metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D224, C7⋊1D32, C32⋊1D7, C224⋊1C2, C8.5D28, C4.1D56, D112⋊1C2, C56.55D4, C14.1D16, C2.3D112, C28.26D8, C16.13D14, C112.14C22, sometimes denoted D448 or Dih224 or Dih448, SmallGroup(448,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D224
G = < a,b | a224=b2=1, bab=a-1 >
Smallest permutation representation of D224
►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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(64 224)(65 223)(66 222)(67 221)(68 220)(69 219)(70 218)(71 217)(72 216)(73 215)(74 214)(75 213)(76 212)(77 211)(78 210)(79 209)(80 208)(81 207)(82 206)(83 205)(84 204)(85 203)(86 202)(87 201)(88 200)(89 199)(90 198)(91 197)(92 196)(93 195)(94 194)(95 193)(96 192)(97 191)(98 190)(99 189)(100 188)(101 187)(102 186)(103 185)(104 184)(105 183)(106 182)(107 181)(108 180)(109 179)(110 178)(111 177)(112 176)(113 175)(114 174)(115 173)(116 172)(117 171)(118 170)(119 169)(120 168)(121 167)(122 166)(123 165)(124 164)(125 163)(126 162)(127 161)(128 160)(129 159)(130 158)(131 157)(132 156)(133 155)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(64,224)(65,223)(66,222)(67,221)(68,220)(69,219)(70,218)(71,217)(72,216)(73,215)(74,214)(75,213)(76,212)(77,211)(78,210)(79,209)(80,208)(81,207)(82,206)(83,205)(84,204)(85,203)(86,202)(87,201)(88,200)(89,199)(90,198)(91,197)(92,196)(93,195)(94,194)(95,193)(96,192)(97,191)(98,190)(99,189)(100,188)(101,187)(102,186)(103,185)(104,184)(105,183)(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)>;
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(64,224)(65,223)(66,222)(67,221)(68,220)(69,219)(70,218)(71,217)(72,216)(73,215)(74,214)(75,213)(76,212)(77,211)(78,210)(79,209)(80,208)(81,207)(82,206)(83,205)(84,204)(85,203)(86,202)(87,201)(88,200)(89,199)(90,198)(91,197)(92,196)(93,195)(94,194)(95,193)(96,192)(97,191)(98,190)(99,189)(100,188)(101,187)(102,186)(103,185)(104,184)(105,183)(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145) );
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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(64,224),(65,223),(66,222),(67,221),(68,220),(69,219),(70,218),(71,217),(72,216),(73,215),(74,214),(75,213),(76,212),(77,211),(78,210),(79,209),(80,208),(81,207),(82,206),(83,205),(84,204),(85,203),(86,202),(87,201),(88,200),(89,199),(90,198),(91,197),(92,196),(93,195),(94,194),(95,193),(96,192),(97,191),(98,190),(99,189),(100,188),(101,187),(102,186),(103,185),(104,184),(105,183),(106,182),(107,181),(108,180),(109,179),(110,178),(111,177),(112,176),(113,175),(114,174),(115,173),(116,172),(117,171),(118,170),(119,169),(120,168),(121,167),(122,166),(123,165),(124,164),(125,163),(126,162),(127,161),(128,160),(129,159),(130,158),(131,157),(132,156),(133,155),(134,154),(135,153),(136,152),(137,151),(138,150),(139,149),(140,148),(141,147),(142,146),(143,145)]])
115 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 32A | ··· | 32H | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 2 | 2 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 112 | 112 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D7 | D8 | D14 | D16 | D28 | D32 | D56 | D112 | D224 |
| kernel | D224 | C224 | D112 | C56 | C32 | C28 | C16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of D224 ►in GL2(𝔽449) generated by
| 202 | 394 |
| 110 | 199 |
| 194 | 376 |
| 85 | 255 |
G:=sub<GL(2,GF(449))| [202,110,394,199],[194,85,376,255] >;
D224 in GAP, Magma, Sage, TeX
D_{224} % in TeX
G:=Group("D224"); // GroupNames label
G:=SmallGroup(448,5);
// by ID
G=gap.SmallGroup(448,5);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,254,142,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b|a^224=b^2=1,b*a*b=a^-1>;
// generators/relations
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