metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊17D4, Dic14⋊16D4, C4⋊D4⋊2D7, C4.99(D4×D7), C7⋊4(D4⋊D4), C4⋊C4.57D14, (C2×D4).37D14, C28.146(C2×D4), (C2×C28).262D4, C14.D8⋊34C2, C14.45C22≀C2, C14.96(C4○D8), C14.Q16⋊33C2, (C22×C14).83D4, C28.55D4⋊11C2, C14.90(C8⋊C22), (C2×C28).356C23, (D4×C14).53C22, (C22×C4).120D14, C23.23(C7⋊D4), C2.13(C23⋊D14), (C2×D28).241C22, C2.15(D4.8D14), C2.11(D4.D14), (C22×C28).160C22, (C2×Dic14).268C22, (C2×D4⋊D7)⋊9C2, (C7×C4⋊D4)⋊2C2, (C2×D4.D7)⋊8C2, (C2×C4○D28)⋊15C2, (C2×C14).487(C2×D4), (C2×C7⋊C8).108C22, (C2×C4).171(C7⋊D4), (C7×C4⋊C4).104C22, (C2×C4).456(C22×D7), C22.162(C2×C7⋊D4), SmallGroup(448,571)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊17D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=a15, ad=da, cbc-1=a21b, dbd=a14b, dcd=c-1 >
Subgroups: 844 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, 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, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, D4⋊D4, C2×C7⋊C8, D4⋊D7, D4.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C14.D8, C14.Q16, C28.55D4, C2×D4⋊D7, C2×D4.D7, C7×C4⋊D4, C2×C4○D28, D28⋊17D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C4○D8, C8⋊C22, C7⋊D4, C22×D7, D4⋊D4, D4×D7, C2×C7⋊D4, D4.D14, C23⋊D14, D4.8D14, D28⋊17D4
►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 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 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 95)(86 94)(87 93)(88 92)(89 91)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(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)(169 175)(170 174)(171 173)(176 196)(177 195)(178 194)(179 193)(180 192)(181 191)(182 190)(183 189)(184 188)(185 187)(197 210)(198 209)(199 208)(200 207)(201 206)(202 205)(203 204)(211 224)(212 223)(213 222)(214 221)(215 220)(216 219)(217 218)
(1 162 204 53)(2 149 205 40)(3 164 206 55)(4 151 207 42)(5 166 208 29)(6 153 209 44)(7 168 210 31)(8 155 211 46)(9 142 212 33)(10 157 213 48)(11 144 214 35)(12 159 215 50)(13 146 216 37)(14 161 217 52)(15 148 218 39)(16 163 219 54)(17 150 220 41)(18 165 221 56)(19 152 222 43)(20 167 223 30)(21 154 224 45)(22 141 197 32)(23 156 198 47)(24 143 199 34)(25 158 200 49)(26 145 201 36)(27 160 202 51)(28 147 203 38)(57 93 140 189)(58 108 113 176)(59 95 114 191)(60 110 115 178)(61 97 116 193)(62 112 117 180)(63 99 118 195)(64 86 119 182)(65 101 120 169)(66 88 121 184)(67 103 122 171)(68 90 123 186)(69 105 124 173)(70 92 125 188)(71 107 126 175)(72 94 127 190)(73 109 128 177)(74 96 129 192)(75 111 130 179)(76 98 131 194)(77 85 132 181)(78 100 133 196)(79 87 134 183)(80 102 135 170)(81 89 136 185)(82 104 137 172)(83 91 138 187)(84 106 139 174)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 112)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 197)(121 198)(122 199)(123 200)(124 201)(125 202)(126 203)(127 204)(128 205)(129 206)(130 207)(131 208)(132 209)(133 210)(134 211)(135 212)(136 213)(137 214)(138 215)(139 216)(140 217)(141 169)(142 170)(143 171)(144 172)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 181)(154 182)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)
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), (1,28)(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)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,95)(86,94)(87,93)(88,92)(89,91)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(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)(169,175)(170,174)(171,173)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(197,210)(198,209)(199,208)(200,207)(201,206)(202,205)(203,204)(211,224)(212,223)(213,222)(214,221)(215,220)(216,219)(217,218), (1,162,204,53)(2,149,205,40)(3,164,206,55)(4,151,207,42)(5,166,208,29)(6,153,209,44)(7,168,210,31)(8,155,211,46)(9,142,212,33)(10,157,213,48)(11,144,214,35)(12,159,215,50)(13,146,216,37)(14,161,217,52)(15,148,218,39)(16,163,219,54)(17,150,220,41)(18,165,221,56)(19,152,222,43)(20,167,223,30)(21,154,224,45)(22,141,197,32)(23,156,198,47)(24,143,199,34)(25,158,200,49)(26,145,201,36)(27,160,202,51)(28,147,203,38)(57,93,140,189)(58,108,113,176)(59,95,114,191)(60,110,115,178)(61,97,116,193)(62,112,117,180)(63,99,118,195)(64,86,119,182)(65,101,120,169)(66,88,121,184)(67,103,122,171)(68,90,123,186)(69,105,124,173)(70,92,125,188)(71,107,126,175)(72,94,127,190)(73,109,128,177)(74,96,129,192)(75,111,130,179)(76,98,131,194)(77,85,132,181)(78,100,133,196)(79,87,134,183)(80,102,135,170)(81,89,136,185)(82,104,137,172)(83,91,138,187)(84,106,139,174), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,197)(121,198)(122,199)(123,200)(124,201)(125,202)(126,203)(127,204)(128,205)(129,206)(130,207)(131,208)(132,209)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)>;
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), (1,28)(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)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,95)(86,94)(87,93)(88,92)(89,91)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(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)(169,175)(170,174)(171,173)(176,196)(177,195)(178,194)(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187)(197,210)(198,209)(199,208)(200,207)(201,206)(202,205)(203,204)(211,224)(212,223)(213,222)(214,221)(215,220)(216,219)(217,218), (1,162,204,53)(2,149,205,40)(3,164,206,55)(4,151,207,42)(5,166,208,29)(6,153,209,44)(7,168,210,31)(8,155,211,46)(9,142,212,33)(10,157,213,48)(11,144,214,35)(12,159,215,50)(13,146,216,37)(14,161,217,52)(15,148,218,39)(16,163,219,54)(17,150,220,41)(18,165,221,56)(19,152,222,43)(20,167,223,30)(21,154,224,45)(22,141,197,32)(23,156,198,47)(24,143,199,34)(25,158,200,49)(26,145,201,36)(27,160,202,51)(28,147,203,38)(57,93,140,189)(58,108,113,176)(59,95,114,191)(60,110,115,178)(61,97,116,193)(62,112,117,180)(63,99,118,195)(64,86,119,182)(65,101,120,169)(66,88,121,184)(67,103,122,171)(68,90,123,186)(69,105,124,173)(70,92,125,188)(71,107,126,175)(72,94,127,190)(73,109,128,177)(74,96,129,192)(75,111,130,179)(76,98,131,194)(77,85,132,181)(78,100,133,196)(79,87,134,183)(80,102,135,170)(81,89,136,185)(82,104,137,172)(83,91,138,187)(84,106,139,174), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,197)(121,198)(122,199)(123,200)(124,201)(125,202)(126,203)(127,204)(128,205)(129,206)(130,207)(131,208)(132,209)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196) );
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)], [(1,28),(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),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(57,72),(58,71),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,95),(86,94),(87,93),(88,92),(89,91),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,120),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(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),(169,175),(170,174),(171,173),(176,196),(177,195),(178,194),(179,193),(180,192),(181,191),(182,190),(183,189),(184,188),(185,187),(197,210),(198,209),(199,208),(200,207),(201,206),(202,205),(203,204),(211,224),(212,223),(213,222),(214,221),(215,220),(216,219),(217,218)], [(1,162,204,53),(2,149,205,40),(3,164,206,55),(4,151,207,42),(5,166,208,29),(6,153,209,44),(7,168,210,31),(8,155,211,46),(9,142,212,33),(10,157,213,48),(11,144,214,35),(12,159,215,50),(13,146,216,37),(14,161,217,52),(15,148,218,39),(16,163,219,54),(17,150,220,41),(18,165,221,56),(19,152,222,43),(20,167,223,30),(21,154,224,45),(22,141,197,32),(23,156,198,47),(24,143,199,34),(25,158,200,49),(26,145,201,36),(27,160,202,51),(28,147,203,38),(57,93,140,189),(58,108,113,176),(59,95,114,191),(60,110,115,178),(61,97,116,193),(62,112,117,180),(63,99,118,195),(64,86,119,182),(65,101,120,169),(66,88,121,184),(67,103,122,171),(68,90,123,186),(69,105,124,173),(70,92,125,188),(71,107,126,175),(72,94,127,190),(73,109,128,177),(74,96,129,192),(75,111,130,179),(76,98,131,194),(77,85,132,181),(78,100,133,196),(79,87,134,183),(80,102,135,170),(81,89,136,185),(82,104,137,172),(83,91,138,187),(84,106,139,174)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,57),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,112),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(113,218),(114,219),(115,220),(116,221),(117,222),(118,223),(119,224),(120,197),(121,198),(122,199),(123,200),(124,201),(125,202),(126,203),(127,204),(128,205),(129,206),(130,207),(131,208),(132,209),(133,210),(134,211),(135,212),(136,213),(137,214),(138,215),(139,216),(140,217),(141,169),(142,170),(143,171),(144,172),(145,173),(146,174),(147,175),(148,176),(149,177),(150,178),(151,179),(152,180),(153,181),(154,182),(155,183),(156,184),(157,185),(158,186),(159,187),(160,188),(161,189),(162,190),(163,191),(164,192),(165,193),(166,194),(167,195),(168,196)]])
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 | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 28 | 28 | 2 | 2 | 2 | 2 | 8 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4×D7 | D4.D14 | D4.8D14 |
| kernel | D28⋊17D4 | C14.D8 | C14.Q16 | C28.55D4 | C2×D4⋊D7 | C2×D4.D7 | C7×C4⋊D4 | C2×C4○D28 | Dic14 | D28 | C2×C28 | C22×C14 | C4⋊D4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2×C4 | C23 | C14 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 1 | 6 | 6 | 6 |
Matrix representation of D28⋊17D4 ►in GL6(𝔽113)
| 103 | 24 | 0 | 0 | 0 | 0 |
| 89 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 111 |
| 0 | 0 | 0 | 0 | 110 | 28 |
| 1 | 89 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 111 |
| 0 | 0 | 0 | 0 | 109 | 28 |
| 29 | 7 | 0 | 0 | 0 | 0 |
| 106 | 84 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 75 | 87 |
| 0 | 0 | 0 | 0 | 86 | 38 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 83 |
| 0 | 0 | 0 | 0 | 68 | 81 |
G:=sub<GL(6,GF(113))| [103,89,0,0,0,0,24,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,85,110,0,0,0,0,111,28],[1,0,0,0,0,0,89,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,85,109,0,0,0,0,111,28],[29,106,0,0,0,0,7,84,0,0,0,0,0,0,0,1,0,0,0,0,112,0,0,0,0,0,0,0,75,86,0,0,0,0,87,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,32,68,0,0,0,0,83,81] >;
D28⋊17D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_{17}D_4 % in TeX
G:=Group("D28:17D4"); // GroupNames label
G:=SmallGroup(448,571);
// by ID
G=gap.SmallGroup(448,571);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^21*b,d*b*d=a^14*b,d*c*d=c^-1>;
// generators/relations