metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic49, C49⋊C4, C98.C2, C2.D49, C7.Dic7, C14.1D7, SmallGroup(196,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — Dic49 |
Generators and relations for Dic49
G = < a,b | a98=1, b2=a49, bab-1=a-1 >
Smallest permutation representation of Dic49
►Regular action on 196 points
Generators in S196
(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)
(1 116 50 165)(2 115 51 164)(3 114 52 163)(4 113 53 162)(5 112 54 161)(6 111 55 160)(7 110 56 159)(8 109 57 158)(9 108 58 157)(10 107 59 156)(11 106 60 155)(12 105 61 154)(13 104 62 153)(14 103 63 152)(15 102 64 151)(16 101 65 150)(17 100 66 149)(18 99 67 148)(19 196 68 147)(20 195 69 146)(21 194 70 145)(22 193 71 144)(23 192 72 143)(24 191 73 142)(25 190 74 141)(26 189 75 140)(27 188 76 139)(28 187 77 138)(29 186 78 137)(30 185 79 136)(31 184 80 135)(32 183 81 134)(33 182 82 133)(34 181 83 132)(35 180 84 131)(36 179 85 130)(37 178 86 129)(38 177 87 128)(39 176 88 127)(40 175 89 126)(41 174 90 125)(42 173 91 124)(43 172 92 123)(44 171 93 122)(45 170 94 121)(46 169 95 120)(47 168 96 119)(48 167 97 118)(49 166 98 117)
G:=sub<Sym(196)| (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), (1,116,50,165)(2,115,51,164)(3,114,52,163)(4,113,53,162)(5,112,54,161)(6,111,55,160)(7,110,56,159)(8,109,57,158)(9,108,58,157)(10,107,59,156)(11,106,60,155)(12,105,61,154)(13,104,62,153)(14,103,63,152)(15,102,64,151)(16,101,65,150)(17,100,66,149)(18,99,67,148)(19,196,68,147)(20,195,69,146)(21,194,70,145)(22,193,71,144)(23,192,72,143)(24,191,73,142)(25,190,74,141)(26,189,75,140)(27,188,76,139)(28,187,77,138)(29,186,78,137)(30,185,79,136)(31,184,80,135)(32,183,81,134)(33,182,82,133)(34,181,83,132)(35,180,84,131)(36,179,85,130)(37,178,86,129)(38,177,87,128)(39,176,88,127)(40,175,89,126)(41,174,90,125)(42,173,91,124)(43,172,92,123)(44,171,93,122)(45,170,94,121)(46,169,95,120)(47,168,96,119)(48,167,97,118)(49,166,98,117)>;
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), (1,116,50,165)(2,115,51,164)(3,114,52,163)(4,113,53,162)(5,112,54,161)(6,111,55,160)(7,110,56,159)(8,109,57,158)(9,108,58,157)(10,107,59,156)(11,106,60,155)(12,105,61,154)(13,104,62,153)(14,103,63,152)(15,102,64,151)(16,101,65,150)(17,100,66,149)(18,99,67,148)(19,196,68,147)(20,195,69,146)(21,194,70,145)(22,193,71,144)(23,192,72,143)(24,191,73,142)(25,190,74,141)(26,189,75,140)(27,188,76,139)(28,187,77,138)(29,186,78,137)(30,185,79,136)(31,184,80,135)(32,183,81,134)(33,182,82,133)(34,181,83,132)(35,180,84,131)(36,179,85,130)(37,178,86,129)(38,177,87,128)(39,176,88,127)(40,175,89,126)(41,174,90,125)(42,173,91,124)(43,172,92,123)(44,171,93,122)(45,170,94,121)(46,169,95,120)(47,168,96,119)(48,167,97,118)(49,166,98,117) );
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)], [(1,116,50,165),(2,115,51,164),(3,114,52,163),(4,113,53,162),(5,112,54,161),(6,111,55,160),(7,110,56,159),(8,109,57,158),(9,108,58,157),(10,107,59,156),(11,106,60,155),(12,105,61,154),(13,104,62,153),(14,103,63,152),(15,102,64,151),(16,101,65,150),(17,100,66,149),(18,99,67,148),(19,196,68,147),(20,195,69,146),(21,194,70,145),(22,193,71,144),(23,192,72,143),(24,191,73,142),(25,190,74,141),(26,189,75,140),(27,188,76,139),(28,187,77,138),(29,186,78,137),(30,185,79,136),(31,184,80,135),(32,183,81,134),(33,182,82,133),(34,181,83,132),(35,180,84,131),(36,179,85,130),(37,178,86,129),(38,177,87,128),(39,176,88,127),(40,175,89,126),(41,174,90,125),(42,173,91,124),(43,172,92,123),(44,171,93,122),(45,170,94,121),(46,169,95,120),(47,168,96,119),(48,167,97,118),(49,166,98,117)]])
Dic49 is a maximal subgroup of
Dic98 C4×D49 C49⋊D4
Dic49 is a maximal quotient of C49⋊C8
52 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 14A | 14B | 14C | 49A | ··· | 49U | 98A | ··· | 98U |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 49 | ··· | 49 | 98 | ··· | 98 |
| size | 1 | 1 | 49 | 49 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | |
| image | C1 | C2 | C4 | D7 | Dic7 | D49 | Dic49 |
| kernel | Dic49 | C98 | C49 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 3 | 3 | 21 | 21 |
Matrix representation of Dic49 ►in GL2(𝔽197) generated by
| 111 | 194 |
| 3 | 23 |
| 21 | 70 |
| 95 | 176 |
G:=sub<GL(2,GF(197))| [111,3,194,23],[21,95,70,176] >;
Dic49 in GAP, Magma, Sage, TeX
{\rm Dic}_{49} % in TeX
G:=Group("Dic49"); // GroupNames label
G:=SmallGroup(196,1);
// by ID
G=gap.SmallGroup(196,1);
# by ID
G:=PCGroup([4,-2,-2,-7,-7,8,626,514,2691]);
// Polycyclic
G:=Group<a,b|a^98=1,b^2=a^49,b*a*b^-1=a^-1>;
// generators/relations
Export