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## G = Dic7.C42order 448 = 26·7

### 3rd non-split extension by Dic7 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Dic7.C42
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — D7×C42 — Dic7.C42
 Lower central C7 — C14 — Dic7.C42
 Upper central C1 — C2×C4 — C8⋊C4

Generators and relations for Dic7.C42
G = < a,b,c,d | a14=c4=1, b2=d4=a7, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a7b, dcd-1=a7c >

Subgroups: 516 in 142 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, D7, C14, C14, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, C28, D14, D14, C2×C14, C4×C8, C8⋊C4, C8⋊C4, C2×C42, C2×M4(2), C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×M4(2), C8⋊D7, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4×C28, C2×C56, C2×C4×D7, C2×C4×D7, C42.D7, C8×Dic7, C7×C8⋊C4, D7×C42, C2×C8⋊D7, Dic7.C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, M4(2), C22×C4, D14, C2×C42, C2×M4(2), C4×D7, C22×D7, C4×M4(2), C2×C4×D7, D7×C42, D7×M4(2), Dic7.C42

Smallest permutation representation of Dic7.C42
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 50 8 43)(2 49 9 56)(3 48 10 55)(4 47 11 54)(5 46 12 53)(6 45 13 52)(7 44 14 51)(15 34 22 41)(16 33 23 40)(17 32 24 39)(18 31 25 38)(19 30 26 37)(20 29 27 36)(21 42 28 35)(57 102 64 109)(58 101 65 108)(59 100 66 107)(60 99 67 106)(61 112 68 105)(62 111 69 104)(63 110 70 103)(71 95 78 88)(72 94 79 87)(73 93 80 86)(74 92 81 85)(75 91 82 98)(76 90 83 97)(77 89 84 96)(113 158 120 165)(114 157 121 164)(115 156 122 163)(116 155 123 162)(117 168 124 161)(118 167 125 160)(119 166 126 159)(127 152 134 145)(128 151 135 144)(129 150 136 143)(130 149 137 142)(131 148 138 141)(132 147 139 154)(133 146 140 153)(169 214 176 221)(170 213 177 220)(171 212 178 219)(172 211 179 218)(173 224 180 217)(174 223 181 216)(175 222 182 215)(183 200 190 207)(184 199 191 206)(185 198 192 205)(186 197 193 204)(187 210 194 203)(188 209 195 202)(189 208 196 201)
(1 127 28 125)(2 128 15 126)(3 129 16 113)(4 130 17 114)(5 131 18 115)(6 132 19 116)(7 133 20 117)(8 134 21 118)(9 135 22 119)(10 136 23 120)(11 137 24 121)(12 138 25 122)(13 139 26 123)(14 140 27 124)(29 168 44 146)(30 155 45 147)(31 156 46 148)(32 157 47 149)(33 158 48 150)(34 159 49 151)(35 160 50 152)(36 161 51 153)(37 162 52 154)(38 163 53 141)(39 164 54 142)(40 165 55 143)(41 166 56 144)(42 167 43 145)(57 184 80 174)(58 185 81 175)(59 186 82 176)(60 187 83 177)(61 188 84 178)(62 189 71 179)(63 190 72 180)(64 191 73 181)(65 192 74 182)(66 193 75 169)(67 194 76 170)(68 195 77 171)(69 196 78 172)(70 183 79 173)(85 222 101 198)(86 223 102 199)(87 224 103 200)(88 211 104 201)(89 212 105 202)(90 213 106 203)(91 214 107 204)(92 215 108 205)(93 216 109 206)(94 217 110 207)(95 218 111 208)(96 219 112 209)(97 220 99 210)(98 221 100 197)
(1 90 35 67 8 97 42 60)(2 91 36 68 9 98 29 61)(3 92 37 69 10 85 30 62)(4 93 38 70 11 86 31 63)(5 94 39 57 12 87 32 64)(6 95 40 58 13 88 33 65)(7 96 41 59 14 89 34 66)(15 107 51 77 22 100 44 84)(16 108 52 78 23 101 45 71)(17 109 53 79 24 102 46 72)(18 110 54 80 25 103 47 73)(19 111 55 81 26 104 48 74)(20 112 56 82 27 105 49 75)(21 99 43 83 28 106 50 76)(113 198 154 179 120 205 147 172)(114 199 141 180 121 206 148 173)(115 200 142 181 122 207 149 174)(116 201 143 182 123 208 150 175)(117 202 144 169 124 209 151 176)(118 203 145 170 125 210 152 177)(119 204 146 171 126 197 153 178)(127 220 160 187 134 213 167 194)(128 221 161 188 135 214 168 195)(129 222 162 189 136 215 155 196)(130 223 163 190 137 216 156 183)(131 224 164 191 138 217 157 184)(132 211 165 192 139 218 158 185)(133 212 166 193 140 219 159 186)

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

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

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

100 conjugacy classes

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

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D7 M4(2) D14 D14 C4×D7 C4×D7 D7×M4(2) kernel Dic7.C42 C42.D7 C8×Dic7 C7×C8⋊C4 D7×C42 C2×C8⋊D7 C8⋊D7 C4×Dic7 C2×C4×D7 C8⋊C4 Dic7 C42 C2×C8 C8 C2×C4 C2 # reps 1 1 2 1 1 2 16 4 4 3 8 3 6 24 12 12

Matrix representation of Dic7.C42 in GL4(𝔽113) generated by

 0 1 0 0 112 9 0 0 0 0 112 0 0 0 0 112
,
 112 0 0 0 104 1 0 0 0 0 98 0 0 0 108 15
,
 98 0 0 0 0 98 0 0 0 0 1 0 0 0 38 112
,
 112 0 0 0 0 112 0 0 0 0 19 112 0 0 37 94
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,112,0,0,0,0,112],[112,104,0,0,0,1,0,0,0,0,98,108,0,0,0,15],[98,0,0,0,0,98,0,0,0,0,1,38,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,19,37,0,0,112,94] >;

Dic7.C42 in GAP, Magma, Sage, TeX

{\rm Dic}_7.C_4^2
% in TeX

G:=Group("Dic7.C4^2");
// GroupNames label

G:=SmallGroup(448,241);
// by ID

G=gap.SmallGroup(448,241);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,387,58,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=c^4=1,b^2=d^4=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=a^7*c>;
// generators/relations

׿
×
𝔽