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## G = C7×C8⋊4D4order 448 = 26·7

### Direct product of C7 and C8⋊4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×C8⋊4D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — D4×C14 — C14×D8 — C7×C8⋊4D4
 Lower central C1 — C2 — C2×C4 — C7×C8⋊4D4
 Upper central C1 — C2×C14 — C4×C28 — C7×C8⋊4D4

Generators and relations for C7×C84D4
G = < a,b,c,d | a7=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 386 in 162 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C14, C14, C14, C42, C2×C8, D8, C2×D4, C2×D4, C28, C2×C14, C2×C14, C4×C8, C41D4, C2×D8, C56, C2×C28, C2×C28, C7×D4, C22×C14, C84D4, C4×C28, C2×C56, C7×D8, D4×C14, D4×C14, C4×C56, C7×C41D4, C14×D8, C7×C84D4
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, C41D4, C2×D8, C7×D4, C22×C14, C84D4, C7×D8, D4×C14, C7×C41D4, C14×D8, C7×C84D4

Smallest permutation representation of C7×C84D4
On 224 points
Generators in S224
(1 195 81 187 73 179 65)(2 196 82 188 74 180 66)(3 197 83 189 75 181 67)(4 198 84 190 76 182 68)(5 199 85 191 77 183 69)(6 200 86 192 78 184 70)(7 193 87 185 79 177 71)(8 194 88 186 80 178 72)(9 33 139 25 131 17 123)(10 34 140 26 132 18 124)(11 35 141 27 133 19 125)(12 36 142 28 134 20 126)(13 37 143 29 135 21 127)(14 38 144 30 136 22 128)(15 39 137 31 129 23 121)(16 40 138 32 130 24 122)(41 147 60 175 165 49 155)(42 148 61 176 166 50 156)(43 149 62 169 167 51 157)(44 150 63 170 168 52 158)(45 151 64 171 161 53 159)(46 152 57 172 162 54 160)(47 145 58 173 163 55 153)(48 146 59 174 164 56 154)(89 114 219 105 211 97 203)(90 115 220 106 212 98 204)(91 116 221 107 213 99 205)(92 117 222 108 214 100 206)(93 118 223 109 215 101 207)(94 119 224 110 216 102 208)(95 120 217 111 209 103 201)(96 113 218 112 210 104 202)
(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)
(1 151 13 89)(2 152 14 90)(3 145 15 91)(4 146 16 92)(5 147 9 93)(6 148 10 94)(7 149 11 95)(8 150 12 96)(17 101 183 155)(18 102 184 156)(19 103 177 157)(20 104 178 158)(21 97 179 159)(22 98 180 160)(23 99 181 153)(24 100 182 154)(25 109 191 165)(26 110 192 166)(27 111 185 167)(28 112 186 168)(29 105 187 161)(30 106 188 162)(31 107 189 163)(32 108 190 164)(33 118 199 60)(34 119 200 61)(35 120 193 62)(36 113 194 63)(37 114 195 64)(38 115 196 57)(39 116 197 58)(40 117 198 59)(41 123 207 69)(42 124 208 70)(43 125 201 71)(44 126 202 72)(45 127 203 65)(46 128 204 66)(47 121 205 67)(48 122 206 68)(49 131 215 77)(50 132 216 78)(51 133 209 79)(52 134 210 80)(53 135 211 73)(54 136 212 74)(55 129 213 75)(56 130 214 76)(81 171 143 219)(82 172 144 220)(83 173 137 221)(84 174 138 222)(85 175 139 223)(86 176 140 224)(87 169 141 217)(88 170 142 218)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 32)(26 31)(27 30)(28 29)(33 40)(34 39)(35 38)(36 37)(41 206)(42 205)(43 204)(44 203)(45 202)(46 201)(47 208)(48 207)(49 214)(50 213)(51 212)(52 211)(53 210)(54 209)(55 216)(56 215)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 88)(82 87)(83 86)(84 85)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 152)(96 151)(97 158)(98 157)(99 156)(100 155)(101 154)(102 153)(103 160)(104 159)(105 168)(106 167)(107 166)(108 165)(109 164)(110 163)(111 162)(112 161)(121 124)(122 123)(125 128)(126 127)(129 132)(130 131)(133 136)(134 135)(137 140)(138 139)(141 144)(142 143)(169 220)(170 219)(171 218)(172 217)(173 224)(174 223)(175 222)(176 221)(177 180)(178 179)(181 184)(182 183)(185 188)(186 187)(189 192)(190 191)(193 196)(194 195)(197 200)(198 199)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 7A ··· 7F 8A ··· 8H 14A ··· 14R 14S ··· 14AP 28A ··· 28AJ 56A ··· 56AV order 1 2 2 2 2 2 2 2 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 8 8 8 8 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2

154 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 C7×C8⋊4D4 C4×C56 C7×C4⋊1D4 C14×D8 C8⋊4D4 C4×C8 C4⋊1D4 C2×D8 C56 C2×C28 C28 C8 C2×C4 C4 # reps 1 1 2 4 6 6 12 24 4 2 8 24 12 48

Matrix representation of C7×C84D4 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 109 0 0 0 0 109
,
 62 62 0 0 82 0 0 0 0 0 51 51 0 0 31 0
,
 112 0 0 0 0 112 0 0 0 0 112 111 0 0 1 1
,
 62 62 0 0 82 51 0 0 0 0 51 51 0 0 31 62
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,109,0,0,0,0,109],[62,82,0,0,62,0,0,0,0,0,51,31,0,0,51,0],[112,0,0,0,0,112,0,0,0,0,112,1,0,0,111,1],[62,82,0,0,62,51,0,0,0,0,51,31,0,0,51,62] >;

C7×C84D4 in GAP, Magma, Sage, TeX

C_7\times C_8\rtimes_4D_4
% in TeX

G:=Group("C7xC8:4D4");
// GroupNames label

G:=SmallGroup(448,901);
// by ID

G=gap.SmallGroup(448,901);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1968,2438,604,9804,172]);
// Polycyclic

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

׿
×
𝔽