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## G = D14⋊C42order 448 = 26·7

### 1st semidirect product of D14 and C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D14⋊C42
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — D7×C22×C4 — D14⋊C42
 Lower central C7 — C14 — D14⋊C42
 Upper central C1 — C23 — C2.C42

Generators and relations for D14⋊C42
G = < a,b,c,d | a14=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a12b, dbd-1=a7b, cd=dc >

Subgroups: 1180 in 258 conjugacy classes, 99 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C4×C22⋊C4, C4×Dic7, D14⋊C4, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C23×D7, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×D14⋊C4, D7×C22×C4, D14⋊C42
Quotients:

Smallest permutation representation of D14⋊C42
On 224 points
Generators in S224
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 42)(26 41)(27 40)(28 39)(57 111)(58 110)(59 109)(60 108)(61 107)(62 106)(63 105)(64 104)(65 103)(66 102)(67 101)(68 100)(69 99)(70 112)(71 87)(72 86)(73 85)(74 98)(75 97)(76 96)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(113 163)(114 162)(115 161)(116 160)(117 159)(118 158)(119 157)(120 156)(121 155)(122 168)(123 167)(124 166)(125 165)(126 164)(127 153)(128 152)(129 151)(130 150)(131 149)(132 148)(133 147)(134 146)(135 145)(136 144)(137 143)(138 142)(139 141)(140 154)(169 213)(170 212)(171 211)(172 224)(173 223)(174 222)(175 221)(176 220)(177 219)(178 218)(179 217)(180 216)(181 215)(182 214)(183 209)(184 208)(185 207)(186 206)(187 205)(188 204)(189 203)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)(196 210)
(1 153 21 161)(2 152 22 160)(3 151 23 159)(4 150 24 158)(5 149 25 157)(6 148 26 156)(7 147 27 155)(8 146 28 168)(9 145 15 167)(10 144 16 166)(11 143 17 165)(12 142 18 164)(13 141 19 163)(14 154 20 162)(29 120 45 132)(30 119 46 131)(31 118 47 130)(32 117 48 129)(33 116 49 128)(34 115 50 127)(35 114 51 140)(36 113 52 139)(37 126 53 138)(38 125 54 137)(39 124 55 136)(40 123 56 135)(41 122 43 134)(42 121 44 133)(57 210 80 222)(58 209 81 221)(59 208 82 220)(60 207 83 219)(61 206 84 218)(62 205 71 217)(63 204 72 216)(64 203 73 215)(65 202 74 214)(66 201 75 213)(67 200 76 212)(68 199 77 211)(69 198 78 224)(70 197 79 223)(85 169 104 191)(86 182 105 190)(87 181 106 189)(88 180 107 188)(89 179 108 187)(90 178 109 186)(91 177 110 185)(92 176 111 184)(93 175 112 183)(94 174 99 196)(95 173 100 195)(96 172 101 194)(97 171 102 193)(98 170 103 192)
(1 94 33 63)(2 95 34 64)(3 96 35 65)(4 97 36 66)(5 98 37 67)(6 85 38 68)(7 86 39 69)(8 87 40 70)(9 88 41 57)(10 89 42 58)(11 90 29 59)(12 91 30 60)(13 92 31 61)(14 93 32 62)(15 107 43 80)(16 108 44 81)(17 109 45 82)(18 110 46 83)(19 111 47 84)(20 112 48 71)(21 99 49 72)(22 100 50 73)(23 101 51 74)(24 102 52 75)(25 103 53 76)(26 104 54 77)(27 105 55 78)(28 106 56 79)(113 201 150 171)(114 202 151 172)(115 203 152 173)(116 204 153 174)(117 205 154 175)(118 206 141 176)(119 207 142 177)(120 208 143 178)(121 209 144 179)(122 210 145 180)(123 197 146 181)(124 198 147 182)(125 199 148 169)(126 200 149 170)(127 215 160 195)(128 216 161 196)(129 217 162 183)(130 218 163 184)(131 219 164 185)(132 220 165 186)(133 221 166 187)(134 222 167 188)(135 223 168 189)(136 224 155 190)(137 211 156 191)(138 212 157 192)(139 213 158 193)(140 214 159 194)

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

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

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

100 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M ··· 4T 4U ··· 4AB 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 14 14 14 14 2 ··· 2 7 ··· 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D7 C4○D4 D14 C4×D7 D4×D7 D4⋊2D7 Q8⋊2D7 kernel D14⋊C42 C14.C42 C7×C2.C42 C2×C4×Dic7 C2×D14⋊C4 D7×C22×C4 D14⋊C4 C2×C4×D7 C2×Dic7 C2.C42 C2×C14 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 1 2 2 1 16 8 4 3 4 9 36 6 3 3

Matrix representation of D14⋊C42 in GL5(𝔽29)

 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 21 21 0 0 0 8 26
,
 1 0 0 0 0 0 28 0 0 0 0 28 1 0 0 0 0 0 21 21 0 0 0 26 8
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 7 17
,
 28 0 0 0 0 0 1 27 0 0 0 0 28 0 0 0 0 0 17 0 0 0 0 0 17

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,21,8,0,0,0,21,26],[1,0,0,0,0,0,28,28,0,0,0,0,1,0,0,0,0,0,21,26,0,0,0,21,8],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,7,0,0,0,0,17],[28,0,0,0,0,0,1,0,0,0,0,27,28,0,0,0,0,0,17,0,0,0,0,0,17] >;

D14⋊C42 in GAP, Magma, Sage, TeX

D_{14}\rtimes C_4^2
% in TeX

G:=Group("D14:C4^2");
// GroupNames label

G:=SmallGroup(448,200);
// by ID

G=gap.SmallGroup(448,200);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,387,58,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^12*b,d*b*d^-1=a^7*b,c*d=d*c>;
// generators/relations

׿
×
𝔽