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## G = C14.D16order 448 = 26·7

### 2nd non-split extension by C14 of D16 acting via D16/D8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C14.D16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C2×D56 — C14.D16
 Lower central C7 — C14 — C28 — C56 — C14.D16
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.D8

Generators and relations for C14.D16
G = < a,b,c | a14=b16=c2=1, bab-1=cac=a-1, cbc=a7b-1 >

Subgroups: 556 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C16, C4⋊C4, C2×C8, D8, C2×D4, C28, C28, D14, C2×C14, C2.D8, C2×C16, C2×D8, C56, D28, C2×C28, C2×C28, C22×D7, C2.D16, C7⋊C16, D56, D56, C7×C4⋊C4, C2×C56, C2×D28, C2×C7⋊C16, C7×C2.D8, C2×D56, C14.D16
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, D16, SD32, C4×D7, D28, C7⋊D4, C2.D16, D14⋊C4, D4⋊D7, Q8⋊D7, C14.D8, C7⋊D16, C7⋊SD32, C14.D16

Smallest permutation representation of C14.D16
On 224 points
Generators in S224
```(1 17 94 222 105 119 185 68 45 159 139 61 208 166)(2 167 193 62 140 160 46 69 186 120 106 223 95 18)(3 19 96 224 107 121 187 70 47 145 141 63 194 168)(4 169 195 64 142 146 48 71 188 122 108 209 81 20)(5 21 82 210 109 123 189 72 33 147 143 49 196 170)(6 171 197 50 144 148 34 73 190 124 110 211 83 22)(7 23 84 212 111 125 191 74 35 149 129 51 198 172)(8 173 199 52 130 150 36 75 192 126 112 213 85 24)(9 25 86 214 97 127 177 76 37 151 131 53 200 174)(10 175 201 54 132 152 38 77 178 128 98 215 87 26)(11 27 88 216 99 113 179 78 39 153 133 55 202 176)(12 161 203 56 134 154 40 79 180 114 100 217 89 28)(13 29 90 218 101 115 181 80 41 155 135 57 204 162)(14 163 205 58 136 156 42 65 182 116 102 219 91 30)(15 31 92 220 103 117 183 66 43 157 137 59 206 164)(16 165 207 60 138 158 44 67 184 118 104 221 93 32)
(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 68)(2 16)(3 66)(4 14)(5 80)(6 12)(7 78)(8 10)(9 76)(11 74)(13 72)(15 70)(17 185)(18 165)(19 183)(20 163)(21 181)(22 161)(23 179)(24 175)(25 177)(26 173)(27 191)(28 171)(29 189)(30 169)(31 187)(32 167)(33 162)(34 180)(35 176)(36 178)(37 174)(38 192)(39 172)(40 190)(41 170)(42 188)(43 168)(44 186)(45 166)(46 184)(47 164)(48 182)(49 135)(50 217)(51 133)(52 215)(53 131)(54 213)(55 129)(56 211)(57 143)(58 209)(59 141)(60 223)(61 139)(62 221)(63 137)(64 219)(65 71)(67 69)(73 79)(75 77)(81 205)(82 115)(83 203)(84 113)(85 201)(86 127)(87 199)(88 125)(89 197)(90 123)(91 195)(92 121)(93 193)(94 119)(95 207)(96 117)(97 214)(98 130)(99 212)(100 144)(101 210)(102 142)(103 224)(104 140)(105 222)(106 138)(107 220)(108 136)(109 218)(110 134)(111 216)(112 132)(114 148)(116 146)(118 160)(120 158)(122 156)(124 154)(126 152)(128 150)(145 206)(147 204)(149 202)(151 200)(153 198)(155 196)(157 194)(159 208)```

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D7 SD16 D8 D14 D16 SD32 C4×D7 D28 C7⋊D4 Q8⋊D7 D4⋊D7 C7⋊D16 C7⋊SD32 kernel C14.D16 C2×C7⋊C16 C7×C2.D8 C2×D56 D56 C56 C2×C28 C2.D8 C28 C2×C14 C2×C8 C14 C14 C8 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 3 2 2 3 4 4 6 6 6 3 3 6 6

Matrix representation of C14.D16 in GL5(𝔽113)

 112 0 0 0 0 0 33 33 0 0 0 80 104 0 0 0 0 0 1 0 0 0 0 0 1
,
 98 0 0 0 0 0 34 5 0 0 0 85 79 0 0 0 0 0 91 13 0 0 0 11 14
,
 1 0 0 0 0 0 1 0 0 0 0 9 112 0 0 0 0 0 1 0 0 0 0 81 112

`G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,33,80,0,0,0,33,104,0,0,0,0,0,1,0,0,0,0,0,1],[98,0,0,0,0,0,34,85,0,0,0,5,79,0,0,0,0,0,91,11,0,0,0,13,14],[1,0,0,0,0,0,1,9,0,0,0,0,112,0,0,0,0,0,1,81,0,0,0,0,112] >;`

C14.D16 in GAP, Magma, Sage, TeX

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

`G:=Group("C14.D16");`
`// GroupNames label`

`G:=SmallGroup(448,48);`
`// by ID`

`G=gap.SmallGroup(448,48);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,675,346,192,1684,851,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^14=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^7*b^-1>;`
`// generators/relations`

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