metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊3D28, Dic14⋊3D4, (C7×D4)⋊2D4, C4.86(D4×D7), C28.2(C2×D4), C4.3(C2×D28), D14⋊C8⋊12C2, C4⋊D28⋊3C2, C7⋊2(D4⋊D4), C4⋊C4.12D14, D4⋊C4⋊13D7, (C2×C8).117D14, C14.Q16⋊6C2, C14.21C22≀C2, (C2×D4).136D14, C14.42(C4○D8), (C22×D7).13D4, C22.179(D4×D7), C2.17(D8⋊D7), C14.35(C8⋊C22), (C2×C56).128C22, (C2×C28).221C23, (C2×Dic7).142D4, (D4×C14).42C22, (C2×D28).51C22, C2.24(C22⋊D28), C2.12(SD16⋊3D7), (C2×Dic14).59C22, (C2×D4⋊D7)⋊4C2, (C2×D4⋊2D7)⋊1C2, (C2×C56⋊C2)⋊15C2, (C2×C7⋊C8).19C22, (C7×D4⋊C4)⋊13C2, (C2×C4×D7).13C22, (C2×C14).234(C2×D4), (C7×C4⋊C4).22C22, (C2×C4).328(C22×D7), SmallGroup(448,315)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊3D28
G = < a,b,c,d | a4=b2=c28=d2=1, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 1012 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, D4⋊D4, C56⋊C2, C2×C7⋊C8, D14⋊C4, D4⋊D7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C2×D28, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C14.Q16, D14⋊C8, C7×D4⋊C4, C4⋊D28, C2×C56⋊C2, C2×D4⋊D7, C2×D4⋊2D7, D4⋊3D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C4○D8, C8⋊C22, D28, C22×D7, D4⋊D4, C2×D28, D4×D7, C22⋊D28, D8⋊D7, SD16⋊3D7, D4⋊3D28
►On 224 points
Generators in S224
(1 121 30 70)(2 71 31 122)(3 123 32 72)(4 73 33 124)(5 125 34 74)(6 75 35 126)(7 127 36 76)(8 77 37 128)(9 129 38 78)(10 79 39 130)(11 131 40 80)(12 81 41 132)(13 133 42 82)(14 83 43 134)(15 135 44 84)(16 57 45 136)(17 137 46 58)(18 59 47 138)(19 139 48 60)(20 61 49 140)(21 113 50 62)(22 63 51 114)(23 115 52 64)(24 65 53 116)(25 117 54 66)(26 67 55 118)(27 119 56 68)(28 69 29 120)(85 182 159 216)(86 217 160 183)(87 184 161 218)(88 219 162 185)(89 186 163 220)(90 221 164 187)(91 188 165 222)(92 223 166 189)(93 190 167 224)(94 197 168 191)(95 192 141 198)(96 199 142 193)(97 194 143 200)(98 201 144 195)(99 196 145 202)(100 203 146 169)(101 170 147 204)(102 205 148 171)(103 172 149 206)(104 207 150 173)(105 174 151 208)(106 209 152 175)(107 176 153 210)(108 211 154 177)(109 178 155 212)(110 213 156 179)(111 180 157 214)(112 215 158 181)
(1 91)(2 189)(3 93)(4 191)(5 95)(6 193)(7 97)(8 195)(9 99)(10 169)(11 101)(12 171)(13 103)(14 173)(15 105)(16 175)(17 107)(18 177)(19 109)(20 179)(21 111)(22 181)(23 85)(24 183)(25 87)(26 185)(27 89)(28 187)(29 221)(30 165)(31 223)(32 167)(33 197)(34 141)(35 199)(36 143)(37 201)(38 145)(39 203)(40 147)(41 205)(42 149)(43 207)(44 151)(45 209)(46 153)(47 211)(48 155)(49 213)(50 157)(51 215)(52 159)(53 217)(54 161)(55 219)(56 163)(57 152)(58 176)(59 154)(60 178)(61 156)(62 180)(63 158)(64 182)(65 160)(66 184)(67 162)(68 186)(69 164)(70 188)(71 166)(72 190)(73 168)(74 192)(75 142)(76 194)(77 144)(78 196)(79 146)(80 170)(81 148)(82 172)(83 150)(84 174)(86 116)(88 118)(90 120)(92 122)(94 124)(96 126)(98 128)(100 130)(102 132)(104 134)(106 136)(108 138)(110 140)(112 114)(113 214)(115 216)(117 218)(119 220)(121 222)(123 224)(125 198)(127 200)(129 202)(131 204)(133 206)(135 208)(137 210)(139 212)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 30)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(83 84)(85 142)(86 141)(87 168)(88 167)(89 166)(90 165)(91 164)(92 163)(93 162)(94 161)(95 160)(96 159)(97 158)(98 157)(99 156)(100 155)(101 154)(102 153)(103 152)(104 151)(105 150)(106 149)(107 148)(108 147)(109 146)(110 145)(111 144)(112 143)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)(169 212)(170 211)(171 210)(172 209)(173 208)(174 207)(175 206)(176 205)(177 204)(178 203)(179 202)(180 201)(181 200)(182 199)(183 198)(184 197)(185 224)(186 223)(187 222)(188 221)(189 220)(190 219)(191 218)(192 217)(193 216)(194 215)(195 214)(196 213)
G:=sub<Sym(224)| (1,121,30,70)(2,71,31,122)(3,123,32,72)(4,73,33,124)(5,125,34,74)(6,75,35,126)(7,127,36,76)(8,77,37,128)(9,129,38,78)(10,79,39,130)(11,131,40,80)(12,81,41,132)(13,133,42,82)(14,83,43,134)(15,135,44,84)(16,57,45,136)(17,137,46,58)(18,59,47,138)(19,139,48,60)(20,61,49,140)(21,113,50,62)(22,63,51,114)(23,115,52,64)(24,65,53,116)(25,117,54,66)(26,67,55,118)(27,119,56,68)(28,69,29,120)(85,182,159,216)(86,217,160,183)(87,184,161,218)(88,219,162,185)(89,186,163,220)(90,221,164,187)(91,188,165,222)(92,223,166,189)(93,190,167,224)(94,197,168,191)(95,192,141,198)(96,199,142,193)(97,194,143,200)(98,201,144,195)(99,196,145,202)(100,203,146,169)(101,170,147,204)(102,205,148,171)(103,172,149,206)(104,207,150,173)(105,174,151,208)(106,209,152,175)(107,176,153,210)(108,211,154,177)(109,178,155,212)(110,213,156,179)(111,180,157,214)(112,215,158,181), (1,91)(2,189)(3,93)(4,191)(5,95)(6,193)(7,97)(8,195)(9,99)(10,169)(11,101)(12,171)(13,103)(14,173)(15,105)(16,175)(17,107)(18,177)(19,109)(20,179)(21,111)(22,181)(23,85)(24,183)(25,87)(26,185)(27,89)(28,187)(29,221)(30,165)(31,223)(32,167)(33,197)(34,141)(35,199)(36,143)(37,201)(38,145)(39,203)(40,147)(41,205)(42,149)(43,207)(44,151)(45,209)(46,153)(47,211)(48,155)(49,213)(50,157)(51,215)(52,159)(53,217)(54,161)(55,219)(56,163)(57,152)(58,176)(59,154)(60,178)(61,156)(62,180)(63,158)(64,182)(65,160)(66,184)(67,162)(68,186)(69,164)(70,188)(71,166)(72,190)(73,168)(74,192)(75,142)(76,194)(77,144)(78,196)(79,146)(80,170)(81,148)(82,172)(83,150)(84,174)(86,116)(88,118)(90,120)(92,122)(94,124)(96,126)(98,128)(100,130)(102,132)(104,134)(106,136)(108,138)(110,140)(112,114)(113,214)(115,216)(117,218)(119,220)(121,222)(123,224)(125,198)(127,200)(129,202)(131,204)(133,206)(135,208)(137,210)(139,212), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,30)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(83,84)(85,142)(86,141)(87,168)(88,167)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(169,212)(170,211)(171,210)(172,209)(173,208)(174,207)(175,206)(176,205)(177,204)(178,203)(179,202)(180,201)(181,200)(182,199)(183,198)(184,197)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)(194,215)(195,214)(196,213)>;
G:=Group( (1,121,30,70)(2,71,31,122)(3,123,32,72)(4,73,33,124)(5,125,34,74)(6,75,35,126)(7,127,36,76)(8,77,37,128)(9,129,38,78)(10,79,39,130)(11,131,40,80)(12,81,41,132)(13,133,42,82)(14,83,43,134)(15,135,44,84)(16,57,45,136)(17,137,46,58)(18,59,47,138)(19,139,48,60)(20,61,49,140)(21,113,50,62)(22,63,51,114)(23,115,52,64)(24,65,53,116)(25,117,54,66)(26,67,55,118)(27,119,56,68)(28,69,29,120)(85,182,159,216)(86,217,160,183)(87,184,161,218)(88,219,162,185)(89,186,163,220)(90,221,164,187)(91,188,165,222)(92,223,166,189)(93,190,167,224)(94,197,168,191)(95,192,141,198)(96,199,142,193)(97,194,143,200)(98,201,144,195)(99,196,145,202)(100,203,146,169)(101,170,147,204)(102,205,148,171)(103,172,149,206)(104,207,150,173)(105,174,151,208)(106,209,152,175)(107,176,153,210)(108,211,154,177)(109,178,155,212)(110,213,156,179)(111,180,157,214)(112,215,158,181), (1,91)(2,189)(3,93)(4,191)(5,95)(6,193)(7,97)(8,195)(9,99)(10,169)(11,101)(12,171)(13,103)(14,173)(15,105)(16,175)(17,107)(18,177)(19,109)(20,179)(21,111)(22,181)(23,85)(24,183)(25,87)(26,185)(27,89)(28,187)(29,221)(30,165)(31,223)(32,167)(33,197)(34,141)(35,199)(36,143)(37,201)(38,145)(39,203)(40,147)(41,205)(42,149)(43,207)(44,151)(45,209)(46,153)(47,211)(48,155)(49,213)(50,157)(51,215)(52,159)(53,217)(54,161)(55,219)(56,163)(57,152)(58,176)(59,154)(60,178)(61,156)(62,180)(63,158)(64,182)(65,160)(66,184)(67,162)(68,186)(69,164)(70,188)(71,166)(72,190)(73,168)(74,192)(75,142)(76,194)(77,144)(78,196)(79,146)(80,170)(81,148)(82,172)(83,150)(84,174)(86,116)(88,118)(90,120)(92,122)(94,124)(96,126)(98,128)(100,130)(102,132)(104,134)(106,136)(108,138)(110,140)(112,114)(113,214)(115,216)(117,218)(119,220)(121,222)(123,224)(125,198)(127,200)(129,202)(131,204)(133,206)(135,208)(137,210)(139,212), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,30)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(83,84)(85,142)(86,141)(87,168)(88,167)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(169,212)(170,211)(171,210)(172,209)(173,208)(174,207)(175,206)(176,205)(177,204)(178,203)(179,202)(180,201)(181,200)(182,199)(183,198)(184,197)(185,224)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)(194,215)(195,214)(196,213) );
G=PermutationGroup([[(1,121,30,70),(2,71,31,122),(3,123,32,72),(4,73,33,124),(5,125,34,74),(6,75,35,126),(7,127,36,76),(8,77,37,128),(9,129,38,78),(10,79,39,130),(11,131,40,80),(12,81,41,132),(13,133,42,82),(14,83,43,134),(15,135,44,84),(16,57,45,136),(17,137,46,58),(18,59,47,138),(19,139,48,60),(20,61,49,140),(21,113,50,62),(22,63,51,114),(23,115,52,64),(24,65,53,116),(25,117,54,66),(26,67,55,118),(27,119,56,68),(28,69,29,120),(85,182,159,216),(86,217,160,183),(87,184,161,218),(88,219,162,185),(89,186,163,220),(90,221,164,187),(91,188,165,222),(92,223,166,189),(93,190,167,224),(94,197,168,191),(95,192,141,198),(96,199,142,193),(97,194,143,200),(98,201,144,195),(99,196,145,202),(100,203,146,169),(101,170,147,204),(102,205,148,171),(103,172,149,206),(104,207,150,173),(105,174,151,208),(106,209,152,175),(107,176,153,210),(108,211,154,177),(109,178,155,212),(110,213,156,179),(111,180,157,214),(112,215,158,181)], [(1,91),(2,189),(3,93),(4,191),(5,95),(6,193),(7,97),(8,195),(9,99),(10,169),(11,101),(12,171),(13,103),(14,173),(15,105),(16,175),(17,107),(18,177),(19,109),(20,179),(21,111),(22,181),(23,85),(24,183),(25,87),(26,185),(27,89),(28,187),(29,221),(30,165),(31,223),(32,167),(33,197),(34,141),(35,199),(36,143),(37,201),(38,145),(39,203),(40,147),(41,205),(42,149),(43,207),(44,151),(45,209),(46,153),(47,211),(48,155),(49,213),(50,157),(51,215),(52,159),(53,217),(54,161),(55,219),(56,163),(57,152),(58,176),(59,154),(60,178),(61,156),(62,180),(63,158),(64,182),(65,160),(66,184),(67,162),(68,186),(69,164),(70,188),(71,166),(72,190),(73,168),(74,192),(75,142),(76,194),(77,144),(78,196),(79,146),(80,170),(81,148),(82,172),(83,150),(84,174),(86,116),(88,118),(90,120),(92,122),(94,124),(96,126),(98,128),(100,130),(102,132),(104,134),(106,136),(108,138),(110,140),(112,114),(113,214),(115,216),(117,218),(119,220),(121,222),(123,224),(125,198),(127,200),(129,202),(131,204),(133,206),(135,208),(137,210),(139,212)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,30),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(57,82),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(83,84),(85,142),(86,141),(87,168),(88,167),(89,166),(90,165),(91,164),(92,163),(93,162),(94,161),(95,160),(96,159),(97,158),(98,157),(99,156),(100,155),(101,154),(102,153),(103,152),(104,151),(105,150),(106,149),(107,148),(108,147),(109,146),(110,145),(111,144),(112,143),(113,128),(114,127),(115,126),(116,125),(117,124),(118,123),(119,122),(120,121),(129,140),(130,139),(131,138),(132,137),(133,136),(134,135),(169,212),(170,211),(171,210),(172,209),(173,208),(174,207),(175,206),(176,205),(177,204),(178,203),(179,202),(180,201),(181,200),(182,199),(183,198),(184,197),(185,224),(186,223),(187,222),(188,221),(189,220),(190,219),(191,218),(192,217),(193,216),(194,215),(195,214),(196,213)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 56 | 2 | 2 | 8 | 14 | 14 | 28 | 28 | 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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | D28 | C8⋊C22 | D4×D7 | D4×D7 | D8⋊D7 | SD16⋊3D7 |
| kernel | D4⋊3D28 | C14.Q16 | D14⋊C8 | C7×D4⋊C4 | C4⋊D28 | C2×C56⋊C2 | C2×D4⋊D7 | C2×D4⋊2D7 | Dic14 | C2×Dic7 | C7×D4 | C22×D7 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | C14 | D4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D4⋊3D28 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 53 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 81 | 16 |
| 0 | 0 | 42 | 32 |
| 23 | 100 | 0 | 0 |
| 26 | 5 | 0 | 0 |
| 0 | 0 | 52 | 87 |
| 0 | 0 | 91 | 61 |
| 71 | 94 | 0 | 0 |
| 69 | 42 | 0 | 0 |
| 0 | 0 | 52 | 87 |
| 0 | 0 | 4 | 61 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,98,53,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,81,42,0,0,16,32],[23,26,0,0,100,5,0,0,0,0,52,91,0,0,87,61],[71,69,0,0,94,42,0,0,0,0,52,4,0,0,87,61] >;
D4⋊3D28 in GAP, Magma, Sage, TeX
D_4\rtimes_3D_{28} % in TeX
G:=Group("D4:3D28"); // GroupNames label
G:=SmallGroup(448,315);
// by ID
G=gap.SmallGroup(448,315);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,254,219,58,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations