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## G = Dic7.D8order 448 = 26·7

### 1st non-split extension by Dic7 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic7.D8
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=a7c-1 >

Subgroups: 532 in 108 conjugacy classes, 43 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, C22×C4, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C2×C14, D4⋊C4, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D4⋊Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C22×Dic7, D4×C14, C28.Q8, Dic7⋊C8, C561C4, D4⋊Dic7, C7×D4⋊C4, C28⋊Q8, D4×Dic7, Dic7.D8
Quotients: C1, C2, C22, D4, Q8, C23, D7, D8, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×D8, C8.C22, Dic14, C22×D7, D4⋊Q8, C2×Dic14, D4×D7, D42D7, C22⋊Dic14, D7×D8, SD16⋊D7, Dic7.D8

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

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + + - - - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D7 D8 C4○D4 D14 D14 D14 Dic14 C8.C22 D4⋊2D7 D4×D7 D7×D8 SD16⋊D7 kernel Dic7.D8 C28.Q8 Dic7⋊C8 C56⋊1C4 D4⋊Dic7 C7×D4⋊C4 C28⋊Q8 D4×Dic7 C2×Dic7 C7×D4 D4⋊C4 Dic7 C28 C4⋊C4 C2×C8 C2×D4 D4 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 3 4 2 3 3 3 12 1 3 3 6 6

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

 1 0 0 0 0 1 0 0 0 0 89 112 0 0 1 0
,
 112 0 0 0 0 112 0 0 0 0 11 56 0 0 18 102
,
 51 99 0 0 105 0 0 0 0 0 96 8 0 0 105 17
,
 1 0 0 0 44 112 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,89,1,0,0,112,0],[112,0,0,0,0,112,0,0,0,0,11,18,0,0,56,102],[51,105,0,0,99,0,0,0,0,0,96,105,0,0,8,17],[1,44,0,0,0,112,0,0,0,0,1,0,0,0,0,1] >;

Dic7.D8 in GAP, Magma, Sage, TeX

{\rm Dic}_7.D_8
% in TeX

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

G:=SmallGroup(448,293);
// by ID

G=gap.SmallGroup(448,293);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,254,219,58,851,438,102,18822]);
// Polycyclic

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

׿
×
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