metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.19D4, C42.35D14, C4⋊C8⋊5D7, (C4×D28)⋊18C2, (C2×D56).4C2, (C2×C4).38D28, C4.131(D4×D7), (C2×C28).244D4, C28.340(C2×D4), (C2×C8).130D14, C7⋊3(D4.2D4), C2.D56⋊12C2, C14.12(C4○D8), C28.44D4⋊8C2, C4.D28⋊12C2, (C4×C28).70C22, (C2×C56).23C22, C28.329(C4○D4), C2.12(C4⋊D28), C14.39(C4⋊D4), C2.18(C8⋊D14), C14.15(C8⋊C22), (C2×C28).754C23, C4.45(Q8⋊2D7), (C2×D28).15C22, C22.117(C2×D28), C2.14(D56⋊7C2), C4⋊Dic7.274C22, (C2×Dic14).15C22, (C7×C4⋊C8)⋊7C2, (C2×C56⋊C2)⋊19C2, (C2×C14).137(C2×D4), (C2×C4).699(C22×D7), SmallGroup(448,378)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.19D4
G = < a,b,c,d | a28=b2=c4=1, d2=a7, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a7b, dcd-1=a14c-1 >
Subgroups: 868 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C2×C28, C22×D7, D4.2D4, C56⋊C2, D56, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C28.44D4, C2.D56, C7×C4⋊C8, C4×D28, C4.D28, C2×C56⋊C2, C2×D56, D28.19D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8⋊C22, D28, C22×D7, D4.2D4, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, D56⋊7C2, C8⋊D14, D28.19D4
►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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 36)(30 35)(31 34)(32 33)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(74 84)(75 83)(76 82)(77 81)(78 80)(85 93)(86 92)(87 91)(88 90)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(113 116)(114 115)(117 140)(118 139)(119 138)(120 137)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(141 161)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)(162 168)(163 167)(164 166)(169 172)(170 171)(173 196)(174 195)(175 194)(176 193)(177 192)(178 191)(179 190)(180 189)(181 188)(182 187)(183 186)(184 185)(197 211)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
(1 40 122 178)(2 41 123 179)(3 42 124 180)(4 43 125 181)(5 44 126 182)(6 45 127 183)(7 46 128 184)(8 47 129 185)(9 48 130 186)(10 49 131 187)(11 50 132 188)(12 51 133 189)(13 52 134 190)(14 53 135 191)(15 54 136 192)(16 55 137 193)(17 56 138 194)(18 29 139 195)(19 30 140 196)(20 31 113 169)(21 32 114 170)(22 33 115 171)(23 34 116 172)(24 35 117 173)(25 36 118 174)(26 37 119 175)(27 38 120 176)(28 39 121 177)(57 109 210 157)(58 110 211 158)(59 111 212 159)(60 112 213 160)(61 85 214 161)(62 86 215 162)(63 87 216 163)(64 88 217 164)(65 89 218 165)(66 90 219 166)(67 91 220 167)(68 92 221 168)(69 93 222 141)(70 94 223 142)(71 95 224 143)(72 96 197 144)(73 97 198 145)(74 98 199 146)(75 99 200 147)(76 100 201 148)(77 101 202 149)(78 102 203 150)(79 103 204 151)(80 104 205 152)(81 105 206 153)(82 106 207 154)(83 107 208 155)(84 108 209 156)
(1 76 8 83 15 62 22 69)(2 77 9 84 16 63 23 70)(3 78 10 57 17 64 24 71)(4 79 11 58 18 65 25 72)(5 80 12 59 19 66 26 73)(6 81 13 60 20 67 27 74)(7 82 14 61 21 68 28 75)(29 151 36 158 43 165 50 144)(30 152 37 159 44 166 51 145)(31 153 38 160 45 167 52 146)(32 154 39 161 46 168 53 147)(33 155 40 162 47 141 54 148)(34 156 41 163 48 142 55 149)(35 157 42 164 49 143 56 150)(85 184 92 191 99 170 106 177)(86 185 93 192 100 171 107 178)(87 186 94 193 101 172 108 179)(88 187 95 194 102 173 109 180)(89 188 96 195 103 174 110 181)(90 189 97 196 104 175 111 182)(91 190 98 169 105 176 112 183)(113 220 120 199 127 206 134 213)(114 221 121 200 128 207 135 214)(115 222 122 201 129 208 136 215)(116 223 123 202 130 209 137 216)(117 224 124 203 131 210 138 217)(118 197 125 204 132 211 139 218)(119 198 126 205 133 212 140 219)
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,36)(30,35)(31,34)(32,33)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,93)(86,92)(87,91)(88,90)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(113,116)(114,115)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(162,168)(163,167)(164,166)(169,172)(170,171)(173,196)(174,195)(175,194)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,40,122,178)(2,41,123,179)(3,42,124,180)(4,43,125,181)(5,44,126,182)(6,45,127,183)(7,46,128,184)(8,47,129,185)(9,48,130,186)(10,49,131,187)(11,50,132,188)(12,51,133,189)(13,52,134,190)(14,53,135,191)(15,54,136,192)(16,55,137,193)(17,56,138,194)(18,29,139,195)(19,30,140,196)(20,31,113,169)(21,32,114,170)(22,33,115,171)(23,34,116,172)(24,35,117,173)(25,36,118,174)(26,37,119,175)(27,38,120,176)(28,39,121,177)(57,109,210,157)(58,110,211,158)(59,111,212,159)(60,112,213,160)(61,85,214,161)(62,86,215,162)(63,87,216,163)(64,88,217,164)(65,89,218,165)(66,90,219,166)(67,91,220,167)(68,92,221,168)(69,93,222,141)(70,94,223,142)(71,95,224,143)(72,96,197,144)(73,97,198,145)(74,98,199,146)(75,99,200,147)(76,100,201,148)(77,101,202,149)(78,102,203,150)(79,103,204,151)(80,104,205,152)(81,105,206,153)(82,106,207,154)(83,107,208,155)(84,108,209,156), (1,76,8,83,15,62,22,69)(2,77,9,84,16,63,23,70)(3,78,10,57,17,64,24,71)(4,79,11,58,18,65,25,72)(5,80,12,59,19,66,26,73)(6,81,13,60,20,67,27,74)(7,82,14,61,21,68,28,75)(29,151,36,158,43,165,50,144)(30,152,37,159,44,166,51,145)(31,153,38,160,45,167,52,146)(32,154,39,161,46,168,53,147)(33,155,40,162,47,141,54,148)(34,156,41,163,48,142,55,149)(35,157,42,164,49,143,56,150)(85,184,92,191,99,170,106,177)(86,185,93,192,100,171,107,178)(87,186,94,193,101,172,108,179)(88,187,95,194,102,173,109,180)(89,188,96,195,103,174,110,181)(90,189,97,196,104,175,111,182)(91,190,98,169,105,176,112,183)(113,220,120,199,127,206,134,213)(114,221,121,200,128,207,135,214)(115,222,122,201,129,208,136,215)(116,223,123,202,130,209,137,216)(117,224,124,203,131,210,138,217)(118,197,125,204,132,211,139,218)(119,198,126,205,133,212,140,219)>;
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,36)(30,35)(31,34)(32,33)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(74,84)(75,83)(76,82)(77,81)(78,80)(85,93)(86,92)(87,91)(88,90)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(113,116)(114,115)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(162,168)(163,167)(164,166)(169,172)(170,171)(173,196)(174,195)(175,194)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,40,122,178)(2,41,123,179)(3,42,124,180)(4,43,125,181)(5,44,126,182)(6,45,127,183)(7,46,128,184)(8,47,129,185)(9,48,130,186)(10,49,131,187)(11,50,132,188)(12,51,133,189)(13,52,134,190)(14,53,135,191)(15,54,136,192)(16,55,137,193)(17,56,138,194)(18,29,139,195)(19,30,140,196)(20,31,113,169)(21,32,114,170)(22,33,115,171)(23,34,116,172)(24,35,117,173)(25,36,118,174)(26,37,119,175)(27,38,120,176)(28,39,121,177)(57,109,210,157)(58,110,211,158)(59,111,212,159)(60,112,213,160)(61,85,214,161)(62,86,215,162)(63,87,216,163)(64,88,217,164)(65,89,218,165)(66,90,219,166)(67,91,220,167)(68,92,221,168)(69,93,222,141)(70,94,223,142)(71,95,224,143)(72,96,197,144)(73,97,198,145)(74,98,199,146)(75,99,200,147)(76,100,201,148)(77,101,202,149)(78,102,203,150)(79,103,204,151)(80,104,205,152)(81,105,206,153)(82,106,207,154)(83,107,208,155)(84,108,209,156), (1,76,8,83,15,62,22,69)(2,77,9,84,16,63,23,70)(3,78,10,57,17,64,24,71)(4,79,11,58,18,65,25,72)(5,80,12,59,19,66,26,73)(6,81,13,60,20,67,27,74)(7,82,14,61,21,68,28,75)(29,151,36,158,43,165,50,144)(30,152,37,159,44,166,51,145)(31,153,38,160,45,167,52,146)(32,154,39,161,46,168,53,147)(33,155,40,162,47,141,54,148)(34,156,41,163,48,142,55,149)(35,157,42,164,49,143,56,150)(85,184,92,191,99,170,106,177)(86,185,93,192,100,171,107,178)(87,186,94,193,101,172,108,179)(88,187,95,194,102,173,109,180)(89,188,96,195,103,174,110,181)(90,189,97,196,104,175,111,182)(91,190,98,169,105,176,112,183)(113,220,120,199,127,206,134,213)(114,221,121,200,128,207,135,214)(115,222,122,201,129,208,136,215)(116,223,123,202,130,209,137,216)(117,224,124,203,131,210,138,217)(118,197,125,204,132,211,139,218)(119,198,126,205,133,212,140,219) );
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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,36),(30,35),(31,34),(32,33),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(57,73),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67),(64,66),(74,84),(75,83),(76,82),(77,81),(78,80),(85,93),(86,92),(87,91),(88,90),(94,112),(95,111),(96,110),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(113,116),(114,115),(117,140),(118,139),(119,138),(120,137),(121,136),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(141,161),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152),(162,168),(163,167),(164,166),(169,172),(170,171),(173,196),(174,195),(175,194),(176,193),(177,192),(178,191),(179,190),(180,189),(181,188),(182,187),(183,186),(184,185),(197,211),(198,210),(199,209),(200,208),(201,207),(202,206),(203,205),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219)], [(1,40,122,178),(2,41,123,179),(3,42,124,180),(4,43,125,181),(5,44,126,182),(6,45,127,183),(7,46,128,184),(8,47,129,185),(9,48,130,186),(10,49,131,187),(11,50,132,188),(12,51,133,189),(13,52,134,190),(14,53,135,191),(15,54,136,192),(16,55,137,193),(17,56,138,194),(18,29,139,195),(19,30,140,196),(20,31,113,169),(21,32,114,170),(22,33,115,171),(23,34,116,172),(24,35,117,173),(25,36,118,174),(26,37,119,175),(27,38,120,176),(28,39,121,177),(57,109,210,157),(58,110,211,158),(59,111,212,159),(60,112,213,160),(61,85,214,161),(62,86,215,162),(63,87,216,163),(64,88,217,164),(65,89,218,165),(66,90,219,166),(67,91,220,167),(68,92,221,168),(69,93,222,141),(70,94,223,142),(71,95,224,143),(72,96,197,144),(73,97,198,145),(74,98,199,146),(75,99,200,147),(76,100,201,148),(77,101,202,149),(78,102,203,150),(79,103,204,151),(80,104,205,152),(81,105,206,153),(82,106,207,154),(83,107,208,155),(84,108,209,156)], [(1,76,8,83,15,62,22,69),(2,77,9,84,16,63,23,70),(3,78,10,57,17,64,24,71),(4,79,11,58,18,65,25,72),(5,80,12,59,19,66,26,73),(6,81,13,60,20,67,27,74),(7,82,14,61,21,68,28,75),(29,151,36,158,43,165,50,144),(30,152,37,159,44,166,51,145),(31,153,38,160,45,167,52,146),(32,154,39,161,46,168,53,147),(33,155,40,162,47,141,54,148),(34,156,41,163,48,142,55,149),(35,157,42,164,49,143,56,150),(85,184,92,191,99,170,106,177),(86,185,93,192,100,171,107,178),(87,186,94,193,101,172,108,179),(88,187,95,194,102,173,109,180),(89,188,96,195,103,174,110,181),(90,189,97,196,104,175,111,182),(91,190,98,169,105,176,112,183),(113,220,120,199,127,206,134,213),(114,221,121,200,128,207,135,214),(115,222,122,201,129,208,136,215),(116,223,123,202,130,209,137,216),(117,224,124,203,131,210,138,217),(118,197,125,204,132,211,139,218),(119,198,126,205,133,212,140,219)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 28 | 56 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 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 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C4○D8 | D28 | D56⋊7C2 | C8⋊C22 | D4×D7 | Q8⋊2D7 | C8⋊D14 |
| kernel | D28.19D4 | C28.44D4 | C2.D56 | C7×C4⋊C8 | C4×D28 | C4.D28 | C2×C56⋊C2 | C2×D56 | D28 | C2×C28 | C4⋊C8 | C28 | C42 | C2×C8 | C14 | C2×C4 | C2 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 6 | 4 | 12 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of D28.19D4 ►in GL4(𝔽113) generated by
| 77 | 32 | 0 | 0 |
| 100 | 90 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 105 | 4 | 0 | 0 |
| 69 | 8 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 35 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 73 | 15 |
| 69 | 65 | 0 | 0 |
| 76 | 106 | 0 | 0 |
| 0 | 0 | 20 | 98 |
| 0 | 0 | 87 | 93 |
G:=sub<GL(4,GF(113))| [77,100,0,0,32,90,0,0,0,0,112,0,0,0,0,112],[105,69,0,0,4,8,0,0,0,0,112,35,0,0,0,1],[15,0,0,0,0,15,0,0,0,0,98,73,0,0,0,15],[69,76,0,0,65,106,0,0,0,0,20,87,0,0,98,93] >;
D28.19D4 in GAP, Magma, Sage, TeX
D_{28}._{19}D_4 % in TeX
G:=Group("D28.19D4"); // GroupNames label
G:=SmallGroup(448,378);
// by ID
G=gap.SmallGroup(448,378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,344,254,219,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^7,b*a*b=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^14*c^-1>;
// generators/relations