metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.7D4, Dic14.7D4, M4(2).6D14, C7⋊C8.27D4, D28.C4.C2, C4.153(D4×D7), (C2×Q8).8D14, C4.10D4⋊4D7, C28.100(C2×D4), C7⋊1(D4.5D4), C4.12D28⋊8C2, C28.53D4⋊4C2, (C2×C28).12C23, C8.D14.1C2, C4○D28.8C22, C14.12(C4⋊D4), C28.C23.1C2, (Q8×C14).10C22, C2.15(D14⋊D4), C4.Dic7.7C22, C22.16(C4○D28), (C2×Dic14).48C22, (C7×M4(2)).15C22, (C2×C7⋊Q16)⋊1C2, (C2×C7⋊C8).4C22, (C7×C4.10D4)⋊2C2, (C2×C4).12(C22×D7), (C2×C14).33(C4○D4), SmallGroup(448,289)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.7D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a21, bab=a-1, cac-1=a15, ad=da, cbc-1=a14b, dbd-1=a7b, dcd-1=a21c3 >
Subgroups: 492 in 100 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C4.10D4, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, D4.5D4, C8×D7, C8⋊D7, C56⋊C2, Dic28, C2×C7⋊C8, C4.Dic7, Q8⋊D7, C7⋊Q16, C7×M4(2), C2×Dic14, C4○D28, Q8×C14, C28.53D4, C4.12D28, C7×C4.10D4, D28.C4, C8.D14, C28.C23, C2×C7⋊Q16, D28.7D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.5D4, C4○D28, D4×D7, D14⋊D4, D28.7D4
►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)
(1 122)(2 121)(3 120)(4 119)(5 118)(6 117)(7 116)(8 115)(9 114)(10 113)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 163)(30 162)(31 161)(32 160)(33 159)(34 158)(35 157)(36 156)(37 155)(38 154)(39 153)(40 152)(41 151)(42 150)(43 149)(44 148)(45 147)(46 146)(47 145)(48 144)(49 143)(50 142)(51 141)(52 168)(53 167)(54 166)(55 165)(56 164)(57 188)(58 187)(59 186)(60 185)(61 184)(62 183)(63 182)(64 181)(65 180)(66 179)(67 178)(68 177)(69 176)(70 175)(71 174)(72 173)(73 172)(74 171)(75 170)(76 169)(77 196)(78 195)(79 194)(80 193)(81 192)(82 191)(83 190)(84 189)(85 207)(86 206)(87 205)(88 204)(89 203)(90 202)(91 201)(92 200)(93 199)(94 198)(95 197)(96 224)(97 223)(98 222)(99 221)(100 220)(101 219)(102 218)(103 217)(104 216)(105 215)(106 214)(107 213)(108 212)(109 211)(110 210)(111 209)(112 208)
(1 80 22 59 15 66 8 73)(2 67 23 74 16 81 9 60)(3 82 24 61 17 68 10 75)(4 69 25 76 18 83 11 62)(5 84 26 63 19 70 12 77)(6 71 27 78 20 57 13 64)(7 58 28 65 21 72 14 79)(29 102 50 109 43 88 36 95)(30 89 51 96 44 103 37 110)(31 104 52 111 45 90 38 97)(32 91 53 98 46 105 39 112)(33 106 54 85 47 92 40 99)(34 93 55 100 48 107 41 86)(35 108 56 87 49 94 42 101)(113 184 120 177 127 170 134 191)(114 171 121 192 128 185 135 178)(115 186 122 179 129 172 136 193)(116 173 123 194 130 187 137 180)(117 188 124 181 131 174 138 195)(118 175 125 196 132 189 139 182)(119 190 126 183 133 176 140 169)(141 210 148 203 155 224 162 217)(142 197 149 218 156 211 163 204)(143 212 150 205 157 198 164 219)(144 199 151 220 158 213 165 206)(145 214 152 207 159 200 166 221)(146 201 153 222 160 215 167 208)(147 216 154 209 161 202 168 223)
(1 96 22 89 15 110 8 103)(2 97 23 90 16 111 9 104)(3 98 24 91 17 112 10 105)(4 99 25 92 18 85 11 106)(5 100 26 93 19 86 12 107)(6 101 27 94 20 87 13 108)(7 102 28 95 21 88 14 109)(29 72 50 65 43 58 36 79)(30 73 51 66 44 59 37 80)(31 74 52 67 45 60 38 81)(32 75 53 68 46 61 39 82)(33 76 54 69 47 62 40 83)(34 77 55 70 48 63 41 84)(35 78 56 71 49 64 42 57)(113 222 134 215 127 208 120 201)(114 223 135 216 128 209 121 202)(115 224 136 217 129 210 122 203)(116 197 137 218 130 211 123 204)(117 198 138 219 131 212 124 205)(118 199 139 220 132 213 125 206)(119 200 140 221 133 214 126 207)(141 186 162 179 155 172 148 193)(142 187 163 180 156 173 149 194)(143 188 164 181 157 174 150 195)(144 189 165 182 158 175 151 196)(145 190 166 183 159 176 152 169)(146 191 167 184 160 177 153 170)(147 192 168 185 161 178 154 171)
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), (1,122)(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,163)(30,162)(31,161)(32,160)(33,159)(34,158)(35,157)(36,156)(37,155)(38,154)(39,153)(40,152)(41,151)(42,150)(43,149)(44,148)(45,147)(46,146)(47,145)(48,144)(49,143)(50,142)(51,141)(52,168)(53,167)(54,166)(55,165)(56,164)(57,188)(58,187)(59,186)(60,185)(61,184)(62,183)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,196)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,189)(85,207)(86,206)(87,205)(88,204)(89,203)(90,202)(91,201)(92,200)(93,199)(94,198)(95,197)(96,224)(97,223)(98,222)(99,221)(100,220)(101,219)(102,218)(103,217)(104,216)(105,215)(106,214)(107,213)(108,212)(109,211)(110,210)(111,209)(112,208), (1,80,22,59,15,66,8,73)(2,67,23,74,16,81,9,60)(3,82,24,61,17,68,10,75)(4,69,25,76,18,83,11,62)(5,84,26,63,19,70,12,77)(6,71,27,78,20,57,13,64)(7,58,28,65,21,72,14,79)(29,102,50,109,43,88,36,95)(30,89,51,96,44,103,37,110)(31,104,52,111,45,90,38,97)(32,91,53,98,46,105,39,112)(33,106,54,85,47,92,40,99)(34,93,55,100,48,107,41,86)(35,108,56,87,49,94,42,101)(113,184,120,177,127,170,134,191)(114,171,121,192,128,185,135,178)(115,186,122,179,129,172,136,193)(116,173,123,194,130,187,137,180)(117,188,124,181,131,174,138,195)(118,175,125,196,132,189,139,182)(119,190,126,183,133,176,140,169)(141,210,148,203,155,224,162,217)(142,197,149,218,156,211,163,204)(143,212,150,205,157,198,164,219)(144,199,151,220,158,213,165,206)(145,214,152,207,159,200,166,221)(146,201,153,222,160,215,167,208)(147,216,154,209,161,202,168,223), (1,96,22,89,15,110,8,103)(2,97,23,90,16,111,9,104)(3,98,24,91,17,112,10,105)(4,99,25,92,18,85,11,106)(5,100,26,93,19,86,12,107)(6,101,27,94,20,87,13,108)(7,102,28,95,21,88,14,109)(29,72,50,65,43,58,36,79)(30,73,51,66,44,59,37,80)(31,74,52,67,45,60,38,81)(32,75,53,68,46,61,39,82)(33,76,54,69,47,62,40,83)(34,77,55,70,48,63,41,84)(35,78,56,71,49,64,42,57)(113,222,134,215,127,208,120,201)(114,223,135,216,128,209,121,202)(115,224,136,217,129,210,122,203)(116,197,137,218,130,211,123,204)(117,198,138,219,131,212,124,205)(118,199,139,220,132,213,125,206)(119,200,140,221,133,214,126,207)(141,186,162,179,155,172,148,193)(142,187,163,180,156,173,149,194)(143,188,164,181,157,174,150,195)(144,189,165,182,158,175,151,196)(145,190,166,183,159,176,152,169)(146,191,167,184,160,177,153,170)(147,192,168,185,161,178,154,171)>;
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), (1,122)(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,163)(30,162)(31,161)(32,160)(33,159)(34,158)(35,157)(36,156)(37,155)(38,154)(39,153)(40,152)(41,151)(42,150)(43,149)(44,148)(45,147)(46,146)(47,145)(48,144)(49,143)(50,142)(51,141)(52,168)(53,167)(54,166)(55,165)(56,164)(57,188)(58,187)(59,186)(60,185)(61,184)(62,183)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,196)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,189)(85,207)(86,206)(87,205)(88,204)(89,203)(90,202)(91,201)(92,200)(93,199)(94,198)(95,197)(96,224)(97,223)(98,222)(99,221)(100,220)(101,219)(102,218)(103,217)(104,216)(105,215)(106,214)(107,213)(108,212)(109,211)(110,210)(111,209)(112,208), (1,80,22,59,15,66,8,73)(2,67,23,74,16,81,9,60)(3,82,24,61,17,68,10,75)(4,69,25,76,18,83,11,62)(5,84,26,63,19,70,12,77)(6,71,27,78,20,57,13,64)(7,58,28,65,21,72,14,79)(29,102,50,109,43,88,36,95)(30,89,51,96,44,103,37,110)(31,104,52,111,45,90,38,97)(32,91,53,98,46,105,39,112)(33,106,54,85,47,92,40,99)(34,93,55,100,48,107,41,86)(35,108,56,87,49,94,42,101)(113,184,120,177,127,170,134,191)(114,171,121,192,128,185,135,178)(115,186,122,179,129,172,136,193)(116,173,123,194,130,187,137,180)(117,188,124,181,131,174,138,195)(118,175,125,196,132,189,139,182)(119,190,126,183,133,176,140,169)(141,210,148,203,155,224,162,217)(142,197,149,218,156,211,163,204)(143,212,150,205,157,198,164,219)(144,199,151,220,158,213,165,206)(145,214,152,207,159,200,166,221)(146,201,153,222,160,215,167,208)(147,216,154,209,161,202,168,223), (1,96,22,89,15,110,8,103)(2,97,23,90,16,111,9,104)(3,98,24,91,17,112,10,105)(4,99,25,92,18,85,11,106)(5,100,26,93,19,86,12,107)(6,101,27,94,20,87,13,108)(7,102,28,95,21,88,14,109)(29,72,50,65,43,58,36,79)(30,73,51,66,44,59,37,80)(31,74,52,67,45,60,38,81)(32,75,53,68,46,61,39,82)(33,76,54,69,47,62,40,83)(34,77,55,70,48,63,41,84)(35,78,56,71,49,64,42,57)(113,222,134,215,127,208,120,201)(114,223,135,216,128,209,121,202)(115,224,136,217,129,210,122,203)(116,197,137,218,130,211,123,204)(117,198,138,219,131,212,124,205)(118,199,139,220,132,213,125,206)(119,200,140,221,133,214,126,207)(141,186,162,179,155,172,148,193)(142,187,163,180,156,173,149,194)(143,188,164,181,157,174,150,195)(144,189,165,182,158,175,151,196)(145,190,166,183,159,176,152,169)(146,191,167,184,160,177,153,170)(147,192,168,185,161,178,154,171) );
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)], [(1,122),(2,121),(3,120),(4,119),(5,118),(6,117),(7,116),(8,115),(9,114),(10,113),(11,140),(12,139),(13,138),(14,137),(15,136),(16,135),(17,134),(18,133),(19,132),(20,131),(21,130),(22,129),(23,128),(24,127),(25,126),(26,125),(27,124),(28,123),(29,163),(30,162),(31,161),(32,160),(33,159),(34,158),(35,157),(36,156),(37,155),(38,154),(39,153),(40,152),(41,151),(42,150),(43,149),(44,148),(45,147),(46,146),(47,145),(48,144),(49,143),(50,142),(51,141),(52,168),(53,167),(54,166),(55,165),(56,164),(57,188),(58,187),(59,186),(60,185),(61,184),(62,183),(63,182),(64,181),(65,180),(66,179),(67,178),(68,177),(69,176),(70,175),(71,174),(72,173),(73,172),(74,171),(75,170),(76,169),(77,196),(78,195),(79,194),(80,193),(81,192),(82,191),(83,190),(84,189),(85,207),(86,206),(87,205),(88,204),(89,203),(90,202),(91,201),(92,200),(93,199),(94,198),(95,197),(96,224),(97,223),(98,222),(99,221),(100,220),(101,219),(102,218),(103,217),(104,216),(105,215),(106,214),(107,213),(108,212),(109,211),(110,210),(111,209),(112,208)], [(1,80,22,59,15,66,8,73),(2,67,23,74,16,81,9,60),(3,82,24,61,17,68,10,75),(4,69,25,76,18,83,11,62),(5,84,26,63,19,70,12,77),(6,71,27,78,20,57,13,64),(7,58,28,65,21,72,14,79),(29,102,50,109,43,88,36,95),(30,89,51,96,44,103,37,110),(31,104,52,111,45,90,38,97),(32,91,53,98,46,105,39,112),(33,106,54,85,47,92,40,99),(34,93,55,100,48,107,41,86),(35,108,56,87,49,94,42,101),(113,184,120,177,127,170,134,191),(114,171,121,192,128,185,135,178),(115,186,122,179,129,172,136,193),(116,173,123,194,130,187,137,180),(117,188,124,181,131,174,138,195),(118,175,125,196,132,189,139,182),(119,190,126,183,133,176,140,169),(141,210,148,203,155,224,162,217),(142,197,149,218,156,211,163,204),(143,212,150,205,157,198,164,219),(144,199,151,220,158,213,165,206),(145,214,152,207,159,200,166,221),(146,201,153,222,160,215,167,208),(147,216,154,209,161,202,168,223)], [(1,96,22,89,15,110,8,103),(2,97,23,90,16,111,9,104),(3,98,24,91,17,112,10,105),(4,99,25,92,18,85,11,106),(5,100,26,93,19,86,12,107),(6,101,27,94,20,87,13,108),(7,102,28,95,21,88,14,109),(29,72,50,65,43,58,36,79),(30,73,51,66,44,59,37,80),(31,74,52,67,45,60,38,81),(32,75,53,68,46,61,39,82),(33,76,54,69,47,62,40,83),(34,77,55,70,48,63,41,84),(35,78,56,71,49,64,42,57),(113,222,134,215,127,208,120,201),(114,223,135,216,128,209,121,202),(115,224,136,217,129,210,122,203),(116,197,137,218,130,211,123,204),(117,198,138,219,131,212,124,205),(118,199,139,220,132,213,125,206),(119,200,140,221,133,214,126,207),(141,186,162,179,155,172,148,193),(142,187,163,180,156,173,149,194),(143,188,164,181,157,174,150,195),(144,189,165,182,158,175,151,196),(145,190,166,183,159,176,152,169),(146,191,167,184,160,177,153,170),(147,192,168,185,161,178,154,171)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 2 | 2 | 8 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C4○D28 | D4.5D4 | D4×D7 | D28.7D4 |
| kernel | D28.7D4 | C28.53D4 | C4.12D28 | C7×C4.10D4 | D28.C4 | C8.D14 | C28.C23 | C2×C7⋊Q16 | C7⋊C8 | Dic14 | D28 | C4.10D4 | C2×C14 | M4(2) | C2×Q8 | C22 | C7 | C4 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 2 | 6 | 3 |
Matrix representation of D28.7D4 ►in GL8(𝔽113)
| 104 | 0 | 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 104 | 0 | 34 | 0 | 0 | 0 | 0 |
| 71 | 0 | 33 | 0 | 0 | 0 | 0 | 0 |
| 0 | 71 | 0 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 85 | 112 | 26 |
| 0 | 0 | 0 | 0 | 78 | 85 | 26 | 1 |
| 31 | 0 | 45 | 0 | 0 | 0 | 0 | 0 |
| 0 | 82 | 0 | 68 | 0 | 0 | 0 | 0 |
| 54 | 0 | 82 | 0 | 0 | 0 | 0 | 0 |
| 0 | 59 | 0 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 104 | 85 | 25 |
| 0 | 0 | 0 | 0 | 13 | 100 | 0 | 25 |
| 0 | 0 | 0 | 0 | 9 | 5 | 84 | 90 |
| 0 | 0 | 0 | 0 | 3 | 7 | 1 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 108 | 11 | 36 | 0 |
| 0 | 0 | 0 | 0 | 10 | 109 | 36 | 81 |
| 0 | 0 | 0 | 0 | 81 | 53 | 107 | 60 |
| 0 | 0 | 0 | 0 | 106 | 0 | 0 | 15 |
| 0 | 58 | 0 | 46 | 0 | 0 | 0 | 0 |
| 55 | 0 | 67 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 55 | 0 | 0 | 0 | 0 |
| 110 | 0 | 58 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 58 | 5 | 51 |
| 0 | 0 | 0 | 0 | 75 | 51 | 78 | 68 |
| 0 | 0 | 0 | 0 | 17 | 65 | 93 | 2 |
| 0 | 0 | 0 | 0 | 28 | 11 | 47 | 45 |
G:=sub<GL(8,GF(113))| [104,0,71,0,0,0,0,0,0,104,0,71,0,0,0,0,34,0,33,0,0,0,0,0,0,34,0,33,0,0,0,0,0,0,0,0,0,112,22,78,0,0,0,0,1,0,85,85,0,0,0,0,0,0,112,26,0,0,0,0,0,0,26,1],[31,0,54,0,0,0,0,0,0,82,0,59,0,0,0,0,45,0,82,0,0,0,0,0,0,68,0,31,0,0,0,0,0,0,0,0,42,13,9,3,0,0,0,0,104,100,5,7,0,0,0,0,85,0,84,1,0,0,0,0,25,25,90,0],[15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,108,10,81,106,0,0,0,0,11,109,53,0,0,0,0,0,36,36,107,0,0,0,0,0,0,81,60,15],[0,55,0,110,0,0,0,0,58,0,3,0,0,0,0,0,0,67,0,58,0,0,0,0,46,0,55,0,0,0,0,0,0,0,0,0,37,75,17,28,0,0,0,0,58,51,65,11,0,0,0,0,5,78,93,47,0,0,0,0,51,68,2,45] >;
D28.7D4 in GAP, Magma, Sage, TeX
D_{28}._7D_4 % in TeX
G:=Group("D28.7D4"); // GroupNames label
G:=SmallGroup(448,289);
// by ID
G=gap.SmallGroup(448,289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,555,184,297,136,1684,851,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^21,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^14*b,d*b*d^-1=a^7*b,d*c*d^-1=a^21*c^3>;
// generators/relations