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G = C42.D14order 448 = 26·7

1st non-split extension by C42 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C42.D14
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4×C28 — C4.D28 — C42.D14
 Lower central C7 — C2×C14 — C2×C28 — C42.D14
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C42.D14
G = < a,b,c,d | a4=b4=1, c14=a-1, d2=a-1b, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=bc13 >

Subgroups: 452 in 70 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C8⋊C4, C8⋊C4, C4.4D4, C7⋊C8, C56, Dic14, D28, C2×Dic7, C2×C28, C22×D7, C42.C22, C2×C7⋊C8, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×D28, C42.D7, C7×C8⋊C4, C4.D28, C42.D14
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C4.D4, C4≀C2, C4×D7, D28, C7⋊D4, C42.C22, D14⋊C4, Dic14⋊C4, C28.46D4, D284C4, C42.D14

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

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

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

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

79 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 28A ··· 28L 28M ··· 28X 56A ··· 56X order 1 2 2 2 2 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 56 2 2 2 2 4 56 2 2 2 4 4 4 4 28 28 28 28 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D7 D14 C4≀C2 C4×D7 D28 C7⋊D4 Dic14⋊C4 C4.D4 C28.46D4 D28⋊4C4 kernel C42.D14 C42.D7 C7×C8⋊C4 C4.D28 C2×Dic14 C2×D28 C2×C28 C8⋊C4 C42 C14 C2×C4 C2×C4 C2×C4 C2 C14 C2 C2 # reps 1 1 1 1 2 2 2 3 3 8 6 6 6 24 1 6 6

Matrix representation of C42.D14 in GL6(𝔽113)

 39 105 0 0 0 0 77 74 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 98 0 0 0 0 0 0 98
,
 20 106 0 0 0 0 25 93 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 112 0
,
 39 57 0 0 0 0 87 58 0 0 0 0 0 0 4 109 0 0 0 0 4 81 0 0 0 0 0 0 106 106 0 0 0 0 106 7
,
 58 56 0 0 0 0 31 55 0 0 0 0 0 0 109 4 0 0 0 0 81 4 0 0 0 0 0 0 106 7 0 0 0 0 106 106

`G:=sub<GL(6,GF(113))| [39,77,0,0,0,0,105,74,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,0,0,0,0,0,0,98],[20,25,0,0,0,0,106,93,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[39,87,0,0,0,0,57,58,0,0,0,0,0,0,4,4,0,0,0,0,109,81,0,0,0,0,0,0,106,106,0,0,0,0,106,7],[58,31,0,0,0,0,56,55,0,0,0,0,0,0,109,81,0,0,0,0,4,4,0,0,0,0,0,0,106,106,0,0,0,0,7,106] >;`

C42.D14 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(448,21);`
`// by ID`

`G=gap.SmallGroup(448,21);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,184,1571,570,192,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^-1,d^2=a^-1*b,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=b*c^13>;`
`// generators/relations`

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