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## G = (C2×C14).D8order 448 = 26·7

### 8th non-split extension by C2×C14 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (C2×C14).D8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4⋊Dic7 — C2×C4⋊Dic7 — (C2×C14).D8
 Lower central C7 — C14 — C2×C28 — (C2×C14).D8
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for (C2×C14).D8
G = < a,b,c,d | a2=b14=c8=1, d2=b7, ab=ba, cac-1=ab7, ad=da, cbc-1=dbd-1=b-1, dcd-1=b7c-1 >

Subgroups: 460 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22.D8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C7×C22⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, D4×C14, D4×C14, C28.Q8, C28.55D4, D4⋊Dic7, C2×C4⋊Dic7, C7×C4⋊D4, (C2×C14).D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C22.D4, C2×D8, C8.C22, C7⋊D4, C22×D7, C22.D8, D4⋊D7, D42D7, C2×C7⋊D4, C2×D4⋊D7, C23.18D14, D4.9D14, (C2×C14).D8

Smallest permutation representation of (C2×C14).D8
On 224 points
Generators in S224
(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)
(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)
(1 180 41 56 223 162 125 136)(2 179 42 55 224 161 126 135)(3 178 29 54 211 160 113 134)(4 177 30 53 212 159 114 133)(5 176 31 52 213 158 115 132)(6 175 32 51 214 157 116 131)(7 174 33 50 215 156 117 130)(8 173 34 49 216 155 118 129)(9 172 35 48 217 168 119 128)(10 171 36 47 218 167 120 127)(11 170 37 46 219 166 121 140)(12 169 38 45 220 165 122 139)(13 182 39 44 221 164 123 138)(14 181 40 43 222 163 124 137)(15 198 74 189 69 112 88 144)(16 197 75 188 70 111 89 143)(17 210 76 187 57 110 90 142)(18 209 77 186 58 109 91 141)(19 208 78 185 59 108 92 154)(20 207 79 184 60 107 93 153)(21 206 80 183 61 106 94 152)(22 205 81 196 62 105 95 151)(23 204 82 195 63 104 96 150)(24 203 83 194 64 103 97 149)(25 202 84 193 65 102 98 148)(26 201 71 192 66 101 85 147)(27 200 72 191 67 100 86 146)(28 199 73 190 68 99 87 145)
(1 25 8 18)(2 24 9 17)(3 23 10 16)(4 22 11 15)(5 21 12 28)(6 20 13 27)(7 19 14 26)(29 96 36 89)(30 95 37 88)(31 94 38 87)(32 93 39 86)(33 92 40 85)(34 91 41 98)(35 90 42 97)(43 108 50 101)(44 107 51 100)(45 106 52 99)(46 105 53 112)(47 104 54 111)(48 103 55 110)(49 102 56 109)(57 224 64 217)(58 223 65 216)(59 222 66 215)(60 221 67 214)(61 220 68 213)(62 219 69 212)(63 218 70 211)(71 117 78 124)(72 116 79 123)(73 115 80 122)(74 114 81 121)(75 113 82 120)(76 126 83 119)(77 125 84 118)(127 204 134 197)(128 203 135 210)(129 202 136 209)(130 201 137 208)(131 200 138 207)(132 199 139 206)(133 198 140 205)(141 173 148 180)(142 172 149 179)(143 171 150 178)(144 170 151 177)(145 169 152 176)(146 182 153 175)(147 181 154 174)(155 193 162 186)(156 192 163 185)(157 191 164 184)(158 190 165 183)(159 189 166 196)(160 188 167 195)(161 187 168 194)

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 14P ··· 14U 28A ··· 28L 28M ··· 28R order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 8 2 2 4 8 28 28 28 28 2 2 2 28 28 28 28 2 ··· 2 4 ··· 4 8 ··· 8 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 D8 D14 D14 D14 C7⋊D4 C7⋊D4 C8.C22 D4⋊2D7 D4⋊D7 D4.9D14 kernel (C2×C14).D8 C28.Q8 C28.55D4 D4⋊Dic7 C2×C4⋊Dic7 C7×C4⋊D4 C2×C28 C22×C14 C4⋊D4 C28 C2×C14 C4⋊C4 C22×C4 C2×D4 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).D8 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 94 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 34 88 0 0 0 0 25 88 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
,
 13 104 0 0 0 0 94 100 0 0 0 0 0 0 90 63 0 0 0 0 101 23 0 0 0 0 0 0 12 82 0 0 0 0 87 39
,
 13 104 0 0 0 0 94 100 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 0 0 0 0 0 104 72 0 0 0 0 24 9

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,94,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,25,0,0,0,0,88,88,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,94,0,0,0,0,104,100,0,0,0,0,0,0,90,101,0,0,0,0,63,23,0,0,0,0,0,0,12,87,0,0,0,0,82,39],[13,94,0,0,0,0,104,100,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,104,24,0,0,0,0,72,9] >;

(C2×C14).D8 in GAP, Magma, Sage, TeX

(C_2\times C_{14}).D_8
% in TeX

G:=Group("(C2xC14).D8");
// GroupNames label

G:=SmallGroup(448,567);
// by ID

G=gap.SmallGroup(448,567);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,254,219,1123,297,136,18822]);
// Polycyclic

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

׿
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