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## G = D14.C42order 448 = 26·7

### 2nd non-split extension by D14 of 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 — C28 — C2×C28 — C2×C4×D7 — C42⋊D7 — D14.C42
 Lower central C7 — C14 — D14.C42
 Upper central C1 — C2×C8 — C4×C8

Generators and relations for D14.C42
G = < a,b,c,d | a14=b2=c4=1, d4=a7, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, cd=dc >

Subgroups: 452 in 130 conjugacy classes, 75 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, D7, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4×C8, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C82M4(2), C8×D7, C8⋊D7, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C42.D7, C8×Dic7, C56⋊C4, C4×C56, C42⋊D7, D7×C2×C8, C2×C8⋊D7, D14.C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, D14, C2×C42, C8○D4, C4×D7, C22×D7, C82M4(2), C2×C4×D7, D7×C42, D28.2C4, D14.C42

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

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 7A 7B 7C 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 14A ··· 14I 28A ··· 28AJ 56A ··· 56AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 7 7 7 8 ··· 8 8 8 8 8 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 14 14 1 1 1 1 2 2 2 2 14 ··· 14 2 2 2 1 ··· 1 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 1 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D7 D14 D14 C8○D4 C4×D7 C4×D7 D28.2C4 kernel D14.C42 C42.D7 C8×Dic7 C56⋊C4 C4×C56 C42⋊D7 D7×C2×C8 C2×C8⋊D7 C8×D7 C8⋊D7 Dic7⋊C4 D14⋊C4 C4×C8 C42 C2×C8 C14 C8 C2×C4 C2 # reps 1 1 1 1 1 1 1 1 8 8 4 4 3 3 6 8 24 12 48

Matrix representation of D14.C42 in GL4(𝔽113) generated by

 9 9 0 0 25 0 0 0 0 0 1 10 0 0 71 33
,
 9 42 0 0 25 104 0 0 0 0 80 112 0 0 71 33
,
 98 0 0 0 0 98 0 0 0 0 55 8 0 0 102 58
,
 98 0 0 0 0 98 0 0 0 0 44 0 0 0 0 44
`G:=sub<GL(4,GF(113))| [9,25,0,0,9,0,0,0,0,0,1,71,0,0,10,33],[9,25,0,0,42,104,0,0,0,0,80,71,0,0,112,33],[98,0,0,0,0,98,0,0,0,0,55,102,0,0,8,58],[98,0,0,0,0,98,0,0,0,0,44,0,0,0,0,44] >;`

D14.C42 in GAP, Magma, Sage, TeX

`D_{14}.C_4^2`
`% in TeX`

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

`G:=SmallGroup(448,223);`
`// by ID`

`G=gap.SmallGroup(448,223);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,58,136,18822]);`
`// Polycyclic`

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

׿
×
𝔽