metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic52, C8.D13, C13⋊1Q16, C2.5D52, C26.3D4, C104.1C2, C4.10D26, C52.10C22, Dic26.1C2, SmallGroup(208,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic52
G = < a,b | a104=1, b2=a52, bab-1=a-1 >
Smallest permutation representation of Dic52
►Regular action on 208 points
Generators in S208
(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)
(1 132 53 184)(2 131 54 183)(3 130 55 182)(4 129 56 181)(5 128 57 180)(6 127 58 179)(7 126 59 178)(8 125 60 177)(9 124 61 176)(10 123 62 175)(11 122 63 174)(12 121 64 173)(13 120 65 172)(14 119 66 171)(15 118 67 170)(16 117 68 169)(17 116 69 168)(18 115 70 167)(19 114 71 166)(20 113 72 165)(21 112 73 164)(22 111 74 163)(23 110 75 162)(24 109 76 161)(25 108 77 160)(26 107 78 159)(27 106 79 158)(28 105 80 157)(29 208 81 156)(30 207 82 155)(31 206 83 154)(32 205 84 153)(33 204 85 152)(34 203 86 151)(35 202 87 150)(36 201 88 149)(37 200 89 148)(38 199 90 147)(39 198 91 146)(40 197 92 145)(41 196 93 144)(42 195 94 143)(43 194 95 142)(44 193 96 141)(45 192 97 140)(46 191 98 139)(47 190 99 138)(48 189 100 137)(49 188 101 136)(50 187 102 135)(51 186 103 134)(52 185 104 133)
G:=sub<Sym(208)| (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), (1,132,53,184)(2,131,54,183)(3,130,55,182)(4,129,56,181)(5,128,57,180)(6,127,58,179)(7,126,59,178)(8,125,60,177)(9,124,61,176)(10,123,62,175)(11,122,63,174)(12,121,64,173)(13,120,65,172)(14,119,66,171)(15,118,67,170)(16,117,68,169)(17,116,69,168)(18,115,70,167)(19,114,71,166)(20,113,72,165)(21,112,73,164)(22,111,74,163)(23,110,75,162)(24,109,76,161)(25,108,77,160)(26,107,78,159)(27,106,79,158)(28,105,80,157)(29,208,81,156)(30,207,82,155)(31,206,83,154)(32,205,84,153)(33,204,85,152)(34,203,86,151)(35,202,87,150)(36,201,88,149)(37,200,89,148)(38,199,90,147)(39,198,91,146)(40,197,92,145)(41,196,93,144)(42,195,94,143)(43,194,95,142)(44,193,96,141)(45,192,97,140)(46,191,98,139)(47,190,99,138)(48,189,100,137)(49,188,101,136)(50,187,102,135)(51,186,103,134)(52,185,104,133)>;
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), (1,132,53,184)(2,131,54,183)(3,130,55,182)(4,129,56,181)(5,128,57,180)(6,127,58,179)(7,126,59,178)(8,125,60,177)(9,124,61,176)(10,123,62,175)(11,122,63,174)(12,121,64,173)(13,120,65,172)(14,119,66,171)(15,118,67,170)(16,117,68,169)(17,116,69,168)(18,115,70,167)(19,114,71,166)(20,113,72,165)(21,112,73,164)(22,111,74,163)(23,110,75,162)(24,109,76,161)(25,108,77,160)(26,107,78,159)(27,106,79,158)(28,105,80,157)(29,208,81,156)(30,207,82,155)(31,206,83,154)(32,205,84,153)(33,204,85,152)(34,203,86,151)(35,202,87,150)(36,201,88,149)(37,200,89,148)(38,199,90,147)(39,198,91,146)(40,197,92,145)(41,196,93,144)(42,195,94,143)(43,194,95,142)(44,193,96,141)(45,192,97,140)(46,191,98,139)(47,190,99,138)(48,189,100,137)(49,188,101,136)(50,187,102,135)(51,186,103,134)(52,185,104,133) );
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)], [(1,132,53,184),(2,131,54,183),(3,130,55,182),(4,129,56,181),(5,128,57,180),(6,127,58,179),(7,126,59,178),(8,125,60,177),(9,124,61,176),(10,123,62,175),(11,122,63,174),(12,121,64,173),(13,120,65,172),(14,119,66,171),(15,118,67,170),(16,117,68,169),(17,116,69,168),(18,115,70,167),(19,114,71,166),(20,113,72,165),(21,112,73,164),(22,111,74,163),(23,110,75,162),(24,109,76,161),(25,108,77,160),(26,107,78,159),(27,106,79,158),(28,105,80,157),(29,208,81,156),(30,207,82,155),(31,206,83,154),(32,205,84,153),(33,204,85,152),(34,203,86,151),(35,202,87,150),(36,201,88,149),(37,200,89,148),(38,199,90,147),(39,198,91,146),(40,197,92,145),(41,196,93,144),(42,195,94,143),(43,194,95,142),(44,193,96,141),(45,192,97,140),(46,191,98,139),(47,190,99,138),(48,189,100,137),(49,188,101,136),(50,187,102,135),(51,186,103,134),(52,185,104,133)]])
Dic52 is a maximal subgroup of
C16⋊D13 Dic104 D8.D13 C13⋊Q32 D104⋊7C2 C8.D26 D8⋊3D13 D4.D26 Q16×D13
Dic52 is a maximal quotient of
C52.44D4 C104⋊5C4
55 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52L | 104A | ··· | 104X |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 52 | 52 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
55 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | Q16 | D13 | D26 | D52 | Dic52 |
| kernel | Dic52 | C104 | Dic26 | C26 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 6 | 6 | 12 | 24 |
Matrix representation of Dic52 ►in GL2(𝔽313) generated by
| 99 | 130 |
| 223 | 296 |
| 256 | 44 |
| 168 | 57 |
G:=sub<GL(2,GF(313))| [99,223,130,296],[256,168,44,57] >;
Dic52 in GAP, Magma, Sage, TeX
{\rm Dic}_{52} % in TeX
G:=Group("Dic52"); // GroupNames label
G:=SmallGroup(208,8);
// by ID
G=gap.SmallGroup(208,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,40,61,66,182,42,4804]);
// Polycyclic
G:=Group<a,b|a^104=1,b^2=a^52,b*a*b^-1=a^-1>;
// generators/relations
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