metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic53, C53⋊2C4, C2.D53, C106.C2, SmallGroup(212,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — Dic53 |
Generators and relations for Dic53
G = < a,b | a106=1, b2=a53, bab-1=a-1 >
Smallest permutation representation of Dic53
►Regular action on 212 points
Generators in S212
(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)
(1 160 54 107)(2 159 55 212)(3 158 56 211)(4 157 57 210)(5 156 58 209)(6 155 59 208)(7 154 60 207)(8 153 61 206)(9 152 62 205)(10 151 63 204)(11 150 64 203)(12 149 65 202)(13 148 66 201)(14 147 67 200)(15 146 68 199)(16 145 69 198)(17 144 70 197)(18 143 71 196)(19 142 72 195)(20 141 73 194)(21 140 74 193)(22 139 75 192)(23 138 76 191)(24 137 77 190)(25 136 78 189)(26 135 79 188)(27 134 80 187)(28 133 81 186)(29 132 82 185)(30 131 83 184)(31 130 84 183)(32 129 85 182)(33 128 86 181)(34 127 87 180)(35 126 88 179)(36 125 89 178)(37 124 90 177)(38 123 91 176)(39 122 92 175)(40 121 93 174)(41 120 94 173)(42 119 95 172)(43 118 96 171)(44 117 97 170)(45 116 98 169)(46 115 99 168)(47 114 100 167)(48 113 101 166)(49 112 102 165)(50 111 103 164)(51 110 104 163)(52 109 105 162)(53 108 106 161)
G:=sub<Sym(212)| (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), (1,160,54,107)(2,159,55,212)(3,158,56,211)(4,157,57,210)(5,156,58,209)(6,155,59,208)(7,154,60,207)(8,153,61,206)(9,152,62,205)(10,151,63,204)(11,150,64,203)(12,149,65,202)(13,148,66,201)(14,147,67,200)(15,146,68,199)(16,145,69,198)(17,144,70,197)(18,143,71,196)(19,142,72,195)(20,141,73,194)(21,140,74,193)(22,139,75,192)(23,138,76,191)(24,137,77,190)(25,136,78,189)(26,135,79,188)(27,134,80,187)(28,133,81,186)(29,132,82,185)(30,131,83,184)(31,130,84,183)(32,129,85,182)(33,128,86,181)(34,127,87,180)(35,126,88,179)(36,125,89,178)(37,124,90,177)(38,123,91,176)(39,122,92,175)(40,121,93,174)(41,120,94,173)(42,119,95,172)(43,118,96,171)(44,117,97,170)(45,116,98,169)(46,115,99,168)(47,114,100,167)(48,113,101,166)(49,112,102,165)(50,111,103,164)(51,110,104,163)(52,109,105,162)(53,108,106,161)>;
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), (1,160,54,107)(2,159,55,212)(3,158,56,211)(4,157,57,210)(5,156,58,209)(6,155,59,208)(7,154,60,207)(8,153,61,206)(9,152,62,205)(10,151,63,204)(11,150,64,203)(12,149,65,202)(13,148,66,201)(14,147,67,200)(15,146,68,199)(16,145,69,198)(17,144,70,197)(18,143,71,196)(19,142,72,195)(20,141,73,194)(21,140,74,193)(22,139,75,192)(23,138,76,191)(24,137,77,190)(25,136,78,189)(26,135,79,188)(27,134,80,187)(28,133,81,186)(29,132,82,185)(30,131,83,184)(31,130,84,183)(32,129,85,182)(33,128,86,181)(34,127,87,180)(35,126,88,179)(36,125,89,178)(37,124,90,177)(38,123,91,176)(39,122,92,175)(40,121,93,174)(41,120,94,173)(42,119,95,172)(43,118,96,171)(44,117,97,170)(45,116,98,169)(46,115,99,168)(47,114,100,167)(48,113,101,166)(49,112,102,165)(50,111,103,164)(51,110,104,163)(52,109,105,162)(53,108,106,161) );
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)], [(1,160,54,107),(2,159,55,212),(3,158,56,211),(4,157,57,210),(5,156,58,209),(6,155,59,208),(7,154,60,207),(8,153,61,206),(9,152,62,205),(10,151,63,204),(11,150,64,203),(12,149,65,202),(13,148,66,201),(14,147,67,200),(15,146,68,199),(16,145,69,198),(17,144,70,197),(18,143,71,196),(19,142,72,195),(20,141,73,194),(21,140,74,193),(22,139,75,192),(23,138,76,191),(24,137,77,190),(25,136,78,189),(26,135,79,188),(27,134,80,187),(28,133,81,186),(29,132,82,185),(30,131,83,184),(31,130,84,183),(32,129,85,182),(33,128,86,181),(34,127,87,180),(35,126,88,179),(36,125,89,178),(37,124,90,177),(38,123,91,176),(39,122,92,175),(40,121,93,174),(41,120,94,173),(42,119,95,172),(43,118,96,171),(44,117,97,170),(45,116,98,169),(46,115,99,168),(47,114,100,167),(48,113,101,166),(49,112,102,165),(50,111,103,164),(51,110,104,163),(52,109,105,162),(53,108,106,161)]])
Dic53 is a maximal subgroup of
C53⋊C8 Dic106 C4×D53 C53⋊D4
Dic53 is a maximal quotient of C53⋊2C8
56 conjugacy classes
| class | 1 | 2 | 4A | 4B | 53A | ··· | 53Z | 106A | ··· | 106Z |
| order | 1 | 2 | 4 | 4 | 53 | ··· | 53 | 106 | ··· | 106 |
| size | 1 | 1 | 53 | 53 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D53 | Dic53 |
| kernel | Dic53 | C106 | C53 | C2 | C1 |
| # reps | 1 | 1 | 2 | 26 | 26 |
Matrix representation of Dic53 ►in GL2(𝔽1061) generated by
| 369 | 1 |
| 1060 | 0 |
| 1023 | 224 |
| 453 | 38 |
G:=sub<GL(2,GF(1061))| [369,1060,1,0],[1023,453,224,38] >;
Dic53 in GAP, Magma, Sage, TeX
{\rm Dic}_{53} % in TeX
G:=Group("Dic53"); // GroupNames label
G:=SmallGroup(212,1);
// by ID
G=gap.SmallGroup(212,1);
# by ID
G:=PCGroup([3,-2,-2,-53,6,1874]);
// Polycyclic
G:=Group<a,b|a^106=1,b^2=a^53,b*a*b^-1=a^-1>;
// generators/relations
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