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## G = Dic7⋊6SD16order 448 = 26·7

### 1st semidirect product of Dic7 and SD16 acting through Inn(Dic7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Dic7⋊6SD16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — D4×Dic7 — Dic7⋊6SD16
 Lower central C7 — C14 — C28 — Dic7⋊6SD16
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic76SD16
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 532 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C4×SD16, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D4.D7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C22×Dic7, D4×C14, C4.Dic14, C8×Dic7, C28.44D4, C7×D4⋊C4, Dic73Q8, C2×D4.D7, D4×Dic7, Dic76SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, SD16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×SD16, C4○D8, C4×D7, C22×D7, C4×SD16, C2×C4×D7, D4×D7, D42D7, Dic74D4, D83D7, D7×SD16, Dic76SD16

Smallest permutation representation of Dic76SD16
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 72 8 79)(2 71 9 78)(3 84 10 77)(4 83 11 76)(5 82 12 75)(6 81 13 74)(7 80 14 73)(15 197 22 204)(16 210 23 203)(17 209 24 202)(18 208 25 201)(19 207 26 200)(20 206 27 199)(21 205 28 198)(29 150 36 143)(30 149 37 142)(31 148 38 141)(32 147 39 154)(33 146 40 153)(34 145 41 152)(35 144 42 151)(43 175 50 182)(44 174 51 181)(45 173 52 180)(46 172 53 179)(47 171 54 178)(48 170 55 177)(49 169 56 176)(57 91 64 98)(58 90 65 97)(59 89 66 96)(60 88 67 95)(61 87 68 94)(62 86 69 93)(63 85 70 92)(99 195 106 188)(100 194 107 187)(101 193 108 186)(102 192 109 185)(103 191 110 184)(104 190 111 183)(105 189 112 196)(113 223 120 216)(114 222 121 215)(115 221 122 214)(116 220 123 213)(117 219 124 212)(118 218 125 211)(119 217 126 224)(127 162 134 155)(128 161 135 168)(129 160 136 167)(130 159 137 166)(131 158 138 165)(132 157 139 164)(133 156 140 163)
(1 213 192 182 163 35 59 210)(2 212 193 181 164 34 60 209)(3 211 194 180 165 33 61 208)(4 224 195 179 166 32 62 207)(5 223 196 178 167 31 63 206)(6 222 183 177 168 30 64 205)(7 221 184 176 155 29 65 204)(8 220 185 175 156 42 66 203)(9 219 186 174 157 41 67 202)(10 218 187 173 158 40 68 201)(11 217 188 172 159 39 69 200)(12 216 189 171 160 38 70 199)(13 215 190 170 161 37 57 198)(14 214 191 169 162 36 58 197)(15 80 122 103 49 127 150 97)(16 79 123 102 50 140 151 96)(17 78 124 101 51 139 152 95)(18 77 125 100 52 138 153 94)(19 76 126 99 53 137 154 93)(20 75 113 112 54 136 141 92)(21 74 114 111 55 135 142 91)(22 73 115 110 56 134 143 90)(23 72 116 109 43 133 144 89)(24 71 117 108 44 132 145 88)(25 84 118 107 45 131 146 87)(26 83 119 106 46 130 147 86)(27 82 120 105 47 129 148 85)(28 81 121 104 48 128 149 98)
(15 150)(16 151)(17 152)(18 153)(19 154)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 204)(30 205)(31 206)(32 207)(33 208)(34 209)(35 210)(36 197)(37 198)(38 199)(39 200)(40 201)(41 202)(42 203)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 113)(55 114)(56 115)(57 190)(58 191)(59 192)(60 193)(61 194)(62 195)(63 196)(64 183)(65 184)(66 185)(67 186)(68 187)(69 188)(70 189)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 99)(94 100)(95 101)(96 102)(97 103)(98 104)(169 214)(170 215)(171 216)(172 217)(173 218)(174 219)(175 220)(176 221)(177 222)(178 223)(179 224)(180 211)(181 212)(182 213)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D7 SD16 C4○D4 D14 D14 D14 C4○D8 C4×D7 D4⋊2D7 D4×D7 D8⋊3D7 D7×SD16 kernel Dic7⋊6SD16 C4.Dic14 C8×Dic7 C28.44D4 C7×D4⋊C4 Dic7⋊3Q8 C2×D4.D7 D4×Dic7 D4.D7 C2×Dic7 D4⋊C4 Dic7 C28 C4⋊C4 C2×C8 C2×D4 C14 D4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 3 4 2 3 3 3 4 12 3 3 6 6

Matrix representation of Dic76SD16 in GL4(𝔽113) generated by

 0 112 0 0 1 89 0 0 0 0 112 0 0 0 0 112
,
 29 7 0 0 25 84 0 0 0 0 15 0 0 0 0 15
,
 96 105 0 0 36 17 0 0 0 0 87 26 0 0 100 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 112
G:=sub<GL(4,GF(113))| [0,1,0,0,112,89,0,0,0,0,112,0,0,0,0,112],[29,25,0,0,7,84,0,0,0,0,15,0,0,0,0,15],[96,36,0,0,105,17,0,0,0,0,87,100,0,0,26,0],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,112] >;

Dic76SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("Dic7:6SD16");
// GroupNames label

G:=SmallGroup(448,292);
// by ID

G=gap.SmallGroup(448,292);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,135,268,570,297,136,18822]);
// Polycyclic

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

׿
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