metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.47D4, D8.10D14, C28.18C24, C56.39C23, Q16.12D14, SD16.2D14, Dic14.47D4, Dic14.12C23, Dic28.16C22, C4○D8⋊6D7, C7⋊3(Q8○D8), (D7×Q16)⋊7C2, C7⋊D4.3D4, C7⋊C8.9C23, D8⋊3D7⋊7C2, C4.145(D4×D7), D4.D7.C22, C4○D4.13D14, D14.31(C2×D4), C28.351(C2×D4), (C2×C8).106D14, SD16⋊D7⋊6C2, (C8×D7).8C22, C4.18(C23×D7), C22.10(D4×D7), C8.18(C22×D7), (C2×Dic28)⋊23C2, D4.9D14⋊8C2, D28.2C4⋊8C2, (Q8×D7).2C22, Dic7.36(C2×D4), (C7×D8).10C22, (C7×D4).12C23, D4.12(C22×D7), (C4×D7).11C23, C8⋊D7.2C22, D4.10D14⋊6C2, (C7×Q8).12C23, Q8.12(C22×D7), C7⋊Q16.2C22, (C2×C56).106C22, (C2×C28).535C23, C4○D28.56C22, D4⋊2D7.2C22, C14.119(C22×D4), (C7×Q16).12C22, (C7×SD16).2C22, C4.Dic7.49C22, (C2×Dic14).200C22, C2.92(C2×D4×D7), (C7×C4○D8)⋊6C2, (C2×C14).15(C2×D4), (C7×C4○D4).23C22, (C2×C4).234(C22×D7), SmallGroup(448,1224)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1108 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×2], D4 [×9], Q8 [×2], Q8 [×11], D7 [×2], C14, C14 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×2], SD16 [×4], Q16, Q16 [×8], C2×Q8 [×8], C4○D4 [×2], C4○D4 [×11], Dic7 [×2], Dic7 [×4], C28 [×2], C28 [×2], D14 [×2], C2×C14, C2×C14 [×2], C8○D4, C2×Q16 [×3], C4○D8, C4○D8 [×2], C8.C22 [×6], 2- (1+4) [×2], C7⋊C8 [×2], C56 [×2], Dic14, Dic14 [×4], Dic14 [×6], C4×D7 [×2], C4×D7 [×4], D28, C2×Dic7 [×6], C7⋊D4 [×2], C7⋊D4 [×4], C2×C28, C2×C28 [×2], C7×D4 [×2], C7×D4 [×2], C7×Q8 [×2], Q8○D8, C8×D7 [×2], C8⋊D7 [×2], Dic28 [×4], C4.Dic7, D4.D7 [×4], C7⋊Q16 [×4], C2×C56, C7×D8, C7×SD16 [×2], C7×Q16, C2×Dic14 [×2], C2×Dic14 [×2], C4○D28, C4○D28 [×2], D4⋊2D7 [×4], D4⋊2D7 [×4], Q8×D7 [×4], C7×C4○D4 [×2], D28.2C4, C2×Dic28, D8⋊3D7 [×2], SD16⋊D7 [×4], D7×Q16 [×2], D4.9D14 [×2], C7×C4○D8, D4.10D14 [×2], D28.47D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], Q8○D8, D4×D7 [×2], C23×D7, C2×D4×D7, D28.47D4
Generators and relations
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a14c3 >
►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 35)(30 34)(31 33)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(82 84)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 126)(121 125)(122 124)(134 140)(135 139)(136 138)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)(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)(197 203)(198 202)(199 201)(204 224)(205 223)(206 222)(207 221)(208 220)(209 219)(210 218)(211 217)(212 216)(213 215)
(1 134 158 80 15 120 144 66)(2 135 159 81 16 121 145 67)(3 136 160 82 17 122 146 68)(4 137 161 83 18 123 147 69)(5 138 162 84 19 124 148 70)(6 139 163 57 20 125 149 71)(7 140 164 58 21 126 150 72)(8 113 165 59 22 127 151 73)(9 114 166 60 23 128 152 74)(10 115 167 61 24 129 153 75)(11 116 168 62 25 130 154 76)(12 117 141 63 26 131 155 77)(13 118 142 64 27 132 156 78)(14 119 143 65 28 133 157 79)(29 103 170 197 43 89 184 211)(30 104 171 198 44 90 185 212)(31 105 172 199 45 91 186 213)(32 106 173 200 46 92 187 214)(33 107 174 201 47 93 188 215)(34 108 175 202 48 94 189 216)(35 109 176 203 49 95 190 217)(36 110 177 204 50 96 191 218)(37 111 178 205 51 97 192 219)(38 112 179 206 52 98 193 220)(39 85 180 207 53 99 194 221)(40 86 181 208 54 100 195 222)(41 87 182 209 55 101 196 223)(42 88 183 210 56 102 169 224)
(1 29 15 43)(2 30 16 44)(3 31 17 45)(4 32 18 46)(5 33 19 47)(6 34 20 48)(7 35 21 49)(8 36 22 50)(9 37 23 51)(10 38 24 52)(11 39 25 53)(12 40 26 54)(13 41 27 55)(14 42 28 56)(57 94 71 108)(58 95 72 109)(59 96 73 110)(60 97 74 111)(61 98 75 112)(62 99 76 85)(63 100 77 86)(64 101 78 87)(65 102 79 88)(66 103 80 89)(67 104 81 90)(68 105 82 91)(69 106 83 92)(70 107 84 93)(113 218 127 204)(114 219 128 205)(115 220 129 206)(116 221 130 207)(117 222 131 208)(118 223 132 209)(119 224 133 210)(120 197 134 211)(121 198 135 212)(122 199 136 213)(123 200 137 214)(124 201 138 215)(125 202 139 216)(126 203 140 217)(141 195 155 181)(142 196 156 182)(143 169 157 183)(144 170 158 184)(145 171 159 185)(146 172 160 186)(147 173 161 187)(148 174 162 188)(149 175 163 189)(150 176 164 190)(151 177 165 191)(152 178 166 192)(153 179 167 193)(154 180 168 194)
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)(29,35)(30,34)(31,33)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(113,133)(114,132)(115,131)(116,130)(117,129)(118,128)(119,127)(120,126)(121,125)(122,124)(134,140)(135,139)(136,138)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(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)(197,203)(198,202)(199,201)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218)(211,217)(212,216)(213,215), (1,134,158,80,15,120,144,66)(2,135,159,81,16,121,145,67)(3,136,160,82,17,122,146,68)(4,137,161,83,18,123,147,69)(5,138,162,84,19,124,148,70)(6,139,163,57,20,125,149,71)(7,140,164,58,21,126,150,72)(8,113,165,59,22,127,151,73)(9,114,166,60,23,128,152,74)(10,115,167,61,24,129,153,75)(11,116,168,62,25,130,154,76)(12,117,141,63,26,131,155,77)(13,118,142,64,27,132,156,78)(14,119,143,65,28,133,157,79)(29,103,170,197,43,89,184,211)(30,104,171,198,44,90,185,212)(31,105,172,199,45,91,186,213)(32,106,173,200,46,92,187,214)(33,107,174,201,47,93,188,215)(34,108,175,202,48,94,189,216)(35,109,176,203,49,95,190,217)(36,110,177,204,50,96,191,218)(37,111,178,205,51,97,192,219)(38,112,179,206,52,98,193,220)(39,85,180,207,53,99,194,221)(40,86,181,208,54,100,195,222)(41,87,182,209,55,101,196,223)(42,88,183,210,56,102,169,224), (1,29,15,43)(2,30,16,44)(3,31,17,45)(4,32,18,46)(5,33,19,47)(6,34,20,48)(7,35,21,49)(8,36,22,50)(9,37,23,51)(10,38,24,52)(11,39,25,53)(12,40,26,54)(13,41,27,55)(14,42,28,56)(57,94,71,108)(58,95,72,109)(59,96,73,110)(60,97,74,111)(61,98,75,112)(62,99,76,85)(63,100,77,86)(64,101,78,87)(65,102,79,88)(66,103,80,89)(67,104,81,90)(68,105,82,91)(69,106,83,92)(70,107,84,93)(113,218,127,204)(114,219,128,205)(115,220,129,206)(116,221,130,207)(117,222,131,208)(118,223,132,209)(119,224,133,210)(120,197,134,211)(121,198,135,212)(122,199,136,213)(123,200,137,214)(124,201,138,215)(125,202,139,216)(126,203,140,217)(141,195,155,181)(142,196,156,182)(143,169,157,183)(144,170,158,184)(145,171,159,185)(146,172,160,186)(147,173,161,187)(148,174,162,188)(149,175,163,189)(150,176,164,190)(151,177,165,191)(152,178,166,192)(153,179,167,193)(154,180,168,194)>;
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)(29,35)(30,34)(31,33)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(113,133)(114,132)(115,131)(116,130)(117,129)(118,128)(119,127)(120,126)(121,125)(122,124)(134,140)(135,139)(136,138)(141,153)(142,152)(143,151)(144,150)(145,149)(146,148)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(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)(197,203)(198,202)(199,201)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218)(211,217)(212,216)(213,215), (1,134,158,80,15,120,144,66)(2,135,159,81,16,121,145,67)(3,136,160,82,17,122,146,68)(4,137,161,83,18,123,147,69)(5,138,162,84,19,124,148,70)(6,139,163,57,20,125,149,71)(7,140,164,58,21,126,150,72)(8,113,165,59,22,127,151,73)(9,114,166,60,23,128,152,74)(10,115,167,61,24,129,153,75)(11,116,168,62,25,130,154,76)(12,117,141,63,26,131,155,77)(13,118,142,64,27,132,156,78)(14,119,143,65,28,133,157,79)(29,103,170,197,43,89,184,211)(30,104,171,198,44,90,185,212)(31,105,172,199,45,91,186,213)(32,106,173,200,46,92,187,214)(33,107,174,201,47,93,188,215)(34,108,175,202,48,94,189,216)(35,109,176,203,49,95,190,217)(36,110,177,204,50,96,191,218)(37,111,178,205,51,97,192,219)(38,112,179,206,52,98,193,220)(39,85,180,207,53,99,194,221)(40,86,181,208,54,100,195,222)(41,87,182,209,55,101,196,223)(42,88,183,210,56,102,169,224), (1,29,15,43)(2,30,16,44)(3,31,17,45)(4,32,18,46)(5,33,19,47)(6,34,20,48)(7,35,21,49)(8,36,22,50)(9,37,23,51)(10,38,24,52)(11,39,25,53)(12,40,26,54)(13,41,27,55)(14,42,28,56)(57,94,71,108)(58,95,72,109)(59,96,73,110)(60,97,74,111)(61,98,75,112)(62,99,76,85)(63,100,77,86)(64,101,78,87)(65,102,79,88)(66,103,80,89)(67,104,81,90)(68,105,82,91)(69,106,83,92)(70,107,84,93)(113,218,127,204)(114,219,128,205)(115,220,129,206)(116,221,130,207)(117,222,131,208)(118,223,132,209)(119,224,133,210)(120,197,134,211)(121,198,135,212)(122,199,136,213)(123,200,137,214)(124,201,138,215)(125,202,139,216)(126,203,140,217)(141,195,155,181)(142,196,156,182)(143,169,157,183)(144,170,158,184)(145,171,159,185)(146,172,160,186)(147,173,161,187)(148,174,162,188)(149,175,163,189)(150,176,164,190)(151,177,165,191)(152,178,166,192)(153,179,167,193)(154,180,168,194) );
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),(29,35),(30,34),(31,33),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(82,84),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(113,133),(114,132),(115,131),(116,130),(117,129),(118,128),(119,127),(120,126),(121,125),(122,124),(134,140),(135,139),(136,138),(141,153),(142,152),(143,151),(144,150),(145,149),(146,148),(154,168),(155,167),(156,166),(157,165),(158,164),(159,163),(160,162),(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),(197,203),(198,202),(199,201),(204,224),(205,223),(206,222),(207,221),(208,220),(209,219),(210,218),(211,217),(212,216),(213,215)], [(1,134,158,80,15,120,144,66),(2,135,159,81,16,121,145,67),(3,136,160,82,17,122,146,68),(4,137,161,83,18,123,147,69),(5,138,162,84,19,124,148,70),(6,139,163,57,20,125,149,71),(7,140,164,58,21,126,150,72),(8,113,165,59,22,127,151,73),(9,114,166,60,23,128,152,74),(10,115,167,61,24,129,153,75),(11,116,168,62,25,130,154,76),(12,117,141,63,26,131,155,77),(13,118,142,64,27,132,156,78),(14,119,143,65,28,133,157,79),(29,103,170,197,43,89,184,211),(30,104,171,198,44,90,185,212),(31,105,172,199,45,91,186,213),(32,106,173,200,46,92,187,214),(33,107,174,201,47,93,188,215),(34,108,175,202,48,94,189,216),(35,109,176,203,49,95,190,217),(36,110,177,204,50,96,191,218),(37,111,178,205,51,97,192,219),(38,112,179,206,52,98,193,220),(39,85,180,207,53,99,194,221),(40,86,181,208,54,100,195,222),(41,87,182,209,55,101,196,223),(42,88,183,210,56,102,169,224)], [(1,29,15,43),(2,30,16,44),(3,31,17,45),(4,32,18,46),(5,33,19,47),(6,34,20,48),(7,35,21,49),(8,36,22,50),(9,37,23,51),(10,38,24,52),(11,39,25,53),(12,40,26,54),(13,41,27,55),(14,42,28,56),(57,94,71,108),(58,95,72,109),(59,96,73,110),(60,97,74,111),(61,98,75,112),(62,99,76,85),(63,100,77,86),(64,101,78,87),(65,102,79,88),(66,103,80,89),(67,104,81,90),(68,105,82,91),(69,106,83,92),(70,107,84,93),(113,218,127,204),(114,219,128,205),(115,220,129,206),(116,221,130,207),(117,222,131,208),(118,223,132,209),(119,224,133,210),(120,197,134,211),(121,198,135,212),(122,199,136,213),(123,200,137,214),(124,201,138,215),(125,202,139,216),(126,203,140,217),(141,195,155,181),(142,196,156,182),(143,169,157,183),(144,170,158,184),(145,171,159,185),(146,172,160,186),(147,173,161,187),(148,174,162,188),(149,175,163,189),(150,176,164,190),(151,177,165,191),(152,178,166,192),(153,179,167,193),(154,180,168,194)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 4 | 36 | 0 | 0 |
| 41 | 2 | 0 | 0 |
| 0 | 0 | 96 | 36 |
| 0 | 0 | 77 | 23 |
| 23 | 103 | 0 | 0 |
| 98 | 90 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 89 | 112 |
| 62 | 0 | 32 | 58 |
| 0 | 62 | 74 | 94 |
| 96 | 73 | 0 | 0 |
| 23 | 100 | 0 | 0 |
| 97 | 0 | 62 | 70 |
| 0 | 97 | 101 | 55 |
| 67 | 38 | 16 | 0 |
| 29 | 18 | 0 | 16 |
G:=sub<GL(4,GF(113))| [4,41,0,0,36,2,0,0,0,0,96,77,0,0,36,23],[23,98,0,0,103,90,0,0,0,0,1,89,0,0,0,112],[62,0,96,23,0,62,73,100,32,74,0,0,58,94,0,0],[97,0,67,29,0,97,38,18,62,101,16,0,70,55,0,16] >;
64 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 | ··· | 14L | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| 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 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | Q8○D8 | D4×D7 | D4×D7 | D28.47D4 |
| kernel | D28.47D4 | D28.2C4 | C2×Dic28 | D8⋊3D7 | SD16⋊D7 | D7×Q16 | D4.9D14 | C7×C4○D8 | D4.10D14 | Dic14 | D28 | C7⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
D_{28}._{47}D_4 % in TeX
G:=Group("D28.47D4"); // GroupNames label
G:=SmallGroup(448,1224);
// by ID
G=gap.SmallGroup(448,1224);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,184,570,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^14*c^3>;
// generators/relations
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