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## G = D8×C2×C14order 448 = 26·7

### Direct product of C2×C14 and D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C2×C14
 Chief series C1 — C2 — C4 — C28 — C7×D4 — C7×D8 — C14×D8 — D8×C2×C14
 Lower central C1 — C2 — C4 — D8×C2×C14
 Upper central C1 — C22×C14 — C22×C28 — D8×C2×C14

Generators and relations for D8×C2×C14
G = < a,b,c,d | a2=b14=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 658 in 338 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, D4, D4, C23, C23, C14, C14, C14, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C28, C28, C2×C14, C2×C14, C22×C8, C2×D8, C22×D4, C56, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C22×D8, C2×C56, C7×D8, C22×C28, D4×C14, D4×C14, C23×C14, C22×C56, C14×D8, D4×C2×C14, D8×C2×C14
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C24, C2×C14, C2×D8, C22×D4, C7×D4, C22×C14, C22×D8, C7×D8, D4×C14, C23×C14, C14×D8, D4×C2×C14, D8×C2×C14

Smallest permutation representation of D8×C2×C14
On 224 points
Generators in S224
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 113)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 155)(82 156)(83 157)(84 158)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)(175 208)(176 209)(177 210)(178 197)(179 198)(180 199)(181 200)(182 201)(183 218)(184 219)(185 220)(186 221)(187 222)(188 223)(189 224)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)
(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 93 168 189 57 198 152 108)(2 94 155 190 58 199 153 109)(3 95 156 191 59 200 154 110)(4 96 157 192 60 201 141 111)(5 97 158 193 61 202 142 112)(6 98 159 194 62 203 143 99)(7 85 160 195 63 204 144 100)(8 86 161 196 64 205 145 101)(9 87 162 183 65 206 146 102)(10 88 163 184 66 207 147 103)(11 89 164 185 67 208 148 104)(12 90 165 186 68 209 149 105)(13 91 166 187 69 210 150 106)(14 92 167 188 70 197 151 107)(15 171 54 38 120 127 72 216)(16 172 55 39 121 128 73 217)(17 173 56 40 122 129 74 218)(18 174 43 41 123 130 75 219)(19 175 44 42 124 131 76 220)(20 176 45 29 125 132 77 221)(21 177 46 30 126 133 78 222)(22 178 47 31 113 134 79 223)(23 179 48 32 114 135 80 224)(24 180 49 33 115 136 81 211)(25 181 50 34 116 137 82 212)(26 182 51 35 117 138 83 213)(27 169 52 36 118 139 84 214)(28 170 53 37 119 140 71 215)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 15)(29 202)(30 203)(31 204)(32 205)(33 206)(34 207)(35 208)(36 209)(37 210)(38 197)(39 198)(40 199)(41 200)(42 201)(43 154)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 166)(72 167)(73 168)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)(82 163)(83 164)(84 165)(85 223)(86 224)(87 211)(88 212)(89 213)(90 214)(91 215)(92 216)(93 217)(94 218)(95 219)(96 220)(97 221)(98 222)(99 177)(100 178)(101 179)(102 180)(103 181)(104 182)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)(127 188)(128 189)(129 190)(130 191)(131 192)(132 193)(133 194)(134 195)(135 196)(136 183)(137 184)(138 185)(139 186)(140 187)

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

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

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

196 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 7A ··· 7F 8A ··· 8H 14A ··· 14AP 14AQ ··· 14CL 28A ··· 28X 56A ··· 56AV order 1 2 ··· 2 2 ··· 2 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 ··· 1 4 ··· 4 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 D4 D8 C7×D4 C7×D4 C7×D8 kernel D8×C2×C14 C22×C56 C14×D8 D4×C2×C14 C22×D8 C22×C8 C2×D8 C22×D4 C2×C28 C22×C14 C2×C14 C2×C4 C23 C22 # reps 1 1 12 2 6 6 72 12 3 1 8 18 6 48

Matrix representation of D8×C2×C14 in GL4(𝔽113) generated by

 112 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 112 0 0 0 0 97 0 0 0 0 97
,
 112 0 0 0 0 1 0 0 0 0 0 51 0 0 31 51
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 112
G:=sub<GL(4,GF(113))| [112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,97,0,0,0,0,97],[112,0,0,0,0,1,0,0,0,0,0,31,0,0,51,51],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,112] >;

D8×C2×C14 in GAP, Magma, Sage, TeX

D_8\times C_2\times C_{14}
% in TeX

G:=Group("D8xC2xC14");
// GroupNames label

G:=SmallGroup(448,1352);
// by ID

G=gap.SmallGroup(448,1352);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,14117,7068,124]);
// Polycyclic

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

׿
×
𝔽