direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D234, C2×D117, C26⋊D9, C18⋊D13, C9⋊2D26, C3.D78, C13⋊2D18, C234⋊1C2, C78.2S3, C6.2D39, C39.2D6, C117⋊2C22, sometimes denoted D468 or Dih234 or Dih468, SmallGroup(468,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C117 — D234 |
Generators and relations for D234
G = < a,b | a234=b2=1, bab=a-1 >
Smallest permutation representation of D234
►On 234 points
Generators in S234
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)
(1 234)(2 233)(3 232)(4 231)(5 230)(6 229)(7 228)(8 227)(9 226)(10 225)(11 224)(12 223)(13 222)(14 221)(15 220)(16 219)(17 218)(18 217)(19 216)(20 215)(21 214)(22 213)(23 212)(24 211)(25 210)(26 209)(27 208)(28 207)(29 206)(30 205)(31 204)(32 203)(33 202)(34 201)(35 200)(36 199)(37 198)(38 197)(39 196)(40 195)(41 194)(42 193)(43 192)(44 191)(45 190)(46 189)(47 188)(48 187)(49 186)(50 185)(51 184)(52 183)(53 182)(54 181)(55 180)(56 179)(57 178)(58 177)(59 176)(60 175)(61 174)(62 173)(63 172)(64 171)(65 170)(66 169)(67 168)(68 167)(69 166)(70 165)(71 164)(72 163)(73 162)(74 161)(75 160)(76 159)(77 158)(78 157)(79 156)(80 155)(81 154)(82 153)(83 152)(84 151)(85 150)(86 149)(87 148)(88 147)(89 146)(90 145)(91 144)(92 143)(93 142)(94 141)(95 140)(96 139)(97 138)(98 137)(99 136)(100 135)(101 134)(102 133)(103 132)(104 131)(105 130)(106 129)(107 128)(108 127)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)
G:=sub<Sym(234)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234), (1,234)(2,233)(3,232)(4,231)(5,230)(6,229)(7,228)(8,227)(9,226)(10,225)(11,224)(12,223)(13,222)(14,221)(15,220)(16,219)(17,218)(18,217)(19,216)(20,215)(21,214)(22,213)(23,212)(24,211)(25,210)(26,209)(27,208)(28,207)(29,206)(30,205)(31,204)(32,203)(33,202)(34,201)(35,200)(36,199)(37,198)(38,197)(39,196)(40,195)(41,194)(42,193)(43,192)(44,191)(45,190)(46,189)(47,188)(48,187)(49,186)(50,185)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,165)(71,164)(72,163)(73,162)(74,161)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234), (1,234)(2,233)(3,232)(4,231)(5,230)(6,229)(7,228)(8,227)(9,226)(10,225)(11,224)(12,223)(13,222)(14,221)(15,220)(16,219)(17,218)(18,217)(19,216)(20,215)(21,214)(22,213)(23,212)(24,211)(25,210)(26,209)(27,208)(28,207)(29,206)(30,205)(31,204)(32,203)(33,202)(34,201)(35,200)(36,199)(37,198)(38,197)(39,196)(40,195)(41,194)(42,193)(43,192)(44,191)(45,190)(46,189)(47,188)(48,187)(49,186)(50,185)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,165)(71,164)(72,163)(73,162)(74,161)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)], [(1,234),(2,233),(3,232),(4,231),(5,230),(6,229),(7,228),(8,227),(9,226),(10,225),(11,224),(12,223),(13,222),(14,221),(15,220),(16,219),(17,218),(18,217),(19,216),(20,215),(21,214),(22,213),(23,212),(24,211),(25,210),(26,209),(27,208),(28,207),(29,206),(30,205),(31,204),(32,203),(33,202),(34,201),(35,200),(36,199),(37,198),(38,197),(39,196),(40,195),(41,194),(42,193),(43,192),(44,191),(45,190),(46,189),(47,188),(48,187),(49,186),(50,185),(51,184),(52,183),(53,182),(54,181),(55,180),(56,179),(57,178),(58,177),(59,176),(60,175),(61,174),(62,173),(63,172),(64,171),(65,170),(66,169),(67,168),(68,167),(69,166),(70,165),(71,164),(72,163),(73,162),(74,161),(75,160),(76,159),(77,158),(78,157),(79,156),(80,155),(81,154),(82,153),(83,152),(84,151),(85,150),(86,149),(87,148),(88,147),(89,146),(90,145),(91,144),(92,143),(93,142),(94,141),(95,140),(96,139),(97,138),(98,137),(99,136),(100,135),(101,134),(102,133),(103,132),(104,131),(105,130),(106,129),(107,128),(108,127),(109,126),(110,125),(111,124),(112,123),(113,122),(114,121),(115,120),(116,119),(117,118)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 13A | ··· | 13F | 18A | 18B | 18C | 26A | ··· | 26F | 39A | ··· | 39L | 78A | ··· | 78L | 117A | ··· | 117AJ | 234A | ··· | 234AJ |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 9 | 9 | 9 | 13 | ··· | 13 | 18 | 18 | 18 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 | 117 | ··· | 117 | 234 | ··· | 234 |
| size | 1 | 1 | 117 | 117 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D9 | D13 | D18 | D26 | D39 | D78 | D117 | D234 |
| kernel | D234 | D117 | C234 | C78 | C39 | C26 | C18 | C13 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 3 | 6 | 3 | 6 | 12 | 12 | 36 | 36 |
Matrix representation of D234 ►in GL2(𝔽937) generated by
| 856 | 923 |
| 14 | 870 |
| 879 | 891 |
| 12 | 58 |
G:=sub<GL(2,GF(937))| [856,14,923,870],[879,12,891,58] >;
D234 in GAP, Magma, Sage, TeX
D_{234} % in TeX
G:=Group("D234"); // GroupNames label
G:=SmallGroup(468,17);
// by ID
G=gap.SmallGroup(468,17);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-3,2462,1182,2883,7804]);
// Polycyclic
G:=Group<a,b|a^234=b^2=1,b*a*b=a^-1>;
// generators/relations
Export