metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic50, C25⋊Q8, C4.D25, C2.3D50, C20.1D5, C5.Dic10, C100.1C2, C10.6D10, C50.1C22, Dic25.1C2, SmallGroup(200,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic50
G = < a,b | a100=1, b2=a50, bab-1=a-1 >
Smallest permutation representation of Dic50
►Regular action on 200 points
Generators in S200
(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)
(1 182 51 132)(2 181 52 131)(3 180 53 130)(4 179 54 129)(5 178 55 128)(6 177 56 127)(7 176 57 126)(8 175 58 125)(9 174 59 124)(10 173 60 123)(11 172 61 122)(12 171 62 121)(13 170 63 120)(14 169 64 119)(15 168 65 118)(16 167 66 117)(17 166 67 116)(18 165 68 115)(19 164 69 114)(20 163 70 113)(21 162 71 112)(22 161 72 111)(23 160 73 110)(24 159 74 109)(25 158 75 108)(26 157 76 107)(27 156 77 106)(28 155 78 105)(29 154 79 104)(30 153 80 103)(31 152 81 102)(32 151 82 101)(33 150 83 200)(34 149 84 199)(35 148 85 198)(36 147 86 197)(37 146 87 196)(38 145 88 195)(39 144 89 194)(40 143 90 193)(41 142 91 192)(42 141 92 191)(43 140 93 190)(44 139 94 189)(45 138 95 188)(46 137 96 187)(47 136 97 186)(48 135 98 185)(49 134 99 184)(50 133 100 183)
G:=sub<Sym(200)| (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), (1,182,51,132)(2,181,52,131)(3,180,53,130)(4,179,54,129)(5,178,55,128)(6,177,56,127)(7,176,57,126)(8,175,58,125)(9,174,59,124)(10,173,60,123)(11,172,61,122)(12,171,62,121)(13,170,63,120)(14,169,64,119)(15,168,65,118)(16,167,66,117)(17,166,67,116)(18,165,68,115)(19,164,69,114)(20,163,70,113)(21,162,71,112)(22,161,72,111)(23,160,73,110)(24,159,74,109)(25,158,75,108)(26,157,76,107)(27,156,77,106)(28,155,78,105)(29,154,79,104)(30,153,80,103)(31,152,81,102)(32,151,82,101)(33,150,83,200)(34,149,84,199)(35,148,85,198)(36,147,86,197)(37,146,87,196)(38,145,88,195)(39,144,89,194)(40,143,90,193)(41,142,91,192)(42,141,92,191)(43,140,93,190)(44,139,94,189)(45,138,95,188)(46,137,96,187)(47,136,97,186)(48,135,98,185)(49,134,99,184)(50,133,100,183)>;
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), (1,182,51,132)(2,181,52,131)(3,180,53,130)(4,179,54,129)(5,178,55,128)(6,177,56,127)(7,176,57,126)(8,175,58,125)(9,174,59,124)(10,173,60,123)(11,172,61,122)(12,171,62,121)(13,170,63,120)(14,169,64,119)(15,168,65,118)(16,167,66,117)(17,166,67,116)(18,165,68,115)(19,164,69,114)(20,163,70,113)(21,162,71,112)(22,161,72,111)(23,160,73,110)(24,159,74,109)(25,158,75,108)(26,157,76,107)(27,156,77,106)(28,155,78,105)(29,154,79,104)(30,153,80,103)(31,152,81,102)(32,151,82,101)(33,150,83,200)(34,149,84,199)(35,148,85,198)(36,147,86,197)(37,146,87,196)(38,145,88,195)(39,144,89,194)(40,143,90,193)(41,142,91,192)(42,141,92,191)(43,140,93,190)(44,139,94,189)(45,138,95,188)(46,137,96,187)(47,136,97,186)(48,135,98,185)(49,134,99,184)(50,133,100,183) );
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)], [(1,182,51,132),(2,181,52,131),(3,180,53,130),(4,179,54,129),(5,178,55,128),(6,177,56,127),(7,176,57,126),(8,175,58,125),(9,174,59,124),(10,173,60,123),(11,172,61,122),(12,171,62,121),(13,170,63,120),(14,169,64,119),(15,168,65,118),(16,167,66,117),(17,166,67,116),(18,165,68,115),(19,164,69,114),(20,163,70,113),(21,162,71,112),(22,161,72,111),(23,160,73,110),(24,159,74,109),(25,158,75,108),(26,157,76,107),(27,156,77,106),(28,155,78,105),(29,154,79,104),(30,153,80,103),(31,152,81,102),(32,151,82,101),(33,150,83,200),(34,149,84,199),(35,148,85,198),(36,147,86,197),(37,146,87,196),(38,145,88,195),(39,144,89,194),(40,143,90,193),(41,142,91,192),(42,141,92,191),(43,140,93,190),(44,139,94,189),(45,138,95,188),(46,137,96,187),(47,136,97,186),(48,135,98,185),(49,134,99,184),(50,133,100,183)]])
Dic50 is a maximal subgroup of
Dic100 C200⋊C2 D4.D25 C25⋊Q16 D100⋊5C2 D4⋊2D25 Q8×D25
Dic50 is a maximal quotient of C50.D4 C4⋊Dic25
53 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 10A | 10B | 20A | 20B | 20C | 20D | 25A | ··· | 25J | 50A | ··· | 50J | 100A | ··· | 100T |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 2 | 50 | 50 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
53 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D5 | D10 | Dic10 | D25 | D50 | Dic50 |
| kernel | Dic50 | Dic25 | C100 | C25 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 20 |
Matrix representation of Dic50 ►in GL2(𝔽101) generated by
| 60 | 69 |
| 32 | 89 |
| 0 | 91 |
| 91 | 0 |
G:=sub<GL(2,GF(101))| [60,32,69,89],[0,91,91,0] >;
Dic50 in GAP, Magma, Sage, TeX
{\rm Dic}_{50} % in TeX
G:=Group("Dic50"); // GroupNames label
G:=SmallGroup(200,4);
// by ID
G=gap.SmallGroup(200,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,61,26,1443,418,4004]);
// Polycyclic
G:=Group<a,b|a^100=1,b^2=a^50,b*a*b^-1=a^-1>;
// generators/relations
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