metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊6C4, Dic7⋊5D8, C7⋊3(C4×D8), C8⋊7(C4×D7), C56⋊4(C2×C4), C2.3(D7×D8), D28⋊7(C2×C4), C2.D8⋊13D7, (C8×Dic7)⋊3C2, (C2×D56).9C2, C14.52(C4×D4), C14.26(C2×D8), D28⋊C4⋊7C2, C4⋊C4.166D14, (C2×C8).225D14, C14.D8⋊19C2, C22.87(D4×D7), C14.73(C4○D8), C28.35(C4○D4), C28.46(C22×C4), (C2×C56).77C22, C4.7(Q8⋊2D7), C2.3(Q8.D14), (C2×C28).288C23, (C2×Dic7).207D4, (C2×D28).80C22, C2.12(D28⋊C4), (C4×Dic7).231C22, C4.43(C2×C4×D7), (C7×C2.D8)⋊2C2, (C2×C14).293(C2×D4), (C7×C4⋊C4).81C22, (C2×C7⋊C8).229C22, (C2×C4).391(C22×D7), SmallGroup(448,406)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊5D8
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 844 in 134 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×D8, D56, C2×C7⋊C8, C4×Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C14.D8, C8×Dic7, C7×C2.D8, D28⋊C4, C2×D56, Dic7⋊5D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, D8, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×D8, C4○D8, C4×D7, C22×D7, C4×D8, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4, D7×D8, Q8.D14, Dic7⋊5D8
►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 20 8 27)(2 19 9 26)(3 18 10 25)(4 17 11 24)(5 16 12 23)(6 15 13 22)(7 28 14 21)(29 100 36 107)(30 99 37 106)(31 112 38 105)(32 111 39 104)(33 110 40 103)(34 109 41 102)(35 108 42 101)(43 166 50 159)(44 165 51 158)(45 164 52 157)(46 163 53 156)(47 162 54 155)(48 161 55 168)(49 160 56 167)(57 126 64 119)(58 125 65 118)(59 124 66 117)(60 123 67 116)(61 122 68 115)(62 121 69 114)(63 120 70 113)(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)(127 150 134 143)(128 149 135 142)(129 148 136 141)(130 147 137 154)(131 146 138 153)(132 145 139 152)(133 144 140 151)(169 208 176 201)(170 207 177 200)(171 206 178 199)(172 205 179 198)(173 204 180 197)(174 203 181 210)(175 202 182 209)(183 214 190 221)(184 213 191 220)(185 212 192 219)(186 211 193 218)(187 224 194 217)(188 223 195 216)(189 222 196 215)
(1 105 198 129 51 211 120 82)(2 106 199 130 52 212 121 83)(3 107 200 131 53 213 122 84)(4 108 201 132 54 214 123 71)(5 109 202 133 55 215 124 72)(6 110 203 134 56 216 125 73)(7 111 204 135 43 217 126 74)(8 112 205 136 44 218 113 75)(9 99 206 137 45 219 114 76)(10 100 207 138 46 220 115 77)(11 101 208 139 47 221 116 78)(12 102 209 140 48 222 117 79)(13 103 210 127 49 223 118 80)(14 104 197 128 50 224 119 81)(15 40 181 143 167 188 65 93)(16 41 182 144 168 189 66 94)(17 42 169 145 155 190 67 95)(18 29 170 146 156 191 68 96)(19 30 171 147 157 192 69 97)(20 31 172 148 158 193 70 98)(21 32 173 149 159 194 57 85)(22 33 174 150 160 195 58 86)(23 34 175 151 161 196 59 87)(24 35 176 152 162 183 60 88)(25 36 177 153 163 184 61 89)(26 37 178 154 164 185 62 90)(27 38 179 141 165 186 63 91)(28 39 180 142 166 187 64 92)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 56)(11 55)(12 54)(13 53)(14 52)(15 163)(16 162)(17 161)(18 160)(19 159)(20 158)(21 157)(22 156)(23 155)(24 168)(25 167)(26 166)(27 165)(28 164)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 154)(40 153)(41 152)(42 151)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(71 222)(72 221)(73 220)(74 219)(75 218)(76 217)(77 216)(78 215)(79 214)(80 213)(81 212)(82 211)(83 224)(84 223)(85 192)(86 191)(87 190)(88 189)(89 188)(90 187)(91 186)(92 185)(93 184)(94 183)(95 196)(96 195)(97 194)(98 193)(99 135)(100 134)(101 133)(102 132)(103 131)(104 130)(105 129)(106 128)(107 127)(108 140)(109 139)(110 138)(111 137)(112 136)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(169 175)(170 174)(171 173)(176 182)(177 181)(178 180)(197 199)(200 210)(201 209)(202 208)(203 207)(204 206)
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), (1,20,8,27)(2,19,9,26)(3,18,10,25)(4,17,11,24)(5,16,12,23)(6,15,13,22)(7,28,14,21)(29,100,36,107)(30,99,37,106)(31,112,38,105)(32,111,39,104)(33,110,40,103)(34,109,41,102)(35,108,42,101)(43,166,50,159)(44,165,51,158)(45,164,52,157)(46,163,53,156)(47,162,54,155)(48,161,55,168)(49,160,56,167)(57,126,64,119)(58,125,65,118)(59,124,66,117)(60,123,67,116)(61,122,68,115)(62,121,69,114)(63,120,70,113)(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)(127,150,134,143)(128,149,135,142)(129,148,136,141)(130,147,137,154)(131,146,138,153)(132,145,139,152)(133,144,140,151)(169,208,176,201)(170,207,177,200)(171,206,178,199)(172,205,179,198)(173,204,180,197)(174,203,181,210)(175,202,182,209)(183,214,190,221)(184,213,191,220)(185,212,192,219)(186,211,193,218)(187,224,194,217)(188,223,195,216)(189,222,196,215), (1,105,198,129,51,211,120,82)(2,106,199,130,52,212,121,83)(3,107,200,131,53,213,122,84)(4,108,201,132,54,214,123,71)(5,109,202,133,55,215,124,72)(6,110,203,134,56,216,125,73)(7,111,204,135,43,217,126,74)(8,112,205,136,44,218,113,75)(9,99,206,137,45,219,114,76)(10,100,207,138,46,220,115,77)(11,101,208,139,47,221,116,78)(12,102,209,140,48,222,117,79)(13,103,210,127,49,223,118,80)(14,104,197,128,50,224,119,81)(15,40,181,143,167,188,65,93)(16,41,182,144,168,189,66,94)(17,42,169,145,155,190,67,95)(18,29,170,146,156,191,68,96)(19,30,171,147,157,192,69,97)(20,31,172,148,158,193,70,98)(21,32,173,149,159,194,57,85)(22,33,174,150,160,195,58,86)(23,34,175,151,161,196,59,87)(24,35,176,152,162,183,60,88)(25,36,177,153,163,184,61,89)(26,37,178,154,164,185,62,90)(27,38,179,141,165,186,63,91)(28,39,180,142,166,187,64,92), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,56)(11,55)(12,54)(13,53)(14,52)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,168)(25,167)(26,166)(27,165)(28,164)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,154)(40,153)(41,152)(42,151)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(71,222)(72,221)(73,220)(74,219)(75,218)(76,217)(77,216)(78,215)(79,214)(80,213)(81,212)(82,211)(83,224)(84,223)(85,192)(86,191)(87,190)(88,189)(89,188)(90,187)(91,186)(92,185)(93,184)(94,183)(95,196)(96,195)(97,194)(98,193)(99,135)(100,134)(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)(108,140)(109,139)(110,138)(111,137)(112,136)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(169,175)(170,174)(171,173)(176,182)(177,181)(178,180)(197,199)(200,210)(201,209)(202,208)(203,207)(204,206)>;
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), (1,20,8,27)(2,19,9,26)(3,18,10,25)(4,17,11,24)(5,16,12,23)(6,15,13,22)(7,28,14,21)(29,100,36,107)(30,99,37,106)(31,112,38,105)(32,111,39,104)(33,110,40,103)(34,109,41,102)(35,108,42,101)(43,166,50,159)(44,165,51,158)(45,164,52,157)(46,163,53,156)(47,162,54,155)(48,161,55,168)(49,160,56,167)(57,126,64,119)(58,125,65,118)(59,124,66,117)(60,123,67,116)(61,122,68,115)(62,121,69,114)(63,120,70,113)(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)(127,150,134,143)(128,149,135,142)(129,148,136,141)(130,147,137,154)(131,146,138,153)(132,145,139,152)(133,144,140,151)(169,208,176,201)(170,207,177,200)(171,206,178,199)(172,205,179,198)(173,204,180,197)(174,203,181,210)(175,202,182,209)(183,214,190,221)(184,213,191,220)(185,212,192,219)(186,211,193,218)(187,224,194,217)(188,223,195,216)(189,222,196,215), (1,105,198,129,51,211,120,82)(2,106,199,130,52,212,121,83)(3,107,200,131,53,213,122,84)(4,108,201,132,54,214,123,71)(5,109,202,133,55,215,124,72)(6,110,203,134,56,216,125,73)(7,111,204,135,43,217,126,74)(8,112,205,136,44,218,113,75)(9,99,206,137,45,219,114,76)(10,100,207,138,46,220,115,77)(11,101,208,139,47,221,116,78)(12,102,209,140,48,222,117,79)(13,103,210,127,49,223,118,80)(14,104,197,128,50,224,119,81)(15,40,181,143,167,188,65,93)(16,41,182,144,168,189,66,94)(17,42,169,145,155,190,67,95)(18,29,170,146,156,191,68,96)(19,30,171,147,157,192,69,97)(20,31,172,148,158,193,70,98)(21,32,173,149,159,194,57,85)(22,33,174,150,160,195,58,86)(23,34,175,151,161,196,59,87)(24,35,176,152,162,183,60,88)(25,36,177,153,163,184,61,89)(26,37,178,154,164,185,62,90)(27,38,179,141,165,186,63,91)(28,39,180,142,166,187,64,92), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,56)(11,55)(12,54)(13,53)(14,52)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,168)(25,167)(26,166)(27,165)(28,164)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,154)(40,153)(41,152)(42,151)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(71,222)(72,221)(73,220)(74,219)(75,218)(76,217)(77,216)(78,215)(79,214)(80,213)(81,212)(82,211)(83,224)(84,223)(85,192)(86,191)(87,190)(88,189)(89,188)(90,187)(91,186)(92,185)(93,184)(94,183)(95,196)(96,195)(97,194)(98,193)(99,135)(100,134)(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)(108,140)(109,139)(110,138)(111,137)(112,136)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(169,175)(170,174)(171,173)(176,182)(177,181)(178,180)(197,199)(200,210)(201,209)(202,208)(203,207)(204,206) );
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)], [(1,20,8,27),(2,19,9,26),(3,18,10,25),(4,17,11,24),(5,16,12,23),(6,15,13,22),(7,28,14,21),(29,100,36,107),(30,99,37,106),(31,112,38,105),(32,111,39,104),(33,110,40,103),(34,109,41,102),(35,108,42,101),(43,166,50,159),(44,165,51,158),(45,164,52,157),(46,163,53,156),(47,162,54,155),(48,161,55,168),(49,160,56,167),(57,126,64,119),(58,125,65,118),(59,124,66,117),(60,123,67,116),(61,122,68,115),(62,121,69,114),(63,120,70,113),(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),(127,150,134,143),(128,149,135,142),(129,148,136,141),(130,147,137,154),(131,146,138,153),(132,145,139,152),(133,144,140,151),(169,208,176,201),(170,207,177,200),(171,206,178,199),(172,205,179,198),(173,204,180,197),(174,203,181,210),(175,202,182,209),(183,214,190,221),(184,213,191,220),(185,212,192,219),(186,211,193,218),(187,224,194,217),(188,223,195,216),(189,222,196,215)], [(1,105,198,129,51,211,120,82),(2,106,199,130,52,212,121,83),(3,107,200,131,53,213,122,84),(4,108,201,132,54,214,123,71),(5,109,202,133,55,215,124,72),(6,110,203,134,56,216,125,73),(7,111,204,135,43,217,126,74),(8,112,205,136,44,218,113,75),(9,99,206,137,45,219,114,76),(10,100,207,138,46,220,115,77),(11,101,208,139,47,221,116,78),(12,102,209,140,48,222,117,79),(13,103,210,127,49,223,118,80),(14,104,197,128,50,224,119,81),(15,40,181,143,167,188,65,93),(16,41,182,144,168,189,66,94),(17,42,169,145,155,190,67,95),(18,29,170,146,156,191,68,96),(19,30,171,147,157,192,69,97),(20,31,172,148,158,193,70,98),(21,32,173,149,159,194,57,85),(22,33,174,150,160,195,58,86),(23,34,175,151,161,196,59,87),(24,35,176,152,162,183,60,88),(25,36,177,153,163,184,61,89),(26,37,178,154,164,185,62,90),(27,38,179,141,165,186,63,91),(28,39,180,142,166,187,64,92)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,56),(11,55),(12,54),(13,53),(14,52),(15,163),(16,162),(17,161),(18,160),(19,159),(20,158),(21,157),(22,156),(23,155),(24,168),(25,167),(26,166),(27,165),(28,164),(29,150),(30,149),(31,148),(32,147),(33,146),(34,145),(35,144),(36,143),(37,142),(38,141),(39,154),(40,153),(41,152),(42,151),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(71,222),(72,221),(73,220),(74,219),(75,218),(76,217),(77,216),(78,215),(79,214),(80,213),(81,212),(82,211),(83,224),(84,223),(85,192),(86,191),(87,190),(88,189),(89,188),(90,187),(91,186),(92,185),(93,184),(94,183),(95,196),(96,195),(97,194),(98,193),(99,135),(100,134),(101,133),(102,132),(103,131),(104,130),(105,129),(106,128),(107,127),(108,140),(109,139),(110,138),(111,137),(112,136),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(169,175),(170,174),(171,173),(176,182),(177,181),(178,180),(197,199),(200,210),(201,209),(202,208),(203,207),(204,206)]])
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 | 28A | ··· | 28F | 28G | ··· | 28R | 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 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 | D14 | C4○D8 | C4×D7 | Q8⋊2D7 | D4×D7 | D7×D8 | Q8.D14 |
| kernel | Dic7⋊5D8 | C14.D8 | C8×Dic7 | C7×C2.D8 | D28⋊C4 | C2×D56 | D56 | C2×Dic7 | C2.D8 | Dic7 | C28 | C4⋊C4 | C2×C8 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 3 | 4 | 2 | 6 | 3 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of Dic7⋊5D8 ►in GL4(𝔽113) generated by
| 0 | 112 | 0 | 0 |
| 1 | 89 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 0 | 0 | 0 |
| 92 | 15 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 51 | 99 |
| 0 | 0 | 105 | 0 |
| 1 | 0 | 0 | 0 |
| 24 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 69 | 1 |
G:=sub<GL(4,GF(113))| [0,1,0,0,112,89,0,0,0,0,1,0,0,0,0,1],[98,92,0,0,0,15,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,51,105,0,0,99,0],[1,24,0,0,0,112,0,0,0,0,112,69,0,0,0,1] >;
Dic7⋊5D8 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes_5D_8 % in TeX
G:=Group("Dic7:5D8"); // GroupNames label
G:=SmallGroup(448,406);
// by ID
G=gap.SmallGroup(448,406);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,120,135,100,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations