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## G = C7×D4.C8order 448 = 26·7

### Direct product of C7 and D4.C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×D4.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C56 — C2×C112 — C7×D4.C8
 Lower central C1 — C2 — C4 — C7×D4.C8
 Upper central C1 — C56 — C2×C56 — C7×D4.C8

Generators and relations for C7×D4.C8
G = < a,b,c,d | a7=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >

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

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

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

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

196 conjugacy classes

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

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C2 C4 C4 C7 C8 C8 C14 C14 C14 C28 C28 C56 C56 D4 M4(2) C7×D4 D4.C8 C7×M4(2) C7×D4.C8 kernel C7×D4.C8 C2×C112 C7×M5(2) C7×C8○D4 C7×M4(2) C7×C4○D4 D4.C8 C7×D4 C7×Q8 C2×C16 M5(2) C8○D4 M4(2) C4○D4 D4 Q8 C56 C2×C14 C8 C7 C22 C1 # reps 1 1 1 1 2 2 6 4 4 6 6 6 12 12 24 24 2 2 12 8 12 48

Matrix representation of C7×D4.C8 in GL2(𝔽113) generated by

 49 0 0 49
,
 15 0 0 98
,
 0 98 15 0
,
 78 0 0 73
G:=sub<GL(2,GF(113))| [49,0,0,49],[15,0,0,98],[0,15,98,0],[78,0,0,73] >;

C7×D4.C8 in GAP, Magma, Sage, TeX

C_7\times D_4.C_8
% in TeX

G:=Group("C7xD4.C8");
// GroupNames label

G:=SmallGroup(448,154);
// by ID

G=gap.SmallGroup(448,154);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3538,248,4911,102,124]);
// Polycyclic

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

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