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## G = C7×C8.D4order 448 = 26·7

### Direct product of C7 and C8.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×C8.D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — Q8×C14 — C14×Q16 — C7×C8.D4
 Lower central C1 — C2 — C2×C4 — C7×C8.D4
 Upper central C1 — C2×C14 — C22×C28 — C7×C8.D4

Generators and relations for C7×C8.D4
G = < a,b,c,d | a7=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 186 in 110 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C56, C56, C2×C28, C2×C28, C7×Q8, C22×C14, C8.D4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C7×Q16, C22×C28, Q8×C14, C7×Q8⋊C4, C7×C4.Q8, C7×C22⋊Q8, C14×M4(2), C14×Q16, C7×C8.D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C8.C22, C7×D4, C22×C14, C8.D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×C8.C22, C7×C8.D4

Smallest permutation representation of C7×C8.D4
On 224 points
Generators in S224
(1 69 16 83 27 75 19)(2 70 9 84 28 76 20)(3 71 10 85 29 77 21)(4 72 11 86 30 78 22)(5 65 12 87 31 79 23)(6 66 13 88 32 80 24)(7 67 14 81 25 73 17)(8 68 15 82 26 74 18)(33 117 105 49 97 41 89)(34 118 106 50 98 42 90)(35 119 107 51 99 43 91)(36 120 108 52 100 44 92)(37 113 109 53 101 45 93)(38 114 110 54 102 46 94)(39 115 111 55 103 47 95)(40 116 112 56 104 48 96)(57 220 168 212 160 204 152)(58 221 161 213 153 205 145)(59 222 162 214 154 206 146)(60 223 163 215 155 207 147)(61 224 164 216 156 208 148)(62 217 165 209 157 201 149)(63 218 166 210 158 202 150)(64 219 167 211 159 203 151)(121 174 197 137 189 129 181)(122 175 198 138 190 130 182)(123 176 199 139 191 131 183)(124 169 200 140 192 132 184)(125 170 193 141 185 133 177)(126 171 194 142 186 134 178)(127 172 195 143 187 135 179)(128 173 196 144 188 136 180)
(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)
(1 149 39 123)(2 152 40 126)(3 147 33 121)(4 150 34 124)(5 145 35 127)(6 148 36 122)(7 151 37 125)(8 146 38 128)(9 220 112 194)(10 223 105 197)(11 218 106 200)(12 221 107 195)(13 224 108 198)(14 219 109 193)(15 222 110 196)(16 217 111 199)(17 203 93 177)(18 206 94 180)(19 201 95 183)(20 204 96 178)(21 207 89 181)(22 202 90 184)(23 205 91 179)(24 208 92 182)(25 211 101 185)(26 214 102 188)(27 209 103 191)(28 212 104 186)(29 215 97 189)(30 210 98 192)(31 213 99 187)(32 216 100 190)(41 129 77 155)(42 132 78 158)(43 135 79 153)(44 130 80 156)(45 133 73 159)(46 136 74 154)(47 131 75 157)(48 134 76 160)(49 137 85 163)(50 140 86 166)(51 143 87 161)(52 138 88 164)(53 141 81 167)(54 144 82 162)(55 139 83 165)(56 142 84 168)(57 116 171 70)(58 119 172 65)(59 114 173 68)(60 117 174 71)(61 120 175 66)(62 115 176 69)(63 118 169 72)(64 113 170 67)
(1 127 5 123)(2 126 6 122)(3 125 7 121)(4 124 8 128)(9 194 13 198)(10 193 14 197)(11 200 15 196)(12 199 16 195)(17 181 21 177)(18 180 22 184)(19 179 23 183)(20 178 24 182)(25 189 29 185)(26 188 30 192)(27 187 31 191)(28 186 32 190)(33 151 37 147)(34 150 38 146)(35 149 39 145)(36 148 40 152)(41 159 45 155)(42 158 46 154)(43 157 47 153)(44 156 48 160)(49 167 53 163)(50 166 54 162)(51 165 55 161)(52 164 56 168)(57 120 61 116)(58 119 62 115)(59 118 63 114)(60 117 64 113)(65 176 69 172)(66 175 70 171)(67 174 71 170)(68 173 72 169)(73 129 77 133)(74 136 78 132)(75 135 79 131)(76 134 80 130)(81 137 85 141)(82 144 86 140)(83 143 87 139)(84 142 88 138)(89 203 93 207)(90 202 94 206)(91 201 95 205)(92 208 96 204)(97 211 101 215)(98 210 102 214)(99 209 103 213)(100 216 104 212)(105 219 109 223)(106 218 110 222)(107 217 111 221)(108 224 112 220)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14X 28A ··· 28L 28M ··· 28R 28S ··· 28AP 56A ··· 56X order 1 2 2 2 2 4 4 4 4 4 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 2 2 4 8 8 8 8 1 ··· 1 4 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 D4 C4○D4 C7×D4 C7×D4 C7×D4 C7×C4○D4 C8.C22 C7×C8.C22 kernel C7×C8.D4 C7×Q8⋊C4 C7×C4.Q8 C7×C22⋊Q8 C14×M4(2) C14×Q16 C8.D4 Q8⋊C4 C4.Q8 C22⋊Q8 C2×M4(2) C2×Q16 C56 C2×C28 C22×C14 C28 C8 C2×C4 C23 C4 C14 C2 # reps 1 2 1 2 1 1 6 12 6 12 6 6 2 1 1 2 12 6 6 12 2 12

Matrix representation of C7×C8.D4 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 109 0 0 0 0 0 0 109 0 0 0 0 0 0 109 0 0 0 0 0 0 109
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 37 20 112 111 0 0 0 112 0 0 0 0 87 31 65 93
,
 99 111 0 0 0 0 42 14 0 0 0 0 0 0 108 58 0 0 0 0 58 5 0 0 0 0 72 100 53 103 0 0 112 71 55 60
,
 99 111 0 0 0 0 41 14 0 0 0 0 0 0 108 58 0 0 0 0 58 5 0 0 0 0 41 13 60 10 0 0 72 28 58 53

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,109,0,0,0,0,0,0,109,0,0,0,0,0,0,109,0,0,0,0,0,0,109],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,37,0,87,0,0,0,20,112,31,0,0,1,112,0,65,0,0,0,111,0,93],[99,42,0,0,0,0,111,14,0,0,0,0,0,0,108,58,72,112,0,0,58,5,100,71,0,0,0,0,53,55,0,0,0,0,103,60],[99,41,0,0,0,0,111,14,0,0,0,0,0,0,108,58,41,72,0,0,58,5,13,28,0,0,0,0,60,58,0,0,0,0,10,53] >;

C7×C8.D4 in GAP, Magma, Sage, TeX

C_7\times C_8.D_4
% in TeX

G:=Group("C7xC8.D4");
// GroupNames label

G:=SmallGroup(448,878);
// by ID

G=gap.SmallGroup(448,878);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1968,2438,2403,9804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations

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