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## G = D8⋊Dic7order 448 = 26·7

### 3rd semidirect product of D8 and Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D8⋊Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a-1, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 580 in 132 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic7, C28, C2×C14, C2×C14, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C7×D4, C22×C14, D8⋊C4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×D8, C22×Dic7, D4×C14, C56⋊C4, C8⋊Dic7, D4⋊Dic7, D4×Dic7, C14×D8, D8⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C8⋊C22, C2×Dic7, C22×D7, D8⋊C4, D4×D7, D42D7, C22×Dic7, D8⋊D7, D4×Dic7, D8⋊Dic7

Smallest permutation representation of D8⋊Dic7
On 224 points
Generators in S224
(1 115 46 159 128 63 79 144)(2 116 47 160 129 64 80 145)(3 117 48 161 130 65 81 146)(4 118 49 162 131 66 82 147)(5 119 50 163 132 67 83 148)(6 120 51 164 133 68 84 149)(7 121 52 165 134 69 71 150)(8 122 53 166 135 70 72 151)(9 123 54 167 136 57 73 152)(10 124 55 168 137 58 74 153)(11 125 56 155 138 59 75 154)(12 126 43 156 139 60 76 141)(13 113 44 157 140 61 77 142)(14 114 45 158 127 62 78 143)(15 190 40 102 221 170 198 98)(16 191 41 103 222 171 199 85)(17 192 42 104 223 172 200 86)(18 193 29 105 224 173 201 87)(19 194 30 106 211 174 202 88)(20 195 31 107 212 175 203 89)(21 196 32 108 213 176 204 90)(22 183 33 109 214 177 205 91)(23 184 34 110 215 178 206 92)(24 185 35 111 216 179 207 93)(25 186 36 112 217 180 208 94)(26 187 37 99 218 181 209 95)(27 188 38 100 219 182 210 96)(28 189 39 101 220 169 197 97)
(15 198)(16 199)(17 200)(18 201)(19 202)(20 203)(21 204)(22 205)(23 206)(24 207)(25 208)(26 209)(27 210)(28 197)(29 224)(30 211)(31 212)(32 213)(33 214)(34 215)(35 216)(36 217)(37 218)(38 219)(39 220)(40 221)(41 222)(42 223)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 71)(53 72)(54 73)(55 74)(56 75)(57 167)(58 168)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 141)(169 189)(170 190)(171 191)(172 192)(173 193)(174 194)(175 195)(176 196)(177 183)(178 184)(179 185)(180 186)(181 187)(182 188)
(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 90 8 97)(2 89 9 96)(3 88 10 95)(4 87 11 94)(5 86 12 93)(6 85 13 92)(7 98 14 91)(15 62 22 69)(16 61 23 68)(17 60 24 67)(18 59 25 66)(19 58 26 65)(20 57 27 64)(21 70 28 63)(29 154 36 147)(30 153 37 146)(31 152 38 145)(32 151 39 144)(33 150 40 143)(34 149 41 142)(35 148 42 141)(43 185 50 192)(44 184 51 191)(45 183 52 190)(46 196 53 189)(47 195 54 188)(48 194 55 187)(49 193 56 186)(71 170 78 177)(72 169 79 176)(73 182 80 175)(74 181 81 174)(75 180 82 173)(76 179 83 172)(77 178 84 171)(99 130 106 137)(100 129 107 136)(101 128 108 135)(102 127 109 134)(103 140 110 133)(104 139 111 132)(105 138 112 131)(113 215 120 222)(114 214 121 221)(115 213 122 220)(116 212 123 219)(117 211 124 218)(118 224 125 217)(119 223 126 216)(155 208 162 201)(156 207 163 200)(157 206 164 199)(158 205 165 198)(159 204 166 197)(160 203 167 210)(161 202 168 209)

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + - + image C1 C2 C2 C2 C2 C2 C4 D4 D7 C4○D4 D14 Dic7 D14 C8⋊C22 D4⋊2D7 D4×D7 D8⋊D7 kernel D8⋊Dic7 C56⋊C4 C8⋊Dic7 D4⋊Dic7 D4×Dic7 C14×D8 C7×D8 C2×Dic7 C2×D8 C28 C2×C8 D8 C2×D4 C14 C4 C22 C2 # reps 1 1 1 2 2 1 8 2 3 2 3 12 6 2 3 3 12

Matrix representation of D8⋊Dic7 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 109 80 4 33 0 0 33 4 80 109 0 0 109 80 109 80 0 0 33 4 33 4
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 0 112 0 0 0 0 1 34 0 0 0 0 0 0 34 112 0 0 0 0 1 0 0 0 0 0 0 0 34 112 0 0 0 0 1 0
,
 14 100 0 0 0 0 76 99 0 0 0 0 0 0 33 17 0 0 0 0 9 80 0 0 0 0 0 0 33 17 0 0 0 0 9 80

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,109,33,109,33,0,0,80,4,80,4,0,0,4,80,109,33,0,0,33,109,80,4],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[0,1,0,0,0,0,112,34,0,0,0,0,0,0,34,1,0,0,0,0,112,0,0,0,0,0,0,0,34,1,0,0,0,0,112,0],[14,76,0,0,0,0,100,99,0,0,0,0,0,0,33,9,0,0,0,0,17,80,0,0,0,0,0,0,33,9,0,0,0,0,17,80] >;

D8⋊Dic7 in GAP, Magma, Sage, TeX

D_8\rtimes {\rm Dic}_7
% in TeX

G:=Group("D8:Dic7");
// GroupNames label

G:=SmallGroup(448,686);
// by ID

G=gap.SmallGroup(448,686);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,758,219,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽