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## G = Q16.D14order 448 = 26·7

### 1st non-split extension by Q16 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — Q16.D14
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — D56⋊7C2 — Q16.D14
 Lower central C7 — C14 — C28 — C56 — Q16.D14
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for Q16.D14
G = < a,b,c,d | a8=1, b2=c14=d2=a4, bab-1=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=a3b, dcd-1=c13 >

Subgroups: 420 in 82 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, M5(2), SD32, Q32, C2×Q16, C4○D8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×Q8, Q32⋊C2, C7⋊C16, C56⋊C2, D56, Dic28, C2×C56, C7×Q16, C7×Q16, C4○D28, Q8×C14, C28.C8, C7⋊SD32, C7⋊Q32, D567C2, C14×Q16, Q16.D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, Q32⋊C2, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, Q16.D14

Smallest permutation representation of Q16.D14
On 224 points
Generators in S224
```(1 56 22 49 15 42 8 35)(2 29 23 50 16 43 9 36)(3 30 24 51 17 44 10 37)(4 31 25 52 18 45 11 38)(5 32 26 53 19 46 12 39)(6 33 27 54 20 47 13 40)(7 34 28 55 21 48 14 41)(57 112 78 105 71 98 64 91)(58 85 79 106 72 99 65 92)(59 86 80 107 73 100 66 93)(60 87 81 108 74 101 67 94)(61 88 82 109 75 102 68 95)(62 89 83 110 76 103 69 96)(63 90 84 111 77 104 70 97)(113 152 120 159 127 166 134 145)(114 153 121 160 128 167 135 146)(115 154 122 161 129 168 136 147)(116 155 123 162 130 141 137 148)(117 156 124 163 131 142 138 149)(118 157 125 164 132 143 139 150)(119 158 126 165 133 144 140 151)(169 197 176 204 183 211 190 218)(170 198 177 205 184 212 191 219)(171 199 178 206 185 213 192 220)(172 200 179 207 186 214 193 221)(173 201 180 208 187 215 194 222)(174 202 181 209 188 216 195 223)(175 203 182 210 189 217 196 224)
(1 72 15 58)(2 59 16 73)(3 74 17 60)(4 61 18 75)(5 76 19 62)(6 63 20 77)(7 78 21 64)(8 65 22 79)(9 80 23 66)(10 67 24 81)(11 82 25 68)(12 69 26 83)(13 84 27 70)(14 71 28 57)(29 93 43 107)(30 108 44 94)(31 95 45 109)(32 110 46 96)(33 97 47 111)(34 112 48 98)(35 99 49 85)(36 86 50 100)(37 101 51 87)(38 88 52 102)(39 103 53 89)(40 90 54 104)(41 105 55 91)(42 92 56 106)(113 222 127 208)(114 209 128 223)(115 224 129 210)(116 211 130 197)(117 198 131 212)(118 213 132 199)(119 200 133 214)(120 215 134 201)(121 202 135 216)(122 217 136 203)(123 204 137 218)(124 219 138 205)(125 206 139 220)(126 221 140 207)(141 169 155 183)(142 184 156 170)(143 171 157 185)(144 186 158 172)(145 173 159 187)(146 188 160 174)(147 175 161 189)(148 190 162 176)(149 177 163 191)(150 192 164 178)(151 179 165 193)(152 194 166 180)(153 181 167 195)(154 196 168 182)
(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)
(1 125 15 139)(2 138 16 124)(3 123 17 137)(4 136 18 122)(5 121 19 135)(6 134 20 120)(7 119 21 133)(8 132 22 118)(9 117 23 131)(10 130 24 116)(11 115 25 129)(12 128 26 114)(13 113 27 127)(14 126 28 140)(29 142 43 156)(30 155 44 141)(31 168 45 154)(32 153 46 167)(33 166 47 152)(34 151 48 165)(35 164 49 150)(36 149 50 163)(37 162 51 148)(38 147 52 161)(39 160 53 146)(40 145 54 159)(41 158 55 144)(42 143 56 157)(57 172 71 186)(58 185 72 171)(59 170 73 184)(60 183 74 169)(61 196 75 182)(62 181 76 195)(63 194 77 180)(64 179 78 193)(65 192 79 178)(66 177 80 191)(67 190 81 176)(68 175 82 189)(69 188 83 174)(70 173 84 187)(85 206 99 220)(86 219 100 205)(87 204 101 218)(88 217 102 203)(89 202 103 216)(90 215 104 201)(91 200 105 214)(92 213 106 199)(93 198 107 212)(94 211 108 197)(95 224 109 210)(96 209 110 223)(97 222 111 208)(98 207 112 221)```

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 14A ··· 14I 16A 16B 16C 16D 28A ··· 28F 28G ··· 28R 56A ··· 56L order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 8 14 ··· 14 16 16 16 16 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 56 2 2 8 8 56 2 2 2 2 2 4 2 ··· 2 28 28 28 28 4 ··· 4 8 ··· 8 4 ··· 4

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 C7⋊D4 C7⋊D4 Q32⋊C2 D4⋊D7 D4⋊D7 Q16.D14 kernel Q16.D14 C28.C8 C7⋊SD32 C7⋊Q32 D56⋊7C2 C14×Q16 C56 C2×C28 C2×Q16 C28 C2×C14 C2×C8 Q16 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 2 2 1 1 1 1 3 2 2 3 6 6 6 2 3 3 12

Matrix representation of Q16.D14 in GL4(𝔽113) generated by

 31 82 0 0 31 31 0 0 51 22 31 31 91 51 82 31
,
 72 76 0 0 76 41 0 0 16 63 102 68 108 64 68 11
,
 0 83 0 0 30 0 0 0 4 71 0 64 110 34 49 0
,
 69 95 0 19 56 6 94 0 1 50 107 56 76 77 95 44
`G:=sub<GL(4,GF(113))| [31,31,51,91,82,31,22,51,0,0,31,82,0,0,31,31],[72,76,16,108,76,41,63,64,0,0,102,68,0,0,68,11],[0,30,4,110,83,0,71,34,0,0,0,49,0,0,64,0],[69,56,1,76,95,6,50,77,0,94,107,95,19,0,56,44] >;`

Q16.D14 in GAP, Magma, Sage, TeX

`Q_{16}.D_{14}`
`% in TeX`

`G:=Group("Q16.D14");`
`// GroupNames label`

`G:=SmallGroup(448,713);`
`// by ID`

`G=gap.SmallGroup(448,713);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,387,184,675,185,192,1684,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=1,b^2=c^14=d^2=a^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^13>;`
`// generators/relations`

׿
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