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## G = C7×D8.C4order 448 = 26·7

### Direct product of C7 and D8.C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×D8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C56 — C7×C8.C4 — C7×D8.C4
 Lower central C1 — C2 — C4 — C8 — C7×D8.C4
 Upper central C1 — C28 — C2×C28 — C2×C56 — C7×D8.C4

Generators and relations for C7×D8.C4
G = < a,b,c,d | a7=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >

Smallest permutation representation of C7×D8.C4
On 224 points
Generators in S224
(1 51 43 35 27 19 11)(2 52 44 36 28 20 12)(3 53 45 37 29 21 13)(4 54 46 38 30 22 14)(5 55 47 39 31 23 15)(6 56 48 40 32 24 16)(7 49 41 33 25 17 9)(8 50 42 34 26 18 10)(57 105 97 89 81 73 65)(58 106 98 90 82 74 66)(59 107 99 91 83 75 67)(60 108 100 92 84 76 68)(61 109 101 93 85 77 69)(62 110 102 94 86 78 70)(63 111 103 95 87 79 71)(64 112 104 96 88 80 72)(113 161 153 145 137 129 121)(114 162 154 146 138 130 122)(115 163 155 147 139 131 123)(116 164 156 148 140 132 124)(117 165 157 149 141 133 125)(118 166 158 150 142 134 126)(119 167 159 151 143 135 127)(120 168 160 152 144 136 128)(169 217 209 201 193 185 177)(170 218 210 202 194 186 178)(171 219 211 203 195 187 179)(172 220 212 204 196 188 180)(173 221 213 205 197 189 181)(174 222 214 206 198 190 182)(175 223 215 207 199 191 183)(176 224 216 208 200 192 184)
(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 60)(2 59)(3 58)(4 57)(5 64)(6 63)(7 62)(8 61)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 72)(16 71)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 80)(24 79)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 88)(32 87)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 96)(40 95)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 104)(48 103)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 112)(56 111)(113 173)(114 172)(115 171)(116 170)(117 169)(118 176)(119 175)(120 174)(121 181)(122 180)(123 179)(124 178)(125 177)(126 184)(127 183)(128 182)(129 189)(130 188)(131 187)(132 186)(133 185)(134 192)(135 191)(136 190)(137 197)(138 196)(139 195)(140 194)(141 193)(142 200)(143 199)(144 198)(145 205)(146 204)(147 203)(148 202)(149 201)(150 208)(151 207)(152 206)(153 213)(154 212)(155 211)(156 210)(157 209)(158 216)(159 215)(160 214)(161 221)(162 220)(163 219)(164 218)(165 217)(166 224)(167 223)(168 222)
(1 115 3 113 5 119 7 117)(2 114 4 120 6 118 8 116)(9 125 11 123 13 121 15 127)(10 124 12 122 14 128 16 126)(17 133 19 131 21 129 23 135)(18 132 20 130 22 136 24 134)(25 141 27 139 29 137 31 143)(26 140 28 138 30 144 32 142)(33 149 35 147 37 145 39 151)(34 148 36 146 38 152 40 150)(41 157 43 155 45 153 47 159)(42 156 44 154 46 160 48 158)(49 165 51 163 53 161 55 167)(50 164 52 162 54 168 56 166)(57 169 63 171 61 173 59 175)(58 176 64 170 62 172 60 174)(65 177 71 179 69 181 67 183)(66 184 72 178 70 180 68 182)(73 185 79 187 77 189 75 191)(74 192 80 186 78 188 76 190)(81 193 87 195 85 197 83 199)(82 200 88 194 86 196 84 198)(89 201 95 203 93 205 91 207)(90 208 96 202 94 204 92 206)(97 209 103 211 101 213 99 215)(98 216 104 210 102 212 100 214)(105 217 111 219 109 221 107 223)(106 224 112 218 110 220 108 222)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 8E 8F 14A ··· 14F 14G ··· 14L 14M ··· 14R 16A ··· 16H 28A ··· 28L 28M ··· 28R 28S ··· 28X 56A ··· 56X 56Y ··· 56AJ 112A ··· 112AV order 1 2 2 2 4 4 4 4 7 ··· 7 8 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 8 1 1 2 8 1 ··· 1 2 2 2 2 8 8 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2 8 ··· 8 2 ··· 2

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 D4 D4 D8 SD16 C7×D4 C7×D4 D8.C4 C7×D8 C7×SD16 C7×D8.C4 kernel C7×D8.C4 C7×C8.C4 C2×C112 C7×C4○D8 C7×D8 C7×Q16 D8.C4 C8.C4 C2×C16 C4○D8 D8 Q16 C56 C2×C28 C28 C2×C14 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 2 2 6 6 6 6 12 12 1 1 2 2 6 6 8 12 12 48

Matrix representation of C7×D8.C4 in GL2(𝔽113) generated by

 109 0 0 109
,
 18 0 0 44
,
 0 15 98 0
,
 0 35 65 0
G:=sub<GL(2,GF(113))| [109,0,0,109],[18,0,0,44],[0,98,15,0],[0,65,35,0] >;

C7×D8.C4 in GAP, Magma, Sage, TeX

C_7\times D_8.C_4
% in TeX

G:=Group("C7xD8.C4");
// GroupNames label

G:=SmallGroup(448,163);
// by ID

G=gap.SmallGroup(448,163);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,1970,360,172,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^5*c>;
// generators/relations

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