metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.9D4, C4.89(D4×D7), D4⋊C4⋊14D7, D28⋊C4⋊4C2, Dic7⋊C8⋊12C2, C4⋊C4.141D14, (C2×D4).32D14, C4.4(C4○D28), (C2×C8).118D14, C28.113(C2×D4), C14.Q16⋊8C2, C7⋊1(D4.2D4), C14.43(C4○D8), C28.12(C4○D4), C28.17D4⋊2C2, (C2×Dic7).25D4, C22.184(D4×D7), C2.19(D8⋊D7), C14.20(C4⋊D4), C14.37(C8⋊C22), (C2×C56).129C22, (C2×C28).227C23, (D4×C14).48C22, (C2×D28).55C22, C2.23(D14⋊D4), (C4×Dic7).14C22, C2.13(SD16⋊3D7), (C2×Dic14).61C22, (C2×D4⋊D7).3C2, (C2×C56⋊C2)⋊16C2, (C2×C7⋊C8).25C22, (C7×D4⋊C4)⋊14C2, (C2×C14).240(C2×D4), (C7×C4⋊C4).28C22, (C2×C4).334(C22×D7), SmallGroup(448,321)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.D4
G = < a,b,c,d | a28=b2=c4=1, d2=a21, bab=a-1, cac-1=a15, ad=da, bc=cb, dbd-1=a7b, dcd-1=a21c-1 >
Subgroups: 724 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, D4.2D4, C56⋊C2, C2×C7⋊C8, C4×Dic7, D14⋊C4, D4⋊D7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, D4×C14, C14.Q16, Dic7⋊C8, C7×D4⋊C4, D28⋊C4, C2×C56⋊C2, C2×D4⋊D7, C28.17D4, D28.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8⋊C22, C22×D7, D4.2D4, C4○D28, D4×D7, D14⋊D4, D8⋊D7, SD16⋊3D7, D28.D4
►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 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 65)(58 64)(59 63)(60 62)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(108 112)(109 111)(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)(141 151)(142 150)(143 149)(144 148)(145 147)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)(169 170)(171 196)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)(181 186)(182 185)(183 184)(197 215)(198 214)(199 213)(200 212)(201 211)(202 210)(203 209)(204 208)(205 207)(216 224)(217 223)(218 222)(219 221)
(1 114 44 191)(2 129 45 178)(3 116 46 193)(4 131 47 180)(5 118 48 195)(6 133 49 182)(7 120 50 169)(8 135 51 184)(9 122 52 171)(10 137 53 186)(11 124 54 173)(12 139 55 188)(13 126 56 175)(14 113 29 190)(15 128 30 177)(16 115 31 192)(17 130 32 179)(18 117 33 194)(19 132 34 181)(20 119 35 196)(21 134 36 183)(22 121 37 170)(23 136 38 185)(24 123 39 172)(25 138 40 187)(26 125 41 174)(27 140 42 189)(28 127 43 176)(57 216 156 106)(58 203 157 93)(59 218 158 108)(60 205 159 95)(61 220 160 110)(62 207 161 97)(63 222 162 112)(64 209 163 99)(65 224 164 86)(66 211 165 101)(67 198 166 88)(68 213 167 103)(69 200 168 90)(70 215 141 105)(71 202 142 92)(72 217 143 107)(73 204 144 94)(74 219 145 109)(75 206 146 96)(76 221 147 111)(77 208 148 98)(78 223 149 85)(79 210 150 100)(80 197 151 87)(81 212 152 102)(82 199 153 89)(83 214 154 104)(84 201 155 91)
(1 86 22 107 15 100 8 93)(2 87 23 108 16 101 9 94)(3 88 24 109 17 102 10 95)(4 89 25 110 18 103 11 96)(5 90 26 111 19 104 12 97)(6 91 27 112 20 105 13 98)(7 92 28 85 21 106 14 99)(29 209 50 202 43 223 36 216)(30 210 51 203 44 224 37 217)(31 211 52 204 45 197 38 218)(32 212 53 205 46 198 39 219)(33 213 54 206 47 199 40 220)(34 214 55 207 48 200 41 221)(35 215 56 208 49 201 42 222)(57 183 78 176 71 169 64 190)(58 184 79 177 72 170 65 191)(59 185 80 178 73 171 66 192)(60 186 81 179 74 172 67 193)(61 187 82 180 75 173 68 194)(62 188 83 181 76 174 69 195)(63 189 84 182 77 175 70 196)(113 156 134 149 127 142 120 163)(114 157 135 150 128 143 121 164)(115 158 136 151 129 144 122 165)(116 159 137 152 130 145 123 166)(117 160 138 153 131 146 124 167)(118 161 139 154 132 147 125 168)(119 162 140 155 133 148 126 141)
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)(29,30)(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)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111)(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)(141,151)(142,150)(143,149)(144,148)(145,147)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(169,170)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,207)(216,224)(217,223)(218,222)(219,221), (1,114,44,191)(2,129,45,178)(3,116,46,193)(4,131,47,180)(5,118,48,195)(6,133,49,182)(7,120,50,169)(8,135,51,184)(9,122,52,171)(10,137,53,186)(11,124,54,173)(12,139,55,188)(13,126,56,175)(14,113,29,190)(15,128,30,177)(16,115,31,192)(17,130,32,179)(18,117,33,194)(19,132,34,181)(20,119,35,196)(21,134,36,183)(22,121,37,170)(23,136,38,185)(24,123,39,172)(25,138,40,187)(26,125,41,174)(27,140,42,189)(28,127,43,176)(57,216,156,106)(58,203,157,93)(59,218,158,108)(60,205,159,95)(61,220,160,110)(62,207,161,97)(63,222,162,112)(64,209,163,99)(65,224,164,86)(66,211,165,101)(67,198,166,88)(68,213,167,103)(69,200,168,90)(70,215,141,105)(71,202,142,92)(72,217,143,107)(73,204,144,94)(74,219,145,109)(75,206,146,96)(76,221,147,111)(77,208,148,98)(78,223,149,85)(79,210,150,100)(80,197,151,87)(81,212,152,102)(82,199,153,89)(83,214,154,104)(84,201,155,91), (1,86,22,107,15,100,8,93)(2,87,23,108,16,101,9,94)(3,88,24,109,17,102,10,95)(4,89,25,110,18,103,11,96)(5,90,26,111,19,104,12,97)(6,91,27,112,20,105,13,98)(7,92,28,85,21,106,14,99)(29,209,50,202,43,223,36,216)(30,210,51,203,44,224,37,217)(31,211,52,204,45,197,38,218)(32,212,53,205,46,198,39,219)(33,213,54,206,47,199,40,220)(34,214,55,207,48,200,41,221)(35,215,56,208,49,201,42,222)(57,183,78,176,71,169,64,190)(58,184,79,177,72,170,65,191)(59,185,80,178,73,171,66,192)(60,186,81,179,74,172,67,193)(61,187,82,180,75,173,68,194)(62,188,83,181,76,174,69,195)(63,189,84,182,77,175,70,196)(113,156,134,149,127,142,120,163)(114,157,135,150,128,143,121,164)(115,158,136,151,129,144,122,165)(116,159,137,152,130,145,123,166)(117,160,138,153,131,146,124,167)(118,161,139,154,132,147,125,168)(119,162,140,155,133,148,126,141)>;
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)(29,30)(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)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111)(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)(141,151)(142,150)(143,149)(144,148)(145,147)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)(169,170)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)(205,207)(216,224)(217,223)(218,222)(219,221), (1,114,44,191)(2,129,45,178)(3,116,46,193)(4,131,47,180)(5,118,48,195)(6,133,49,182)(7,120,50,169)(8,135,51,184)(9,122,52,171)(10,137,53,186)(11,124,54,173)(12,139,55,188)(13,126,56,175)(14,113,29,190)(15,128,30,177)(16,115,31,192)(17,130,32,179)(18,117,33,194)(19,132,34,181)(20,119,35,196)(21,134,36,183)(22,121,37,170)(23,136,38,185)(24,123,39,172)(25,138,40,187)(26,125,41,174)(27,140,42,189)(28,127,43,176)(57,216,156,106)(58,203,157,93)(59,218,158,108)(60,205,159,95)(61,220,160,110)(62,207,161,97)(63,222,162,112)(64,209,163,99)(65,224,164,86)(66,211,165,101)(67,198,166,88)(68,213,167,103)(69,200,168,90)(70,215,141,105)(71,202,142,92)(72,217,143,107)(73,204,144,94)(74,219,145,109)(75,206,146,96)(76,221,147,111)(77,208,148,98)(78,223,149,85)(79,210,150,100)(80,197,151,87)(81,212,152,102)(82,199,153,89)(83,214,154,104)(84,201,155,91), (1,86,22,107,15,100,8,93)(2,87,23,108,16,101,9,94)(3,88,24,109,17,102,10,95)(4,89,25,110,18,103,11,96)(5,90,26,111,19,104,12,97)(6,91,27,112,20,105,13,98)(7,92,28,85,21,106,14,99)(29,209,50,202,43,223,36,216)(30,210,51,203,44,224,37,217)(31,211,52,204,45,197,38,218)(32,212,53,205,46,198,39,219)(33,213,54,206,47,199,40,220)(34,214,55,207,48,200,41,221)(35,215,56,208,49,201,42,222)(57,183,78,176,71,169,64,190)(58,184,79,177,72,170,65,191)(59,185,80,178,73,171,66,192)(60,186,81,179,74,172,67,193)(61,187,82,180,75,173,68,194)(62,188,83,181,76,174,69,195)(63,189,84,182,77,175,70,196)(113,156,134,149,127,142,120,163)(114,157,135,150,128,143,121,164)(115,158,136,151,129,144,122,165)(116,159,137,152,130,145,123,166)(117,160,138,153,131,146,124,167)(118,161,139,154,132,147,125,168)(119,162,140,155,133,148,126,141) );
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),(29,30),(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),(57,65),(58,64),(59,63),(60,62),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(108,112),(109,111),(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),(141,151),(142,150),(143,149),(144,148),(145,147),(152,168),(153,167),(154,166),(155,165),(156,164),(157,163),(158,162),(159,161),(169,170),(171,196),(172,195),(173,194),(174,193),(175,192),(176,191),(177,190),(178,189),(179,188),(180,187),(181,186),(182,185),(183,184),(197,215),(198,214),(199,213),(200,212),(201,211),(202,210),(203,209),(204,208),(205,207),(216,224),(217,223),(218,222),(219,221)], [(1,114,44,191),(2,129,45,178),(3,116,46,193),(4,131,47,180),(5,118,48,195),(6,133,49,182),(7,120,50,169),(8,135,51,184),(9,122,52,171),(10,137,53,186),(11,124,54,173),(12,139,55,188),(13,126,56,175),(14,113,29,190),(15,128,30,177),(16,115,31,192),(17,130,32,179),(18,117,33,194),(19,132,34,181),(20,119,35,196),(21,134,36,183),(22,121,37,170),(23,136,38,185),(24,123,39,172),(25,138,40,187),(26,125,41,174),(27,140,42,189),(28,127,43,176),(57,216,156,106),(58,203,157,93),(59,218,158,108),(60,205,159,95),(61,220,160,110),(62,207,161,97),(63,222,162,112),(64,209,163,99),(65,224,164,86),(66,211,165,101),(67,198,166,88),(68,213,167,103),(69,200,168,90),(70,215,141,105),(71,202,142,92),(72,217,143,107),(73,204,144,94),(74,219,145,109),(75,206,146,96),(76,221,147,111),(77,208,148,98),(78,223,149,85),(79,210,150,100),(80,197,151,87),(81,212,152,102),(82,199,153,89),(83,214,154,104),(84,201,155,91)], [(1,86,22,107,15,100,8,93),(2,87,23,108,16,101,9,94),(3,88,24,109,17,102,10,95),(4,89,25,110,18,103,11,96),(5,90,26,111,19,104,12,97),(6,91,27,112,20,105,13,98),(7,92,28,85,21,106,14,99),(29,209,50,202,43,223,36,216),(30,210,51,203,44,224,37,217),(31,211,52,204,45,197,38,218),(32,212,53,205,46,198,39,219),(33,213,54,206,47,199,40,220),(34,214,55,207,48,200,41,221),(35,215,56,208,49,201,42,222),(57,183,78,176,71,169,64,190),(58,184,79,177,72,170,65,191),(59,185,80,178,73,171,66,192),(60,186,81,179,74,172,67,193),(61,187,82,180,75,173,68,194),(62,188,83,181,76,174,69,195),(63,189,84,182,77,175,70,196),(113,156,134,149,127,142,120,163),(114,157,135,150,128,143,121,164),(115,158,136,151,129,144,122,165),(116,159,137,152,130,145,123,166),(117,160,138,153,131,146,124,167),(118,161,139,154,132,147,125,168),(119,162,140,155,133,148,126,141)]])
61 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 | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 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 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D8 | C4○D28 | C8⋊C22 | D4×D7 | D4×D7 | D8⋊D7 | SD16⋊3D7 |
| kernel | D28.D4 | C14.Q16 | Dic7⋊C8 | C7×D4⋊C4 | D28⋊C4 | C2×C56⋊C2 | C2×D4⋊D7 | C28.17D4 | D28 | C2×Dic7 | D4⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×D4 | C14 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D28.D4 ►in GL4(𝔽113) generated by
| 88 | 79 | 0 | 0 |
| 34 | 1 | 0 | 0 |
| 0 | 0 | 112 | 106 |
| 0 | 0 | 81 | 1 |
| 25 | 34 | 0 | 0 |
| 88 | 88 | 0 | 0 |
| 0 | 0 | 112 | 106 |
| 0 | 0 | 0 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 105 |
| 0 | 0 | 0 | 98 |
| 9 | 67 | 0 | 0 |
| 46 | 104 | 0 | 0 |
| 0 | 0 | 87 | 22 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(113))| [88,34,0,0,79,1,0,0,0,0,112,81,0,0,106,1],[25,88,0,0,34,88,0,0,0,0,112,0,0,0,106,1],[15,0,0,0,0,15,0,0,0,0,15,0,0,0,105,98],[9,46,0,0,67,104,0,0,0,0,87,36,0,0,22,0] >;
D28.D4 in GAP, Magma, Sage, TeX
D_{28}.D_4 % in TeX
G:=Group("D28.D4"); // GroupNames label
G:=SmallGroup(448,321);
// by ID
G=gap.SmallGroup(448,321);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,555,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^21,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=a^21*c^-1>;
// generators/relations