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## G = D14.4C42order 448 = 26·7

### 4th non-split extension by D14 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D14.4C42
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C42⋊D7 — D14.4C42
 Lower central C7 — C14 — D14.4C42
 Upper central C1 — C2×C4 — C8⋊C4

Generators and relations for D14.4C42
G = < a,b,c,d | a14=b2=c4=1, d4=a7, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=a7c >

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, C8⋊C4, 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, C4×C7⋊C8, C8×Dic7, C56⋊C4, C7×C8⋊C4, C42⋊D7, D7×C2×C8, C2×C8⋊D7, D14.4C42
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.C4, D14.4C42

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

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

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

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

100 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 ··· 8P 8Q 8R 8S 8T 14A ··· 14I 28A ··· 28L 28M ··· 28X 56A ··· 56X 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 28 ··· 28 56 ··· 56 size 1 1 1 1 14 14 1 1 1 1 2 2 2 2 14 ··· 14 2 2 2 2 ··· 2 7 ··· 7 14 14 14 14 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D7 D14 D14 C8○D4 C4×D7 C4×D7 D28.C4 kernel D14.4C42 C4×C7⋊C8 C8×Dic7 C56⋊C4 C7×C8⋊C4 C42⋊D7 D7×C2×C8 C2×C8⋊D7 C8×D7 C8⋊D7 Dic7⋊C4 D14⋊C4 C8⋊C4 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 12

Matrix representation of D14.4C42 in GL4(𝔽113) generated by

 103 103 0 0 10 89 0 0 0 0 112 0 0 0 0 112
,
 10 10 0 0 24 103 0 0 0 0 1 0 0 0 111 112
,
 98 0 0 0 0 98 0 0 0 0 14 14 0 0 107 99
,
 15 0 0 0 0 15 0 0 0 0 69 0 0 0 88 44
`G:=sub<GL(4,GF(113))| [103,10,0,0,103,89,0,0,0,0,112,0,0,0,0,112],[10,24,0,0,10,103,0,0,0,0,1,111,0,0,0,112],[98,0,0,0,0,98,0,0,0,0,14,107,0,0,14,99],[15,0,0,0,0,15,0,0,0,0,69,88,0,0,0,44] >;`

D14.4C42 in GAP, Magma, Sage, TeX

`D_{14}._4C_4^2`
`% in TeX`

`G:=Group("D14.4C4^2");`
`// GroupNames label`

`G:=SmallGroup(448,242);`
`// by ID`

`G=gap.SmallGroup(448,242);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,387,58,136,18822]);`
`// 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,d*c*d^-1=a^7*c>;`
`// generators/relations`

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