metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic51, C51⋊3C4, C34.S3, C6.D17, C2.D51, C3⋊Dic17, C17⋊2Dic3, C102.1C2, SmallGroup(204,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — Dic51 |
Generators and relations for Dic51
G = < a,b | a102=1, b2=a51, bab-1=a-1 >
Smallest permutation representation of Dic51
►Regular action on 204 points
Generators in S204
(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)
(1 122 52 173)(2 121 53 172)(3 120 54 171)(4 119 55 170)(5 118 56 169)(6 117 57 168)(7 116 58 167)(8 115 59 166)(9 114 60 165)(10 113 61 164)(11 112 62 163)(12 111 63 162)(13 110 64 161)(14 109 65 160)(15 108 66 159)(16 107 67 158)(17 106 68 157)(18 105 69 156)(19 104 70 155)(20 103 71 154)(21 204 72 153)(22 203 73 152)(23 202 74 151)(24 201 75 150)(25 200 76 149)(26 199 77 148)(27 198 78 147)(28 197 79 146)(29 196 80 145)(30 195 81 144)(31 194 82 143)(32 193 83 142)(33 192 84 141)(34 191 85 140)(35 190 86 139)(36 189 87 138)(37 188 88 137)(38 187 89 136)(39 186 90 135)(40 185 91 134)(41 184 92 133)(42 183 93 132)(43 182 94 131)(44 181 95 130)(45 180 96 129)(46 179 97 128)(47 178 98 127)(48 177 99 126)(49 176 100 125)(50 175 101 124)(51 174 102 123)
G:=sub<Sym(204)| (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), (1,122,52,173)(2,121,53,172)(3,120,54,171)(4,119,55,170)(5,118,56,169)(6,117,57,168)(7,116,58,167)(8,115,59,166)(9,114,60,165)(10,113,61,164)(11,112,62,163)(12,111,63,162)(13,110,64,161)(14,109,65,160)(15,108,66,159)(16,107,67,158)(17,106,68,157)(18,105,69,156)(19,104,70,155)(20,103,71,154)(21,204,72,153)(22,203,73,152)(23,202,74,151)(24,201,75,150)(25,200,76,149)(26,199,77,148)(27,198,78,147)(28,197,79,146)(29,196,80,145)(30,195,81,144)(31,194,82,143)(32,193,83,142)(33,192,84,141)(34,191,85,140)(35,190,86,139)(36,189,87,138)(37,188,88,137)(38,187,89,136)(39,186,90,135)(40,185,91,134)(41,184,92,133)(42,183,93,132)(43,182,94,131)(44,181,95,130)(45,180,96,129)(46,179,97,128)(47,178,98,127)(48,177,99,126)(49,176,100,125)(50,175,101,124)(51,174,102,123)>;
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), (1,122,52,173)(2,121,53,172)(3,120,54,171)(4,119,55,170)(5,118,56,169)(6,117,57,168)(7,116,58,167)(8,115,59,166)(9,114,60,165)(10,113,61,164)(11,112,62,163)(12,111,63,162)(13,110,64,161)(14,109,65,160)(15,108,66,159)(16,107,67,158)(17,106,68,157)(18,105,69,156)(19,104,70,155)(20,103,71,154)(21,204,72,153)(22,203,73,152)(23,202,74,151)(24,201,75,150)(25,200,76,149)(26,199,77,148)(27,198,78,147)(28,197,79,146)(29,196,80,145)(30,195,81,144)(31,194,82,143)(32,193,83,142)(33,192,84,141)(34,191,85,140)(35,190,86,139)(36,189,87,138)(37,188,88,137)(38,187,89,136)(39,186,90,135)(40,185,91,134)(41,184,92,133)(42,183,93,132)(43,182,94,131)(44,181,95,130)(45,180,96,129)(46,179,97,128)(47,178,98,127)(48,177,99,126)(49,176,100,125)(50,175,101,124)(51,174,102,123) );
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)], [(1,122,52,173),(2,121,53,172),(3,120,54,171),(4,119,55,170),(5,118,56,169),(6,117,57,168),(7,116,58,167),(8,115,59,166),(9,114,60,165),(10,113,61,164),(11,112,62,163),(12,111,63,162),(13,110,64,161),(14,109,65,160),(15,108,66,159),(16,107,67,158),(17,106,68,157),(18,105,69,156),(19,104,70,155),(20,103,71,154),(21,204,72,153),(22,203,73,152),(23,202,74,151),(24,201,75,150),(25,200,76,149),(26,199,77,148),(27,198,78,147),(28,197,79,146),(29,196,80,145),(30,195,81,144),(31,194,82,143),(32,193,83,142),(33,192,84,141),(34,191,85,140),(35,190,86,139),(36,189,87,138),(37,188,88,137),(38,187,89,136),(39,186,90,135),(40,185,91,134),(41,184,92,133),(42,183,93,132),(43,182,94,131),(44,181,95,130),(45,180,96,129),(46,179,97,128),(47,178,98,127),(48,177,99,126),(49,176,100,125),(50,175,101,124),(51,174,102,123)]])
Dic51 is a maximal subgroup of
Dic3×D17 S3×Dic17 C51⋊D4 C51⋊Q8 Dic102 C4×D51 C51⋊7D4
Dic51 is a maximal quotient of C51⋊5C8
54 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 102A | ··· | 102P |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 2 | 51 | 51 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D17 | Dic17 | D51 | Dic51 |
| kernel | Dic51 | C102 | C51 | C34 | C17 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 8 | 8 | 16 | 16 |
Matrix representation of Dic51 ►in GL3(𝔽409) generated by
| 408 | 0 | 0 |
| 0 | 143 | 239 |
| 0 | 170 | 21 |
| 266 | 0 | 0 |
| 0 | 388 | 239 |
| 0 | 171 | 21 |
G:=sub<GL(3,GF(409))| [408,0,0,0,143,170,0,239,21],[266,0,0,0,388,171,0,239,21] >;
Dic51 in GAP, Magma, Sage, TeX
{\rm Dic}_{51} % in TeX
G:=Group("Dic51"); // GroupNames label
G:=SmallGroup(204,3);
// by ID
G=gap.SmallGroup(204,3);
# by ID
G:=PCGroup([4,-2,-2,-3,-17,8,98,3075]);
// Polycyclic
G:=Group<a,b|a^102=1,b^2=a^51,b*a*b^-1=a^-1>;
// generators/relations
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