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## G = C28.439(C2×D4)  order 448 = 26·7

### 31st non-split extension by C28 of C2×D4 acting via C2×D4/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.439(C2×D4)
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C23.21D14 — C28.439(C2×D4)
 Lower central C7 — C2×C14 — C28.439(C2×D4)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C28.439(C2×D4)
G = < a,b,c,d | a4=b14=1, c2=b7, d4=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b7c >

Subgroups: 356 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C42.6C22, C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C22×C28, Dic7⋊C8, C22×C7⋊C8, C23.21D14, C14×M4(2), C28.439(C2×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C8○D4, Dic14, C4×D7, C7⋊D4, C22×D7, C42.6C22, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, D28.C4, C2×Dic7⋊C4, C28.439(C2×D4)

Smallest permutation representation of C28.439(C2×D4)
On 224 points
Generators in S224
(1 54 79 223)(2 55 80 224)(3 56 81 211)(4 43 82 212)(5 44 83 213)(6 45 84 214)(7 46 71 215)(8 47 72 216)(9 48 73 217)(10 49 74 218)(11 50 75 219)(12 51 76 220)(13 52 77 221)(14 53 78 222)(15 111 96 210)(16 112 97 197)(17 99 98 198)(18 100 85 199)(19 101 86 200)(20 102 87 201)(21 103 88 202)(22 104 89 203)(23 105 90 204)(24 106 91 205)(25 107 92 206)(26 108 93 207)(27 109 94 208)(28 110 95 209)(29 128 154 165)(30 129 141 166)(31 130 142 167)(32 131 143 168)(33 132 144 155)(34 133 145 156)(35 134 146 157)(36 135 147 158)(37 136 148 159)(38 137 149 160)(39 138 150 161)(40 139 151 162)(41 140 152 163)(42 127 153 164)(57 182 120 189)(58 169 121 190)(59 170 122 191)(60 171 123 192)(61 172 124 193)(62 173 125 194)(63 174 126 195)(64 175 113 196)(65 176 114 183)(66 177 115 184)(67 178 116 185)(68 179 117 186)(69 180 118 187)(70 181 119 188)
(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)
(1 152 8 145)(2 151 9 144)(3 150 10 143)(4 149 11 142)(5 148 12 141)(6 147 13 154)(7 146 14 153)(15 173 22 180)(16 172 23 179)(17 171 24 178)(18 170 25 177)(19 169 26 176)(20 182 27 175)(21 181 28 174)(29 84 36 77)(30 83 37 76)(31 82 38 75)(32 81 39 74)(33 80 40 73)(34 79 41 72)(35 78 42 71)(43 137 50 130)(44 136 51 129)(45 135 52 128)(46 134 53 127)(47 133 54 140)(48 132 55 139)(49 131 56 138)(57 109 64 102)(58 108 65 101)(59 107 66 100)(60 106 67 99)(61 105 68 112)(62 104 69 111)(63 103 70 110)(85 191 92 184)(86 190 93 183)(87 189 94 196)(88 188 95 195)(89 187 96 194)(90 186 97 193)(91 185 98 192)(113 201 120 208)(114 200 121 207)(115 199 122 206)(116 198 123 205)(117 197 124 204)(118 210 125 203)(119 209 126 202)(155 224 162 217)(156 223 163 216)(157 222 164 215)(158 221 165 214)(159 220 166 213)(160 219 167 212)(161 218 168 211)
(1 125 54 173 79 62 223 194)(2 126 55 174 80 63 224 195)(3 113 56 175 81 64 211 196)(4 114 43 176 82 65 212 183)(5 115 44 177 83 66 213 184)(6 116 45 178 84 67 214 185)(7 117 46 179 71 68 215 186)(8 118 47 180 72 69 216 187)(9 119 48 181 73 70 217 188)(10 120 49 182 74 57 218 189)(11 121 50 169 75 58 219 190)(12 122 51 170 76 59 220 191)(13 123 52 171 77 60 221 192)(14 124 53 172 78 61 222 193)(15 41 111 163 96 152 210 140)(16 42 112 164 97 153 197 127)(17 29 99 165 98 154 198 128)(18 30 100 166 85 141 199 129)(19 31 101 167 86 142 200 130)(20 32 102 168 87 143 201 131)(21 33 103 155 88 144 202 132)(22 34 104 156 89 145 203 133)(23 35 105 157 90 146 204 134)(24 36 106 158 91 147 205 135)(25 37 107 159 92 148 206 136)(26 38 108 160 93 149 207 137)(27 39 109 161 94 150 208 138)(28 40 110 162 95 151 209 139)

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

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

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

88 conjugacy classes

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

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D7 D14 D14 C8○D4 Dic14 C4×D7 C7⋊D4 C4×D7 D28.C4 kernel C28.439(C2×D4) Dic7⋊C8 C22×C7⋊C8 C23.21D14 C14×M4(2) C4⋊Dic7 C23.D7 C2×C28 C2×C28 C2×M4(2) C2×C8 C22×C4 C14 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 4 2 2 3 6 3 8 12 6 12 6 12

Matrix representation of C28.439(C2×D4) in GL6(𝔽113)

 98 0 0 0 0 0 112 15 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 1 24 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 18 25 0 0 0 0 82 95 0 0 0 0 0 0 77 23 0 0 0 0 96 36 0 0 0 0 0 0 31 18 0 0 0 0 47 82
,
 15 2 0 0 0 0 106 98 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 107 26 0 0 0 0 3 6

G:=sub<GL(6,GF(113))| [98,112,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,112,24,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[18,82,0,0,0,0,25,95,0,0,0,0,0,0,77,96,0,0,0,0,23,36,0,0,0,0,0,0,31,47,0,0,0,0,18,82],[15,106,0,0,0,0,2,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,107,3,0,0,0,0,26,6] >;

C28.439(C2×D4) in GAP, Magma, Sage, TeX

C_{28}._{439}(C_2\times D_4)
% in TeX

G:=Group("C28.439(C2xD4)");
// GroupNames label

G:=SmallGroup(448,653);
// by ID

G=gap.SmallGroup(448,653);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,422,387,58,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^14=1,c^2=b^7,d^4=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations

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