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## G = C4.Dic14order 224 = 25·7

### 2nd non-split extension by C4 of Dic14 acting via Dic14/Dic7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C4.Dic14
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C7⋊C8 — C4.Dic14
 Lower central C7 — C14 — C28 — C4.Dic14
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C4.Dic14
G = < a,b,c | a14=b8=1, c2=a7, bab-1=a-1, ac=ca, cbc-1=b3 >

Smallest permutation representation of C4.Dic14
Regular action on 224 points
Generators in S224
(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)
(1 187 47 171 69 114 71 107)(2 186 48 170 70 113 72 106)(3 185 49 169 57 126 73 105)(4 184 50 182 58 125 74 104)(5 183 51 181 59 124 75 103)(6 196 52 180 60 123 76 102)(7 195 53 179 61 122 77 101)(8 194 54 178 62 121 78 100)(9 193 55 177 63 120 79 99)(10 192 56 176 64 119 80 112)(11 191 43 175 65 118 81 111)(12 190 44 174 66 117 82 110)(13 189 45 173 67 116 83 109)(14 188 46 172 68 115 84 108)(15 96 152 135 39 217 201 167)(16 95 153 134 40 216 202 166)(17 94 154 133 41 215 203 165)(18 93 141 132 42 214 204 164)(19 92 142 131 29 213 205 163)(20 91 143 130 30 212 206 162)(21 90 144 129 31 211 207 161)(22 89 145 128 32 224 208 160)(23 88 146 127 33 223 209 159)(24 87 147 140 34 222 210 158)(25 86 148 139 35 221 197 157)(26 85 149 138 36 220 198 156)(27 98 150 137 37 219 199 155)(28 97 151 136 38 218 200 168)
(1 35 8 42)(2 36 9 29)(3 37 10 30)(4 38 11 31)(5 39 12 32)(6 40 13 33)(7 41 14 34)(15 66 22 59)(16 67 23 60)(17 68 24 61)(18 69 25 62)(19 70 26 63)(20 57 27 64)(21 58 28 65)(43 144 50 151)(44 145 51 152)(45 146 52 153)(46 147 53 154)(47 148 54 141)(48 149 55 142)(49 150 56 143)(71 197 78 204)(72 198 79 205)(73 199 80 206)(74 200 81 207)(75 201 82 208)(76 202 83 209)(77 203 84 210)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)(113 138 120 131)(114 139 121 132)(115 140 122 133)(116 127 123 134)(117 128 124 135)(118 129 125 136)(119 130 126 137)(155 192 162 185)(156 193 163 186)(157 194 164 187)(158 195 165 188)(159 196 166 189)(160 183 167 190)(161 184 168 191)(169 219 176 212)(170 220 177 213)(171 221 178 214)(172 222 179 215)(173 223 180 216)(174 224 181 217)(175 211 182 218)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + - + + + - - + image C1 C2 C2 C2 C4 Q8 D4 D7 SD16 D14 Dic14 C4×D7 C7⋊D4 D4.D7 Q8⋊D7 kernel C4.Dic14 C2×C7⋊C8 C4⋊Dic7 C7×C4⋊C4 C7⋊C8 C28 C2×C14 C4⋊C4 C14 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 3 4 3 6 6 6 3 3

Matrix representation of C4.Dic14 in GL4(𝔽113) generated by

 80 80 0 0 33 9 0 0 0 0 112 0 0 0 0 112
,
 81 38 0 0 89 32 0 0 0 0 0 1 0 0 1 87
,
 55 38 0 0 75 58 0 0 0 0 68 97 0 0 56 45
G:=sub<GL(4,GF(113))| [80,33,0,0,80,9,0,0,0,0,112,0,0,0,0,112],[81,89,0,0,38,32,0,0,0,0,0,1,0,0,1,87],[55,75,0,0,38,58,0,0,0,0,68,56,0,0,97,45] >;

C4.Dic14 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{14}
% in TeX

G:=Group("C4.Dic14");
// GroupNames label

G:=SmallGroup(224,14);
// by ID

G=gap.SmallGroup(224,14);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,313,31,297,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^14=b^8=1,c^2=a^7,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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