metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D231, C7⋊D33, C3⋊D77, C11⋊D21, C77⋊1S3, C33⋊1D7, C231⋊1C2, C21⋊1D11, sometimes denoted D462 or Dih231 or Dih462, SmallGroup(462,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C231 — D231 |
Generators and relations for D231
G = < a,b | a231=b2=1, bab=a-1 >
Smallest permutation representation of D231
►On 231 points
Generators in S231
(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)
(2 231)(3 230)(4 229)(5 228)(6 227)(7 226)(8 225)(9 224)(10 223)(11 222)(12 221)(13 220)(14 219)(15 218)(16 217)(17 216)(18 215)(19 214)(20 213)(21 212)(22 211)(23 210)(24 209)(25 208)(26 207)(27 206)(28 205)(29 204)(30 203)(31 202)(32 201)(33 200)(34 199)(35 198)(36 197)(37 196)(38 195)(39 194)(40 193)(41 192)(42 191)(43 190)(44 189)(45 188)(46 187)(47 186)(48 185)(49 184)(50 183)(51 182)(52 181)(53 180)(54 179)(55 178)(56 177)(57 176)(58 175)(59 174)(60 173)(61 172)(62 171)(63 170)(64 169)(65 168)(66 167)(67 166)(68 165)(69 164)(70 163)(71 162)(72 161)(73 160)(74 159)(75 158)(76 157)(77 156)(78 155)(79 154)(80 153)(81 152)(82 151)(83 150)(84 149)(85 148)(86 147)(87 146)(88 145)(89 144)(90 143)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 135)(99 134)(100 133)(101 132)(102 131)(103 130)(104 129)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 117)
G:=sub<Sym(231)| (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), (2,231)(3,230)(4,229)(5,228)(6,227)(7,226)(8,225)(9,224)(10,223)(11,222)(12,221)(13,220)(14,219)(15,218)(16,217)(17,216)(18,215)(19,214)(20,213)(21,212)(22,211)(23,210)(24,209)(25,208)(26,207)(27,206)(28,205)(29,204)(30,203)(31,202)(32,201)(33,200)(34,199)(35,198)(36,197)(37,196)(38,195)(39,194)(40,193)(41,192)(42,191)(43,190)(44,189)(45,188)(46,187)(47,186)(48,185)(49,184)(50,183)(51,182)(52,181)(53,180)(54,179)(55,178)(56,177)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,160)(74,159)(75,158)(76,157)(77,156)(78,155)(79,154)(80,153)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,117)>;
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), (2,231)(3,230)(4,229)(5,228)(6,227)(7,226)(8,225)(9,224)(10,223)(11,222)(12,221)(13,220)(14,219)(15,218)(16,217)(17,216)(18,215)(19,214)(20,213)(21,212)(22,211)(23,210)(24,209)(25,208)(26,207)(27,206)(28,205)(29,204)(30,203)(31,202)(32,201)(33,200)(34,199)(35,198)(36,197)(37,196)(38,195)(39,194)(40,193)(41,192)(42,191)(43,190)(44,189)(45,188)(46,187)(47,186)(48,185)(49,184)(50,183)(51,182)(52,181)(53,180)(54,179)(55,178)(56,177)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,160)(74,159)(75,158)(76,157)(77,156)(78,155)(79,154)(80,153)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,117) );
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)], [(2,231),(3,230),(4,229),(5,228),(6,227),(7,226),(8,225),(9,224),(10,223),(11,222),(12,221),(13,220),(14,219),(15,218),(16,217),(17,216),(18,215),(19,214),(20,213),(21,212),(22,211),(23,210),(24,209),(25,208),(26,207),(27,206),(28,205),(29,204),(30,203),(31,202),(32,201),(33,200),(34,199),(35,198),(36,197),(37,196),(38,195),(39,194),(40,193),(41,192),(42,191),(43,190),(44,189),(45,188),(46,187),(47,186),(48,185),(49,184),(50,183),(51,182),(52,181),(53,180),(54,179),(55,178),(56,177),(57,176),(58,175),(59,174),(60,173),(61,172),(62,171),(63,170),(64,169),(65,168),(66,167),(67,166),(68,165),(69,164),(70,163),(71,162),(72,161),(73,160),(74,159),(75,158),(76,157),(77,156),(78,155),(79,154),(80,153),(81,152),(82,151),(83,150),(84,149),(85,148),(86,147),(87,146),(88,145),(89,144),(90,143),(91,142),(92,141),(93,140),(94,139),(95,138),(96,137),(97,136),(98,135),(99,134),(100,133),(101,132),(102,131),(103,130),(104,129),(105,128),(106,127),(107,126),(108,125),(109,124),(110,123),(111,122),(112,121),(113,120),(114,119),(115,118),(116,117)]])
117 conjugacy classes
| class | 1 | 2 | 3 | 7A | 7B | 7C | 11A | ··· | 11E | 21A | ··· | 21F | 33A | ··· | 33J | 77A | ··· | 77AD | 231A | ··· | 231BH |
| order | 1 | 2 | 3 | 7 | 7 | 7 | 11 | ··· | 11 | 21 | ··· | 21 | 33 | ··· | 33 | 77 | ··· | 77 | 231 | ··· | 231 |
| size | 1 | 231 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D7 | D11 | D21 | D33 | D77 | D231 |
| kernel | D231 | C231 | C77 | C33 | C21 | C11 | C7 | C3 | C1 |
| # reps | 1 | 1 | 1 | 3 | 5 | 6 | 10 | 30 | 60 |
Matrix representation of D231 ►in GL2(𝔽463) generated by
| 372 | 66 |
| 397 | 114 |
| 1 | 0 |
| 417 | 462 |
G:=sub<GL(2,GF(463))| [372,397,66,114],[1,417,0,462] >;
D231 in GAP, Magma, Sage, TeX
D_{231} % in TeX
G:=Group("D231"); // GroupNames label
G:=SmallGroup(462,11);
// by ID
G=gap.SmallGroup(462,11);
# by ID
G:=PCGroup([4,-2,-3,-7,-11,33,434,6723]);
// Polycyclic
G:=Group<a,b|a^231=b^2=1,b*a*b=a^-1>;
// generators/relations
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