metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.44D4, C56.7C23, Q16.6D14, C28.26C24, SD16.1D14, Dic14.44D4, D28.19C23, M4(2).18D14, Dic28.1C22, Dic14.19C23, C7⋊4(Q8○D8), (D7×Q16)⋊2C2, C7⋊D4.7D4, D4⋊D7.C22, D28.C4⋊4C2, C4.118(D4×D7), C7⋊C8.28C23, C56⋊C2.C22, C8.C22⋊5D7, C8⋊D7.C22, C8.7(C22×D7), Q16⋊D7⋊4C2, C4○D4.15D14, D14.35(C2×D4), C8.D14⋊4C2, SD16⋊D7⋊4C2, C28.247(C2×D4), (C2×Q8).92D14, (C8×D7).2C22, C22.17(D4×D7), C4.26(C23×D7), Q8⋊D7.2C22, (C7×SD16).C22, SD16⋊3D7⋊4C2, D4.8D14⋊6C2, (Q8×D7).3C22, Dic7.40(C2×D4), (C7×D4).19C23, (C4×D7).17C23, D4.19(C22×D7), D4.D7.2C22, D4.10D14⋊8C2, (C7×Q16).1C22, Q8.19(C22×D7), (C7×Q8).19C23, C7⋊Q16.3C22, (C2×C28).117C23, Q8.10D14⋊6C2, C4○D28.32C22, D4⋊2D7.3C22, C14.127(C22×D4), Q8⋊2D7.3C22, (Q8×C14).153C22, (C7×M4(2)).1C22, (C2×Dic14).201C22, C2.100(C2×D4×D7), (C2×C7⋊Q16)⋊29C2, (C2×C14).72(C2×D4), (C7×C8.C22)⋊4C2, (C2×C7⋊C8).182C22, (C7×C4○D4).28C22, (C2×C4).101(C22×D7), SmallGroup(448,1232)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.44D4
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, cac-1=dad-1=a15, cbc-1=dbd-1=a14b, dcd-1=a14c3 >
Subgroups: 1132 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×Q16, C4○D8, C8.C22, C8.C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, Q8○D8, C8×D7, C8⋊D7, C56⋊C2, Dic28, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C4○D28, C4○D28, D4⋊2D7, D4⋊2D7, Q8×D7, Q8×D7, Q8⋊2D7, Q8⋊2D7, Q8×C14, C7×C4○D4, D28.C4, C8.D14, SD16⋊D7, SD16⋊3D7, D7×Q16, Q16⋊D7, C2×C7⋊Q16, D4.8D14, C7×C8.C22, Q8.10D14, D4.10D14, D28.44D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, Q8○D8, D4×D7, C23×D7, C2×D4×D7, D28.44D4
►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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(57 63)(58 62)(59 61)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(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)(113 139)(114 138)(115 137)(116 136)(117 135)(118 134)(119 133)(120 132)(121 131)(122 130)(123 129)(124 128)(125 127)(141 145)(142 144)(146 168)(147 167)(148 166)(149 165)(150 164)(151 163)(152 162)(153 161)(154 160)(155 159)(156 158)(169 177)(170 176)(171 175)(172 174)(178 196)(179 195)(180 194)(181 193)(182 192)(183 191)(184 190)(185 189)(186 188)(197 209)(198 208)(199 207)(200 206)(201 205)(202 204)(210 224)(211 223)(212 222)(213 221)(214 220)(215 219)(216 218)
(1 108 46 221 15 94 32 207)(2 95 47 208 16 109 33 222)(3 110 48 223 17 96 34 209)(4 97 49 210 18 111 35 224)(5 112 50 197 19 98 36 211)(6 99 51 212 20 85 37 198)(7 86 52 199 21 100 38 213)(8 101 53 214 22 87 39 200)(9 88 54 201 23 102 40 215)(10 103 55 216 24 89 41 202)(11 90 56 203 25 104 42 217)(12 105 29 218 26 91 43 204)(13 92 30 205 27 106 44 219)(14 107 31 220 28 93 45 206)(57 161 170 116 71 147 184 130)(58 148 171 131 72 162 185 117)(59 163 172 118 73 149 186 132)(60 150 173 133 74 164 187 119)(61 165 174 120 75 151 188 134)(62 152 175 135 76 166 189 121)(63 167 176 122 77 153 190 136)(64 154 177 137 78 168 191 123)(65 141 178 124 79 155 192 138)(66 156 179 139 80 142 193 125)(67 143 180 126 81 157 194 140)(68 158 181 113 82 144 195 127)(69 145 182 128 83 159 196 114)(70 160 183 115 84 146 169 129)
(1 64 15 78)(2 79 16 65)(3 66 17 80)(4 81 18 67)(5 68 19 82)(6 83 20 69)(7 70 21 84)(8 57 22 71)(9 72 23 58)(10 59 24 73)(11 74 25 60)(12 61 26 75)(13 76 27 62)(14 63 28 77)(29 188 43 174)(30 175 44 189)(31 190 45 176)(32 177 46 191)(33 192 47 178)(34 179 48 193)(35 194 49 180)(36 181 50 195)(37 196 51 182)(38 183 52 169)(39 170 53 184)(40 185 54 171)(41 172 55 186)(42 187 56 173)(85 114 99 128)(86 129 100 115)(87 116 101 130)(88 131 102 117)(89 118 103 132)(90 133 104 119)(91 120 105 134)(92 135 106 121)(93 122 107 136)(94 137 108 123)(95 124 109 138)(96 139 110 125)(97 126 111 140)(98 113 112 127)(141 222 155 208)(142 209 156 223)(143 224 157 210)(144 211 158 197)(145 198 159 212)(146 213 160 199)(147 200 161 214)(148 215 162 201)(149 202 163 216)(150 217 164 203)(151 204 165 218)(152 219 166 205)(153 206 167 220)(154 221 168 207)
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(57,63)(58,62)(59,61)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(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)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(141,145)(142,144)(146,168)(147,167)(148,166)(149,165)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(156,158)(169,177)(170,176)(171,175)(172,174)(178,196)(179,195)(180,194)(181,193)(182,192)(183,191)(184,190)(185,189)(186,188)(197,209)(198,208)(199,207)(200,206)(201,205)(202,204)(210,224)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218), (1,108,46,221,15,94,32,207)(2,95,47,208,16,109,33,222)(3,110,48,223,17,96,34,209)(4,97,49,210,18,111,35,224)(5,112,50,197,19,98,36,211)(6,99,51,212,20,85,37,198)(7,86,52,199,21,100,38,213)(8,101,53,214,22,87,39,200)(9,88,54,201,23,102,40,215)(10,103,55,216,24,89,41,202)(11,90,56,203,25,104,42,217)(12,105,29,218,26,91,43,204)(13,92,30,205,27,106,44,219)(14,107,31,220,28,93,45,206)(57,161,170,116,71,147,184,130)(58,148,171,131,72,162,185,117)(59,163,172,118,73,149,186,132)(60,150,173,133,74,164,187,119)(61,165,174,120,75,151,188,134)(62,152,175,135,76,166,189,121)(63,167,176,122,77,153,190,136)(64,154,177,137,78,168,191,123)(65,141,178,124,79,155,192,138)(66,156,179,139,80,142,193,125)(67,143,180,126,81,157,194,140)(68,158,181,113,82,144,195,127)(69,145,182,128,83,159,196,114)(70,160,183,115,84,146,169,129), (1,64,15,78)(2,79,16,65)(3,66,17,80)(4,81,18,67)(5,68,19,82)(6,83,20,69)(7,70,21,84)(8,57,22,71)(9,72,23,58)(10,59,24,73)(11,74,25,60)(12,61,26,75)(13,76,27,62)(14,63,28,77)(29,188,43,174)(30,175,44,189)(31,190,45,176)(32,177,46,191)(33,192,47,178)(34,179,48,193)(35,194,49,180)(36,181,50,195)(37,196,51,182)(38,183,52,169)(39,170,53,184)(40,185,54,171)(41,172,55,186)(42,187,56,173)(85,114,99,128)(86,129,100,115)(87,116,101,130)(88,131,102,117)(89,118,103,132)(90,133,104,119)(91,120,105,134)(92,135,106,121)(93,122,107,136)(94,137,108,123)(95,124,109,138)(96,139,110,125)(97,126,111,140)(98,113,112,127)(141,222,155,208)(142,209,156,223)(143,224,157,210)(144,211,158,197)(145,198,159,212)(146,213,160,199)(147,200,161,214)(148,215,162,201)(149,202,163,216)(150,217,164,203)(151,204,165,218)(152,219,166,205)(153,206,167,220)(154,221,168,207)>;
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(57,63)(58,62)(59,61)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(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)(113,139)(114,138)(115,137)(116,136)(117,135)(118,134)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(141,145)(142,144)(146,168)(147,167)(148,166)(149,165)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(156,158)(169,177)(170,176)(171,175)(172,174)(178,196)(179,195)(180,194)(181,193)(182,192)(183,191)(184,190)(185,189)(186,188)(197,209)(198,208)(199,207)(200,206)(201,205)(202,204)(210,224)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218), (1,108,46,221,15,94,32,207)(2,95,47,208,16,109,33,222)(3,110,48,223,17,96,34,209)(4,97,49,210,18,111,35,224)(5,112,50,197,19,98,36,211)(6,99,51,212,20,85,37,198)(7,86,52,199,21,100,38,213)(8,101,53,214,22,87,39,200)(9,88,54,201,23,102,40,215)(10,103,55,216,24,89,41,202)(11,90,56,203,25,104,42,217)(12,105,29,218,26,91,43,204)(13,92,30,205,27,106,44,219)(14,107,31,220,28,93,45,206)(57,161,170,116,71,147,184,130)(58,148,171,131,72,162,185,117)(59,163,172,118,73,149,186,132)(60,150,173,133,74,164,187,119)(61,165,174,120,75,151,188,134)(62,152,175,135,76,166,189,121)(63,167,176,122,77,153,190,136)(64,154,177,137,78,168,191,123)(65,141,178,124,79,155,192,138)(66,156,179,139,80,142,193,125)(67,143,180,126,81,157,194,140)(68,158,181,113,82,144,195,127)(69,145,182,128,83,159,196,114)(70,160,183,115,84,146,169,129), (1,64,15,78)(2,79,16,65)(3,66,17,80)(4,81,18,67)(5,68,19,82)(6,83,20,69)(7,70,21,84)(8,57,22,71)(9,72,23,58)(10,59,24,73)(11,74,25,60)(12,61,26,75)(13,76,27,62)(14,63,28,77)(29,188,43,174)(30,175,44,189)(31,190,45,176)(32,177,46,191)(33,192,47,178)(34,179,48,193)(35,194,49,180)(36,181,50,195)(37,196,51,182)(38,183,52,169)(39,170,53,184)(40,185,54,171)(41,172,55,186)(42,187,56,173)(85,114,99,128)(86,129,100,115)(87,116,101,130)(88,131,102,117)(89,118,103,132)(90,133,104,119)(91,120,105,134)(92,135,106,121)(93,122,107,136)(94,137,108,123)(95,124,109,138)(96,139,110,125)(97,126,111,140)(98,113,112,127)(141,222,155,208)(142,209,156,223)(143,224,157,210)(144,211,158,197)(145,198,159,212)(146,213,160,199)(147,200,161,214)(148,215,162,201)(149,202,163,216)(150,217,164,203)(151,204,165,218)(152,219,166,205)(153,206,167,220)(154,221,168,207) );
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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(57,63),(58,62),(59,61),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(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),(113,139),(114,138),(115,137),(116,136),(117,135),(118,134),(119,133),(120,132),(121,131),(122,130),(123,129),(124,128),(125,127),(141,145),(142,144),(146,168),(147,167),(148,166),(149,165),(150,164),(151,163),(152,162),(153,161),(154,160),(155,159),(156,158),(169,177),(170,176),(171,175),(172,174),(178,196),(179,195),(180,194),(181,193),(182,192),(183,191),(184,190),(185,189),(186,188),(197,209),(198,208),(199,207),(200,206),(201,205),(202,204),(210,224),(211,223),(212,222),(213,221),(214,220),(215,219),(216,218)], [(1,108,46,221,15,94,32,207),(2,95,47,208,16,109,33,222),(3,110,48,223,17,96,34,209),(4,97,49,210,18,111,35,224),(5,112,50,197,19,98,36,211),(6,99,51,212,20,85,37,198),(7,86,52,199,21,100,38,213),(8,101,53,214,22,87,39,200),(9,88,54,201,23,102,40,215),(10,103,55,216,24,89,41,202),(11,90,56,203,25,104,42,217),(12,105,29,218,26,91,43,204),(13,92,30,205,27,106,44,219),(14,107,31,220,28,93,45,206),(57,161,170,116,71,147,184,130),(58,148,171,131,72,162,185,117),(59,163,172,118,73,149,186,132),(60,150,173,133,74,164,187,119),(61,165,174,120,75,151,188,134),(62,152,175,135,76,166,189,121),(63,167,176,122,77,153,190,136),(64,154,177,137,78,168,191,123),(65,141,178,124,79,155,192,138),(66,156,179,139,80,142,193,125),(67,143,180,126,81,157,194,140),(68,158,181,113,82,144,195,127),(69,145,182,128,83,159,196,114),(70,160,183,115,84,146,169,129)], [(1,64,15,78),(2,79,16,65),(3,66,17,80),(4,81,18,67),(5,68,19,82),(6,83,20,69),(7,70,21,84),(8,57,22,71),(9,72,23,58),(10,59,24,73),(11,74,25,60),(12,61,26,75),(13,76,27,62),(14,63,28,77),(29,188,43,174),(30,175,44,189),(31,190,45,176),(32,177,46,191),(33,192,47,178),(34,179,48,193),(35,194,49,180),(36,181,50,195),(37,196,51,182),(38,183,52,169),(39,170,53,184),(40,185,54,171),(41,172,55,186),(42,187,56,173),(85,114,99,128),(86,129,100,115),(87,116,101,130),(88,131,102,117),(89,118,103,132),(90,133,104,119),(91,120,105,134),(92,135,106,121),(93,122,107,136),(94,137,108,123),(95,124,109,138),(96,139,110,125),(97,126,111,140),(98,113,112,127),(141,222,155,208),(142,209,156,223),(143,224,157,210),(144,211,158,197),(145,198,159,212),(146,213,160,199),(147,200,161,214),(148,215,162,201),(149,202,163,216),(150,217,164,203),(151,204,165,218),(152,219,166,205),(153,206,167,220),(154,221,168,207)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | Q8○D8 | D4×D7 | D4×D7 | D28.44D4 |
| kernel | D28.44D4 | D28.C4 | C8.D14 | SD16⋊D7 | SD16⋊3D7 | D7×Q16 | Q16⋊D7 | C2×C7⋊Q16 | D4.8D14 | C7×C8.C22 | Q8.10D14 | D4.10D14 | Dic14 | D28 | C7⋊D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
Matrix representation of D28.44D4 ►in GL8(𝔽113)
| 10 | 89 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 112 | 11 | 60 |
| 0 | 0 | 0 | 0 | 63 | 36 | 88 | 100 |
| 0 | 0 | 0 | 0 | 34 | 46 | 41 | 49 |
| 0 | 0 | 0 | 0 | 112 | 76 | 53 | 91 |
| 10 | 89 | 0 | 0 | 0 | 0 | 0 | 0 |
| 103 | 103 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 105 | 64 | 0 | 1 |
| 87 | 0 | 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 98 | 0 | 0 | 0 | 0 |
| 15 | 0 | 26 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 52 | 102 | 44 |
| 0 | 0 | 0 | 0 | 76 | 77 | 7 | 13 |
| 0 | 0 | 0 | 0 | 3 | 103 | 72 | 51 |
| 0 | 0 | 0 | 0 | 54 | 19 | 71 | 99 |
| 15 | 0 | 26 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 26 | 0 | 0 | 0 | 0 |
| 87 | 0 | 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 25 | 26 | 0 | 22 |
| 0 | 0 | 0 | 0 | 41 | 0 | 52 | 0 |
| 0 | 0 | 0 | 0 | 9 | 36 | 31 | 87 |
G:=sub<GL(8,GF(113))| [10,24,0,0,0,0,0,0,89,112,0,0,0,0,0,0,0,0,10,24,0,0,0,0,0,0,89,112,0,0,0,0,0,0,0,0,58,63,34,112,0,0,0,0,112,36,46,76,0,0,0,0,11,88,41,53,0,0,0,0,60,100,49,91],[10,103,0,0,0,0,0,0,89,103,0,0,0,0,0,0,0,0,10,103,0,0,0,0,0,0,89,103,0,0,0,0,0,0,0,0,112,0,85,105,0,0,0,0,0,112,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[87,0,15,0,0,0,0,0,0,87,0,15,0,0,0,0,98,0,26,0,0,0,0,0,0,98,0,26,0,0,0,0,0,0,0,0,91,76,3,54,0,0,0,0,52,77,103,19,0,0,0,0,102,7,72,71,0,0,0,0,44,13,51,99],[15,0,87,0,0,0,0,0,0,15,0,87,0,0,0,0,26,0,98,0,0,0,0,0,0,26,0,98,0,0,0,0,0,0,0,0,61,25,41,9,0,0,0,0,0,26,0,36,0,0,0,0,36,0,52,31,0,0,0,0,0,22,0,87] >;
D28.44D4 in GAP, Magma, Sage, TeX
D_{28}._{44}D_4 % in TeX
G:=Group("D28.44D4"); // GroupNames label
G:=SmallGroup(448,1232);
// by ID
G=gap.SmallGroup(448,1232);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,184,570,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^15,c*b*c^-1=d*b*d^-1=a^14*b,d*c*d^-1=a^14*c^3>;
// generators/relations