrix representation of C2×D14⋊C8 in GL5(𝔽113)

 112 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 112 0 0 0 0 0 112
,
 1 0 0 0 0 0 112 0 0 0 0 0 112 0 0 0 0 0 24 80 0 0 0 24 0
,
 1 0 0 0 0 0 112 0 0 0 0 0 1 0 0 0 0 0 10 104 0 0 0 11 103
,
 1 0 0 0 0 0 0 112 0 0 0 98 0 0 0 0 0 0 95 0 0 0 0 0 95

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,24,24,0,0,0,80,0],[1,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,10,11,0,0,0,104,103],[1,0,0,0,0,0,0,98,0,0,0,112,0,0,0,0,0,0,95,0,0,0,0,0,95] >;

C2×D14⋊C8 in GAP, Magma, Sage, TeX

C_2\times D_{14}\rtimes C_8
% in TeX

G:=Group("C2xD14:C8");
// GroupNames label

G:=SmallGroup(448,642);
// by ID

G=gap.SmallGroup(448,642);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,136,18822]);
// Polycyclic

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

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