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## G = Dic7⋊C8order 224 = 25·7

### The semidirect product of Dic7 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Dic7⋊C8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — Dic7⋊C8
 Lower central C7 — C14 — Dic7⋊C8
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for Dic7⋊C8
G = < a,b,c | a14=c8=1, b2=a7, bab-1=a-1, ac=ca, cbc-1=a7b >

Smallest permutation representation of Dic7⋊C8
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)
(1 130 8 137)(2 129 9 136)(3 128 10 135)(4 127 11 134)(5 140 12 133)(6 139 13 132)(7 138 14 131)(15 179 22 172)(16 178 23 171)(17 177 24 170)(18 176 25 169)(19 175 26 182)(20 174 27 181)(21 173 28 180)(29 62 36 69)(30 61 37 68)(31 60 38 67)(32 59 39 66)(33 58 40 65)(34 57 41 64)(35 70 42 63)(43 119 50 126)(44 118 51 125)(45 117 52 124)(46 116 53 123)(47 115 54 122)(48 114 55 121)(49 113 56 120)(71 92 78 85)(72 91 79 98)(73 90 80 97)(74 89 81 96)(75 88 82 95)(76 87 83 94)(77 86 84 93)(99 141 106 148)(100 154 107 147)(101 153 108 146)(102 152 109 145)(103 151 110 144)(104 150 111 143)(105 149 112 142)(155 191 162 184)(156 190 163 183)(157 189 164 196)(158 188 165 195)(159 187 166 194)(160 186 167 193)(161 185 168 192)(197 216 204 223)(198 215 205 222)(199 214 206 221)(200 213 207 220)(201 212 208 219)(202 211 209 218)(203 224 210 217)
(1 47 222 36 158 101 95 19)(2 48 223 37 159 102 96 20)(3 49 224 38 160 103 97 21)(4 50 211 39 161 104 98 22)(5 51 212 40 162 105 85 23)(6 52 213 41 163 106 86 24)(7 53 214 42 164 107 87 25)(8 54 215 29 165 108 88 26)(9 55 216 30 166 109 89 27)(10 56 217 31 167 110 90 28)(11 43 218 32 168 111 91 15)(12 44 219 33 155 112 92 16)(13 45 220 34 156 99 93 17)(14 46 221 35 157 100 94 18)(57 183 141 84 177 139 117 207)(58 184 142 71 178 140 118 208)(59 185 143 72 179 127 119 209)(60 186 144 73 180 128 120 210)(61 187 145 74 181 129 121 197)(62 188 146 75 182 130 122 198)(63 189 147 76 169 131 123 199)(64 190 148 77 170 132 124 200)(65 191 149 78 171 133 125 201)(66 192 150 79 172 134 126 202)(67 193 151 80 173 135 113 203)(68 194 152 81 174 136 114 204)(69 195 153 82 175 137 115 205)(70 196 154 83 176 138 116 206)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C4 C8 D4 Q8 D7 M4(2) D14 Dic14 C7⋊D4 C4×D7 C8×D7 C8⋊D7 kernel Dic7⋊C8 C2×C7⋊C8 C4×Dic7 C2×C56 C2×Dic7 Dic7 C28 C28 C2×C8 C14 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 8 1 1 3 2 3 6 6 6 12 12

Matrix representation of Dic7⋊C8 in GL4(𝔽113) generated by

 0 1 0 0 112 79 0 0 0 0 80 112 0 0 72 9
,
 1 0 0 0 79 112 0 0 0 0 15 0 0 0 48 98
,
 95 0 0 0 0 95 0 0 0 0 86 31 0 0 28 27
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,80,72,0,0,112,9],[1,79,0,0,0,112,0,0,0,0,15,48,0,0,0,98],[95,0,0,0,0,95,0,0,0,0,86,28,0,0,31,27] >;

Dic7⋊C8 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes C_8
% in TeX

G:=Group("Dic7:C8");
// GroupNames label

G:=SmallGroup(224,20);
// by ID

G=gap.SmallGroup(224,20);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,121,31,86,6917]);
// Polycyclic

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

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