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## G = Dic14.D4order 448 = 26·7

### 8th non-split extension by Dic14 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — Dic14.D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — Dic7⋊3Q8 — Dic14.D4
 Lower central C7 — C14 — C2×C28 — Dic14.D4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic14.D4
G = < a,b,c,d | a28=c4=1, b2=d2=a14, bab-1=dad-1=a-1, cac-1=a13, bc=cb, dbd-1=a7b, dcd-1=a14c-1 >

Subgroups: 532 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, Q8.D4, Dic28, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, D4.D7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, D4×C14, C14.Q16, Dic7⋊C8, C7×D4⋊C4, Dic73Q8, C2×Dic28, C2×D4.D7, C28.17D4, Dic14.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8.C22, C22×D7, Q8.D4, C4○D28, D4×D7, D14⋊D4, D83D7, SD16⋊D7, Dic14.D4

Smallest permutation representation of Dic14.D4
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)
(1 130 15 116)(2 129 16 115)(3 128 17 114)(4 127 18 113)(5 126 19 140)(6 125 20 139)(7 124 21 138)(8 123 22 137)(9 122 23 136)(10 121 24 135)(11 120 25 134)(12 119 26 133)(13 118 27 132)(14 117 28 131)(29 65 43 79)(30 64 44 78)(31 63 45 77)(32 62 46 76)(33 61 47 75)(34 60 48 74)(35 59 49 73)(36 58 50 72)(37 57 51 71)(38 84 52 70)(39 83 53 69)(40 82 54 68)(41 81 55 67)(42 80 56 66)(85 167 99 153)(86 166 100 152)(87 165 101 151)(88 164 102 150)(89 163 103 149)(90 162 104 148)(91 161 105 147)(92 160 106 146)(93 159 107 145)(94 158 108 144)(95 157 109 143)(96 156 110 142)(97 155 111 141)(98 154 112 168)(169 212 183 198)(170 211 184 197)(171 210 185 224)(172 209 186 223)(173 208 187 222)(174 207 188 221)(175 206 189 220)(176 205 190 219)(177 204 191 218)(178 203 192 217)(179 202 193 216)(180 201 194 215)(181 200 195 214)(182 199 196 213)
(1 31 170 155)(2 44 171 168)(3 29 172 153)(4 42 173 166)(5 55 174 151)(6 40 175 164)(7 53 176 149)(8 38 177 162)(9 51 178 147)(10 36 179 160)(11 49 180 145)(12 34 181 158)(13 47 182 143)(14 32 183 156)(15 45 184 141)(16 30 185 154)(17 43 186 167)(18 56 187 152)(19 41 188 165)(20 54 189 150)(21 39 190 163)(22 52 191 148)(23 37 192 161)(24 50 193 146)(25 35 194 159)(26 48 195 144)(27 33 196 157)(28 46 169 142)(57 217 105 136)(58 202 106 121)(59 215 107 134)(60 200 108 119)(61 213 109 132)(62 198 110 117)(63 211 111 130)(64 224 112 115)(65 209 85 128)(66 222 86 113)(67 207 87 126)(68 220 88 139)(69 205 89 124)(70 218 90 137)(71 203 91 122)(72 216 92 135)(73 201 93 120)(74 214 94 133)(75 199 95 118)(76 212 96 131)(77 197 97 116)(78 210 98 129)(79 223 99 114)(80 208 100 127)(81 221 101 140)(82 206 102 125)(83 219 103 138)(84 204 104 123)
(1 141 15 155)(2 168 16 154)(3 167 17 153)(4 166 18 152)(5 165 19 151)(6 164 20 150)(7 163 21 149)(8 162 22 148)(9 161 23 147)(10 160 24 146)(11 159 25 145)(12 158 26 144)(13 157 27 143)(14 156 28 142)(29 172 43 186)(30 171 44 185)(31 170 45 184)(32 169 46 183)(33 196 47 182)(34 195 48 181)(35 194 49 180)(36 193 50 179)(37 192 51 178)(38 191 52 177)(39 190 53 176)(40 189 54 175)(41 188 55 174)(42 187 56 173)(57 210 71 224)(58 209 72 223)(59 208 73 222)(60 207 74 221)(61 206 75 220)(62 205 76 219)(63 204 77 218)(64 203 78 217)(65 202 79 216)(66 201 80 215)(67 200 81 214)(68 199 82 213)(69 198 83 212)(70 197 84 211)(85 121 99 135)(86 120 100 134)(87 119 101 133)(88 118 102 132)(89 117 103 131)(90 116 104 130)(91 115 105 129)(92 114 106 128)(93 113 107 127)(94 140 108 126)(95 139 109 125)(96 138 110 124)(97 137 111 123)(98 136 112 122)

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 D14 C4○D8 C4○D28 C8.C22 D4×D7 D4×D7 D8⋊3D7 SD16⋊D7 kernel Dic14.D4 C14.Q16 Dic7⋊C8 C7×D4⋊C4 Dic7⋊3Q8 C2×Dic28 C2×D4.D7 C28.17D4 Dic14 C2×Dic7 D4⋊C4 C28 C4⋊C4 C2×C8 C2×D4 C14 C4 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 3 2 3 3 3 4 12 1 3 3 6 6

Matrix representation of Dic14.D4 in GL4(𝔽113) generated by

 83 0 0 0 41 64 0 0 0 0 30 72 0 0 33 83
,
 112 17 0 0 0 1 0 0 0 0 49 81 0 0 68 64
,
 98 29 0 0 0 15 0 0 0 0 98 0 0 0 0 98
,
 98 29 0 0 78 15 0 0 0 0 98 0 0 0 69 15
G:=sub<GL(4,GF(113))| [83,41,0,0,0,64,0,0,0,0,30,33,0,0,72,83],[112,0,0,0,17,1,0,0,0,0,49,68,0,0,81,64],[98,0,0,0,29,15,0,0,0,0,98,0,0,0,0,98],[98,78,0,0,29,15,0,0,0,0,98,69,0,0,0,15] >;

Dic14.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{14}.D_4
% in TeX

G:=Group("Dic14.D4");
// GroupNames label

G:=SmallGroup(448,301);
// by ID

G=gap.SmallGroup(448,301);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,344,1094,135,570,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=c^4=1,b^2=d^2=a^14,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^13,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=a^14*c^-1>;
// generators/relations

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