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## G = Dic14.C8order 448 = 26·7

### 2nd non-split extension by Dic14 of C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Dic14.C8
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — D28.2C4 — Dic14.C8
 Lower central C7 — C14 — C28 — Dic14.C8
 Upper central C1 — C8 — C2×C8 — M5(2)

Generators and relations for Dic14.C8
G = < a,b,c | a28=1, b2=c8=a14, bab-1=a-1, cac-1=a15, cbc-1=a21b >

Smallest permutation representation of Dic14.C8
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)
(1 127 15 113)(2 126 16 140)(3 125 17 139)(4 124 18 138)(5 123 19 137)(6 122 20 136)(7 121 21 135)(8 120 22 134)(9 119 23 133)(10 118 24 132)(11 117 25 131)(12 116 26 130)(13 115 27 129)(14 114 28 128)(29 158 43 144)(30 157 44 143)(31 156 45 142)(32 155 46 141)(33 154 47 168)(34 153 48 167)(35 152 49 166)(36 151 50 165)(37 150 51 164)(38 149 52 163)(39 148 53 162)(40 147 54 161)(41 146 55 160)(42 145 56 159)(57 178 71 192)(58 177 72 191)(59 176 73 190)(60 175 74 189)(61 174 75 188)(62 173 76 187)(63 172 77 186)(64 171 78 185)(65 170 79 184)(66 169 80 183)(67 196 81 182)(68 195 82 181)(69 194 83 180)(70 193 84 179)(85 206 99 220)(86 205 100 219)(87 204 101 218)(88 203 102 217)(89 202 103 216)(90 201 104 215)(91 200 105 214)(92 199 106 213)(93 198 107 212)(94 197 108 211)(95 224 109 210)(96 223 110 209)(97 222 111 208)(98 221 112 207)
(1 87 46 80 8 108 53 73 15 101 32 66 22 94 39 59)(2 102 47 67 9 95 54 60 16 88 33 81 23 109 40 74)(3 89 48 82 10 110 55 75 17 103 34 68 24 96 41 61)(4 104 49 69 11 97 56 62 18 90 35 83 25 111 42 76)(5 91 50 84 12 112 29 77 19 105 36 70 26 98 43 63)(6 106 51 71 13 99 30 64 20 92 37 57 27 85 44 78)(7 93 52 58 14 86 31 79 21 107 38 72 28 100 45 65)(113 197 155 176 134 204 148 183 127 211 141 190 120 218 162 169)(114 212 156 191 135 219 149 170 128 198 142 177 121 205 163 184)(115 199 157 178 136 206 150 185 129 213 143 192 122 220 164 171)(116 214 158 193 137 221 151 172 130 200 144 179 123 207 165 186)(117 201 159 180 138 208 152 187 131 215 145 194 124 222 166 173)(118 216 160 195 139 223 153 174 132 202 146 181 125 209 167 188)(119 203 161 182 140 210 154 189 133 217 147 196 126 224 168 175)

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

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

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

88 conjugacy classes

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

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 D4 D7 M4(2) D14 D28 C7⋊D4 C4×D7 D4.C8 C8×D7 C8⋊D7 Dic14.C8 kernel Dic14.C8 C2×C7⋊C16 C7×M5(2) D28.2C4 C4.Dic7 C4○D28 Dic14 D28 C56 M5(2) C2×C14 C2×C8 C8 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 2 3 2 3 6 6 6 8 12 12 12

Matrix representation of Dic14.C8 in GL4(𝔽113) generated by

 24 111 0 0 1 80 0 0 0 0 15 0 0 0 0 98
,
 89 10 0 0 112 24 0 0 0 0 0 40 0 0 48 0
,
 55 12 0 0 107 58 0 0 0 0 0 1 0 0 18 0
G:=sub<GL(4,GF(113))| [24,1,0,0,111,80,0,0,0,0,15,0,0,0,0,98],[89,112,0,0,10,24,0,0,0,0,0,48,0,0,40,0],[55,107,0,0,12,58,0,0,0,0,0,18,0,0,1,0] >;

Dic14.C8 in GAP, Magma, Sage, TeX

{\rm Dic}_{14}.C_8
% in TeX

G:=Group("Dic14.C8");
// GroupNames label

G:=SmallGroup(448,72);
// by ID

G=gap.SmallGroup(448,72);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,100,1123,570,136,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^28=1,b^2=c^8=a^14,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^21*b>;
// generators/relations

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