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## G = C2×C16⋊D7order 448 = 26·7

### Direct product of C2 and C16⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C16⋊D7
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D7×C2×C8 — C2×C16⋊D7
 Lower central C7 — C14 — C2×C16⋊D7
 Upper central C1 — C2×C8 — C2×C16

Generators and relations for C2×C16⋊D7
G = < a,b,c,d | a2=b16=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b9, dcd=c-1 >

Subgroups: 308 in 90 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C16, C16, C2×C8, C2×C8, C22×C4, Dic7, C28, D14, D14, C2×C14, C2×C16, C2×C16, M5(2), C22×C8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C2×M5(2), C7⋊C16, C112, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C16⋊D7, C2×C7⋊C16, C2×C112, D7×C2×C8, C2×C16⋊D7
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, D14, M5(2), C22×C8, C4×D7, C22×D7, C2×M5(2), C8×D7, C2×C4×D7, C16⋊D7, D7×C2×C8, C2×C16⋊D7

Smallest permutation representation of C2×C16⋊D7
On 224 points
Generators in S224
(1 175)(2 176)(3 161)(4 162)(5 163)(6 164)(7 165)(8 166)(9 167)(10 168)(11 169)(12 170)(13 171)(14 172)(15 173)(16 174)(17 117)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 113)(30 114)(31 115)(32 116)(33 96)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 129)(62 130)(63 131)(64 132)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 97)(79 98)(80 99)(145 216)(146 217)(147 218)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 209)(155 210)(156 211)(157 212)(158 213)(159 214)(160 215)(177 203)(178 204)(179 205)(180 206)(181 207)(182 208)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)(191 201)(192 202)
(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 23 38 139 160 71 177)(2 24 39 140 145 72 178)(3 25 40 141 146 73 179)(4 26 41 142 147 74 180)(5 27 42 143 148 75 181)(6 28 43 144 149 76 182)(7 29 44 129 150 77 183)(8 30 45 130 151 78 184)(9 31 46 131 152 79 185)(10 32 47 132 153 80 186)(11 17 48 133 154 65 187)(12 18 33 134 155 66 188)(13 19 34 135 156 67 189)(14 20 35 136 157 68 190)(15 21 36 137 158 69 191)(16 22 37 138 159 70 192)(49 209 100 197 169 117 95)(50 210 101 198 170 118 96)(51 211 102 199 171 119 81)(52 212 103 200 172 120 82)(53 213 104 201 173 121 83)(54 214 105 202 174 122 84)(55 215 106 203 175 123 85)(56 216 107 204 176 124 86)(57 217 108 205 161 125 87)(58 218 109 206 162 126 88)(59 219 110 207 163 127 89)(60 220 111 208 164 128 90)(61 221 112 193 165 113 91)(62 222 97 194 166 114 92)(63 223 98 195 167 115 93)(64 224 99 196 168 116 94)
(1 177)(2 186)(3 179)(4 188)(5 181)(6 190)(7 183)(8 192)(9 185)(10 178)(11 187)(12 180)(13 189)(14 182)(15 191)(16 184)(17 65)(18 74)(19 67)(20 76)(21 69)(22 78)(23 71)(24 80)(25 73)(26 66)(27 75)(28 68)(29 77)(30 70)(31 79)(32 72)(33 147)(34 156)(35 149)(36 158)(37 151)(38 160)(39 153)(40 146)(41 155)(42 148)(43 157)(44 150)(45 159)(46 152)(47 145)(48 154)(50 58)(52 60)(54 62)(56 64)(81 211)(82 220)(83 213)(84 222)(85 215)(86 224)(87 217)(88 210)(89 219)(90 212)(91 221)(92 214)(93 223)(94 216)(95 209)(96 218)(97 122)(98 115)(99 124)(100 117)(101 126)(102 119)(103 128)(104 121)(105 114)(106 123)(107 116)(108 125)(109 118)(110 127)(111 120)(112 113)(130 138)(132 140)(134 142)(136 144)(161 205)(162 198)(163 207)(164 200)(165 193)(166 202)(167 195)(168 204)(169 197)(170 206)(171 199)(172 208)(173 201)(174 194)(175 203)(176 196)

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A ··· 8H 8I 8J 8K 8L 14A ··· 14I 16A ··· 16H 16I ··· 16P 28A ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 ··· 8 8 8 8 8 14 ··· 14 16 ··· 16 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 1 1 14 14 1 1 1 1 14 14 2 2 2 1 ··· 1 14 14 14 14 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 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D7 D14 D14 M5(2) C4×D7 C4×D7 C8×D7 C8×D7 C16⋊D7 kernel C2×C16⋊D7 C16⋊D7 C2×C7⋊C16 C2×C112 D7×C2×C8 C8×D7 C2×C7⋊C8 C2×C4×D7 C4×D7 C2×Dic7 C22×D7 C2×C16 C16 C2×C8 C14 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 8 4 4 3 6 3 8 6 6 12 12 48

Matrix representation of C2×C16⋊D7 in GL3(𝔽113) generated by

 112 0 0 0 112 0 0 0 112
,
 18 0 0 0 72 16 0 97 41
,
 1 0 0 0 0 112 0 1 9
,
 112 0 0 0 9 80 0 112 104
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[18,0,0,0,72,97,0,16,41],[1,0,0,0,0,1,0,112,9],[112,0,0,0,9,112,0,80,104] >;

C2×C16⋊D7 in GAP, Magma, Sage, TeX

C_2\times C_{16}\rtimes D_7
% in TeX

G:=Group("C2xC16:D7");
// GroupNames label

G:=SmallGroup(448,434);
// by ID

G=gap.SmallGroup(448,434);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,58,80,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^16=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽