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## G = C16.D14order 448 = 26·7

### 1st non-split extension by C16 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C16.D14
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — D56⋊7C2 — C16.D14
 Lower central C7 — C14 — C28 — C56 — C16.D14
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for C16.D14
G = < a,b,c | a16=1, b14=c2=a8, bab-1=a9, cac-1=a-1, cbc-1=b13 >

Subgroups: 564 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, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, M5(2), SD32, Q32, C2×Q16, C4○D8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, Q32⋊C2, C112, C56⋊C2, D56, Dic28, Dic28, Dic28, C2×C56, C2×Dic14, C4○D28, C112⋊C2, Dic56, C7×M5(2), D567C2, C2×Dic28, C16.D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, D28, C22×D7, Q32⋊C2, D56, C2×D28, C2×D56, C16.D14

Smallest permutation representation of C16.D14
On 224 points
Generators in S224
```(1 88 36 71 22 95 29 78 15 102 50 57 8 109 43 64)(2 103 37 58 23 110 30 65 16 89 51 72 9 96 44 79)(3 90 38 73 24 97 31 80 17 104 52 59 10 111 45 66)(4 105 39 60 25 112 32 67 18 91 53 74 11 98 46 81)(5 92 40 75 26 99 33 82 19 106 54 61 12 85 47 68)(6 107 41 62 27 86 34 69 20 93 55 76 13 100 48 83)(7 94 42 77 28 101 35 84 21 108 56 63 14 87 49 70)(113 180 142 222 120 173 149 215 127 194 156 208 134 187 163 201)(114 195 143 209 121 188 150 202 128 181 157 223 135 174 164 216)(115 182 144 224 122 175 151 217 129 196 158 210 136 189 165 203)(116 169 145 211 123 190 152 204 130 183 159 197 137 176 166 218)(117 184 146 198 124 177 153 219 131 170 160 212 138 191 167 205)(118 171 147 213 125 192 154 206 132 185 161 199 139 178 168 220)(119 186 148 200 126 179 155 221 133 172 162 214 140 193 141 207)
(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)
(1 120 15 134)(2 133 16 119)(3 118 17 132)(4 131 18 117)(5 116 19 130)(6 129 20 115)(7 114 21 128)(8 127 22 113)(9 140 23 126)(10 125 24 139)(11 138 25 124)(12 123 26 137)(13 136 27 122)(14 121 28 135)(29 163 43 149)(30 148 44 162)(31 161 45 147)(32 146 46 160)(33 159 47 145)(34 144 48 158)(35 157 49 143)(36 142 50 156)(37 155 51 141)(38 168 52 154)(39 153 53 167)(40 166 54 152)(41 151 55 165)(42 164 56 150)(57 194 71 180)(58 179 72 193)(59 192 73 178)(60 177 74 191)(61 190 75 176)(62 175 76 189)(63 188 77 174)(64 173 78 187)(65 186 79 172)(66 171 80 185)(67 184 81 170)(68 169 82 183)(69 182 83 196)(70 195 84 181)(85 211 99 197)(86 224 100 210)(87 209 101 223)(88 222 102 208)(89 207 103 221)(90 220 104 206)(91 205 105 219)(92 218 106 204)(93 203 107 217)(94 216 108 202)(95 201 109 215)(96 214 110 200)(97 199 111 213)(98 212 112 198)```

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

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

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

76 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 14A 14B 14C 14D 14E 14F 16A 16B 16C 16D 28A ··· 28F 28G 28H 28I 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 8 14 14 14 14 14 14 16 16 16 16 28 ··· 28 28 28 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 56 2 2 56 56 56 2 2 2 2 2 4 2 2 2 4 4 4 4 4 4 4 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

76 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 D28 D28 D56 D56 Q32⋊C2 C16.D14 kernel C16.D14 C112⋊C2 Dic56 C7×M5(2) D56⋊7C2 C2×Dic28 C56 C2×C28 M5(2) C28 C2×C14 C16 C2×C8 C8 C2×C4 C4 C22 C7 C1 # reps 1 2 2 1 1 1 1 1 3 2 2 6 3 6 6 12 12 2 12

Matrix representation of C16.D14 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 101 36 0 0 0 0 0 1 1 0 0 91 1 55 38 0 0 30 112 58 75
,
 9 80 0 0 0 0 33 80 0 0 0 0 0 0 1 77 0 0 0 0 44 112 0 0 0 0 47 96 0 1 0 0 23 93 112 0
,
 81 109 0 0 0 0 58 32 0 0 0 0 0 0 8 112 45 42 0 0 82 18 106 47 0 0 102 104 66 92 0 0 66 56 47 21

`G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,91,30,0,0,101,0,1,112,0,0,36,1,55,58,0,0,0,1,38,75],[9,33,0,0,0,0,80,80,0,0,0,0,0,0,1,44,47,23,0,0,77,112,96,93,0,0,0,0,0,112,0,0,0,0,1,0],[81,58,0,0,0,0,109,32,0,0,0,0,0,0,8,82,102,66,0,0,112,18,104,56,0,0,45,106,66,47,0,0,42,47,92,21] >;`

C16.D14 in GAP, Magma, Sage, TeX

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

`G:=Group("C16.D14");`
`// GroupNames label`

`G:=SmallGroup(448,443);`
`// by ID`

`G=gap.SmallGroup(448,443);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,387,142,1571,80,1684,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^16=1,b^14=c^2=a^8,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=b^13>;`
`// generators/relations`

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