metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D228, C4⋊D57, C57⋊4D4, C3⋊1D76, C76⋊1S3, C19⋊1D12, C228⋊1C2, C12⋊1D19, D114⋊1C2, C38.10D6, C6.10D38, C2.4D114, C114.10C22, sometimes denoted D456 or Dih228 or Dih456, SmallGroup(456,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D228
G = < a,b | a228=b2=1, bab=a-1 >
Smallest permutation representation of D228
►On 228 points
Generators in S228
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228)
(1 171)(2 170)(3 169)(4 168)(5 167)(6 166)(7 165)(8 164)(9 163)(10 162)(11 161)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(21 151)(22 150)(23 149)(24 148)(25 147)(26 146)(27 145)(28 144)(29 143)(30 142)(31 141)(32 140)(33 139)(34 138)(35 137)(36 136)(37 135)(38 134)(39 133)(40 132)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 123)(50 122)(51 121)(52 120)(53 119)(54 118)(55 117)(56 116)(57 115)(58 114)(59 113)(60 112)(61 111)(62 110)(63 109)(64 108)(65 107)(66 106)(67 105)(68 104)(69 103)(70 102)(71 101)(72 100)(73 99)(74 98)(75 97)(76 96)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)(172 228)(173 227)(174 226)(175 225)(176 224)(177 223)(178 222)(179 221)(180 220)(181 219)(182 218)(183 217)(184 216)(185 215)(186 214)(187 213)(188 212)(189 211)(190 210)(191 209)(192 208)(193 207)(194 206)(195 205)(196 204)(197 203)(198 202)(199 201)
G:=sub<Sym(228)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228), (1,171)(2,170)(3,169)(4,168)(5,167)(6,166)(7,165)(8,164)(9,163)(10,162)(11,161)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,151)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(31,141)(32,140)(33,139)(34,138)(35,137)(36,136)(37,135)(38,134)(39,133)(40,132)(41,131)(42,130)(43,129)(44,128)(45,127)(46,126)(47,125)(48,124)(49,123)(50,122)(51,121)(52,120)(53,119)(54,118)(55,117)(56,116)(57,115)(58,114)(59,113)(60,112)(61,111)(62,110)(63,109)(64,108)(65,107)(66,106)(67,105)(68,104)(69,103)(70,102)(71,101)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(172,228)(173,227)(174,226)(175,225)(176,224)(177,223)(178,222)(179,221)(180,220)(181,219)(182,218)(183,217)(184,216)(185,215)(186,214)(187,213)(188,212)(189,211)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228), (1,171)(2,170)(3,169)(4,168)(5,167)(6,166)(7,165)(8,164)(9,163)(10,162)(11,161)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,151)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(31,141)(32,140)(33,139)(34,138)(35,137)(36,136)(37,135)(38,134)(39,133)(40,132)(41,131)(42,130)(43,129)(44,128)(45,127)(46,126)(47,125)(48,124)(49,123)(50,122)(51,121)(52,120)(53,119)(54,118)(55,117)(56,116)(57,115)(58,114)(59,113)(60,112)(61,111)(62,110)(63,109)(64,108)(65,107)(66,106)(67,105)(68,104)(69,103)(70,102)(71,101)(72,100)(73,99)(74,98)(75,97)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(172,228)(173,227)(174,226)(175,225)(176,224)(177,223)(178,222)(179,221)(180,220)(181,219)(182,218)(183,217)(184,216)(185,215)(186,214)(187,213)(188,212)(189,211)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)], [(1,171),(2,170),(3,169),(4,168),(5,167),(6,166),(7,165),(8,164),(9,163),(10,162),(11,161),(12,160),(13,159),(14,158),(15,157),(16,156),(17,155),(18,154),(19,153),(20,152),(21,151),(22,150),(23,149),(24,148),(25,147),(26,146),(27,145),(28,144),(29,143),(30,142),(31,141),(32,140),(33,139),(34,138),(35,137),(36,136),(37,135),(38,134),(39,133),(40,132),(41,131),(42,130),(43,129),(44,128),(45,127),(46,126),(47,125),(48,124),(49,123),(50,122),(51,121),(52,120),(53,119),(54,118),(55,117),(56,116),(57,115),(58,114),(59,113),(60,112),(61,111),(62,110),(63,109),(64,108),(65,107),(66,106),(67,105),(68,104),(69,103),(70,102),(71,101),(72,100),(73,99),(74,98),(75,97),(76,96),(77,95),(78,94),(79,93),(80,92),(81,91),(82,90),(83,89),(84,88),(85,87),(172,228),(173,227),(174,226),(175,225),(176,224),(177,223),(178,222),(179,221),(180,220),(181,219),(182,218),(183,217),(184,216),(185,215),(186,214),(187,213),(188,212),(189,211),(190,210),(191,209),(192,208),(193,207),(194,206),(195,205),(196,204),(197,203),(198,202),(199,201)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 114 | 114 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D12 | D19 | D38 | D57 | D76 | D114 | D228 |
| kernel | D228 | C228 | D114 | C76 | C57 | C38 | C19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of D228 ►in GL2(𝔽229) generated by
| 87 | 82 |
| 147 | 207 |
| 140 | 221 |
| 74 | 89 |
G:=sub<GL(2,GF(229))| [87,147,82,207],[140,74,221,89] >;
D228 in GAP, Magma, Sage, TeX
D_{228} % in TeX
G:=Group("D228"); // GroupNames label
G:=SmallGroup(456,36);
// by ID
G=gap.SmallGroup(456,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,61,26,323,10804]);
// Polycyclic
G:=Group<a,b|a^228=b^2=1,b*a*b=a^-1>;
// generators/relations
Export