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## G = D8×Dic7order 448 = 26·7

### Direct product of D8 and Dic7

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, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 580 in 134 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic7, Dic7, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C7×D4, C22×C14, C4×D8, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×D8, C22×Dic7, D4×C14, C8×Dic7, C561C4, D4⋊Dic7, D4×Dic7, C14×D8, D8×Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, D8, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C2×D8, C4○D8, C2×Dic7, C22×D7, C4×D8, D4×D7, D42D7, C22×Dic7, D7×D8, D83D7, D4×Dic7, D8×Dic7

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

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

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

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

70 conjugacy classes

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

70 irreducible representations

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

Matrix representation of D8×Dic7 in GL4(𝔽113) generated by

 31 82 0 0 31 31 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 10 112 0 0 100 24
,
 112 0 0 0 0 112 0 0 0 0 101 19 0 0 34 12
G:=sub<GL(4,GF(113))| [31,31,0,0,82,31,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,10,100,0,0,112,24],[112,0,0,0,0,112,0,0,0,0,101,34,0,0,19,12] >;

D8×Dic7 in GAP, Magma, Sage, TeX

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

G:=Group("D8xDic7");
// GroupNames label

G:=SmallGroup(448,683);
// by ID

G=gap.SmallGroup(448,683);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,219,851,438,102,18822]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽