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## G = (C2×C14).Q16order 448 = 26·7

### 5th non-split extension by C2×C14 of Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (C2×C14).Q16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4⋊Dic7 — C2×C4⋊Dic7 — (C2×C14).Q16
 Lower central C7 — C14 — C2×C28 — (C2×C14).Q16
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for (C2×C14).Q16
G = < a,b,c,d | a2=b14=c8=1, d2=b7c4, ab=ba, cac-1=dad-1=ab7, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 412 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×C14, C23.48D4, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, Q8×C14, C28.Q8, C28.55D4, Q8⋊Dic7, C2×C4⋊Dic7, C7×C22⋊Q8, (C2×C14).Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, C4○D4, D14, C22.D4, C2×Q16, C8⋊C22, C7⋊D4, C22×D7, C23.48D4, C7⋊Q16, D42D7, C2×C7⋊D4, C23.18D14, C2×C7⋊Q16, D4⋊D14, (C2×C14).Q16

Smallest permutation representation of (C2×C14).Q16
On 224 points
Generators in S224
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(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)
(1 206 100 221 83 51 167 94)(2 205 101 220 84 50 168 93)(3 204 102 219 71 49 155 92)(4 203 103 218 72 48 156 91)(5 202 104 217 73 47 157 90)(6 201 105 216 74 46 158 89)(7 200 106 215 75 45 159 88)(8 199 107 214 76 44 160 87)(9 198 108 213 77 43 161 86)(10 197 109 212 78 56 162 85)(11 210 110 211 79 55 163 98)(12 209 111 224 80 54 164 97)(13 208 112 223 81 53 165 96)(14 207 99 222 82 52 166 95)(15 194 138 124 181 62 37 141)(16 193 139 123 182 61 38 154)(17 192 140 122 169 60 39 153)(18 191 127 121 170 59 40 152)(19 190 128 120 171 58 41 151)(20 189 129 119 172 57 42 150)(21 188 130 118 173 70 29 149)(22 187 131 117 174 69 30 148)(23 186 132 116 175 68 31 147)(24 185 133 115 176 67 32 146)(25 184 134 114 177 66 33 145)(26 183 135 113 178 65 34 144)(27 196 136 126 179 64 35 143)(28 195 137 125 180 63 36 142)
(1 179 76 20)(2 180 77 21)(3 181 78 22)(4 182 79 23)(5 169 80 24)(6 170 81 25)(7 171 82 26)(8 172 83 27)(9 173 84 28)(10 174 71 15)(11 175 72 16)(12 176 73 17)(13 177 74 18)(14 178 75 19)(29 101 137 161)(30 102 138 162)(31 103 139 163)(32 104 140 164)(33 105 127 165)(34 106 128 166)(35 107 129 167)(36 108 130 168)(37 109 131 155)(38 110 132 156)(39 111 133 157)(40 112 134 158)(41 99 135 159)(42 100 136 160)(43 149 205 125)(44 150 206 126)(45 151 207 113)(46 152 208 114)(47 153 209 115)(48 154 210 116)(49 141 197 117)(50 142 198 118)(51 143 199 119)(52 144 200 120)(53 145 201 121)(54 146 202 122)(55 147 203 123)(56 148 204 124)(57 221 196 87)(58 222 183 88)(59 223 184 89)(60 224 185 90)(61 211 186 91)(62 212 187 92)(63 213 188 93)(64 214 189 94)(65 215 190 95)(66 216 191 96)(67 217 192 97)(68 218 193 98)(69 219 194 85)(70 220 195 86)

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + + - - + image C1 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 Q16 D14 D14 D14 C7⋊D4 C7⋊D4 C8⋊C22 D4⋊2D7 C7⋊Q16 D4⋊D14 kernel (C2×C14).Q16 C28.Q8 C28.55D4 Q8⋊Dic7 C2×C4⋊Dic7 C7×C22⋊Q8 C2×C28 C22×C14 C22⋊Q8 C28 C2×C14 C4⋊C4 C22×C4 C2×Q8 C2×C4 C23 C14 C4 C22 C2 # reps 1 2 1 2 1 1 1 1 3 4 4 3 3 3 6 6 1 6 6 6

Matrix representation of (C2×C14).Q16 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 65 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 33 33 0 0 0 0 80 104 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 75 0 0 0 0 24 81 0 0 0 0 0 0 42 30 0 0 0 0 73 71 0 0 0 0 0 0 0 107 0 0 0 0 19 62
,
 34 5 0 0 0 0 108 79 0 0 0 0 0 0 65 111 0 0 0 0 79 48 0 0 0 0 0 0 47 100 0 0 0 0 57 66

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,65,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[33,80,0,0,0,0,33,104,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,24,0,0,0,0,75,81,0,0,0,0,0,0,42,73,0,0,0,0,30,71,0,0,0,0,0,0,0,19,0,0,0,0,107,62],[34,108,0,0,0,0,5,79,0,0,0,0,0,0,65,79,0,0,0,0,111,48,0,0,0,0,0,0,47,57,0,0,0,0,100,66] >;

(C2×C14).Q16 in GAP, Magma, Sage, TeX

(C_2\times C_{14}).Q_{16}
% in TeX

G:=Group("(C2xC14).Q16");
// GroupNames label

G:=SmallGroup(448,578);
// by ID

G=gap.SmallGroup(448,578);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,336,254,219,268,1123,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^8=1,d^2=b^7*c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^7,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
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