metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D196, C4⋊D49, C49⋊1D4, C7.D28, C196⋊1C2, D98⋊1C2, C28.2D7, C2.4D98, C14.8D14, C98.3C22, sometimes denoted D392 or Dih196 or Dih392, SmallGroup(392,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D196
G = < a,b | a196=b2=1, bab=a-1 >
Smallest permutation representation of D196
►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)
(1 147)(2 146)(3 145)(4 144)(5 143)(6 142)(7 141)(8 140)(9 139)(10 138)(11 137)(12 136)(13 135)(14 134)(15 133)(16 132)(17 131)(18 130)(19 129)(20 128)(21 127)(22 126)(23 125)(24 124)(25 123)(26 122)(27 121)(28 120)(29 119)(30 118)(31 117)(32 116)(33 115)(34 114)(35 113)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(148 196)(149 195)(150 194)(151 193)(152 192)(153 191)(154 190)(155 189)(156 188)(157 187)(158 186)(159 185)(160 184)(161 183)(162 182)(163 181)(164 180)(165 179)(166 178)(167 177)(168 176)(169 175)(170 174)(171 173)
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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(148,196)(149,195)(150,194)(151,193)(152,192)(153,191)(154,190)(155,189)(156,188)(157,187)(158,186)(159,185)(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173)>;
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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(43,105)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(148,196)(149,195)(150,194)(151,193)(152,192)(153,191)(154,190)(155,189)(156,188)(157,187)(158,186)(159,185)(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173) );
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)], [(1,147),(2,146),(3,145),(4,144),(5,143),(6,142),(7,141),(8,140),(9,139),(10,138),(11,137),(12,136),(13,135),(14,134),(15,133),(16,132),(17,131),(18,130),(19,129),(20,128),(21,127),(22,126),(23,125),(24,124),(25,123),(26,122),(27,121),(28,120),(29,119),(30,118),(31,117),(32,116),(33,115),(34,114),(35,113),(36,112),(37,111),(38,110),(39,109),(40,108),(41,107),(42,106),(43,105),(44,104),(45,103),(46,102),(47,101),(48,100),(49,99),(50,98),(51,97),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,88),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(148,196),(149,195),(150,194),(151,193),(152,192),(153,191),(154,190),(155,189),(156,188),(157,187),(158,186),(159,185),(160,184),(161,183),(162,182),(163,181),(164,180),(165,179),(166,178),(167,177),(168,176),(169,175),(170,174),(171,173)]])
101 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | 7B | 7C | 14A | 14B | 14C | 28A | ··· | 28F | 49A | ··· | 49U | 98A | ··· | 98U | 196A | ··· | 196AP |
| order | 1 | 2 | 2 | 2 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 28 | ··· | 28 | 49 | ··· | 49 | 98 | ··· | 98 | 196 | ··· | 196 |
| size | 1 | 1 | 98 | 98 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
101 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D7 | D14 | D28 | D49 | D98 | D196 |
| kernel | D196 | C196 | D98 | C49 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 3 | 3 | 6 | 21 | 21 | 42 |
Matrix representation of D196 ►in GL2(𝔽197) generated by
| 178 | 174 |
| 23 | 90 |
| 105 | 131 |
| 173 | 92 |
G:=sub<GL(2,GF(197))| [178,23,174,90],[105,173,131,92] >;
D196 in GAP, Magma, Sage, TeX
D_{196} % in TeX
G:=Group("D196"); // GroupNames label
G:=SmallGroup(392,5);
// by ID
G=gap.SmallGroup(392,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,61,26,2083,858,8404]);
// Polycyclic
G:=Group<a,b|a^196=b^2=1,b*a*b=a^-1>;
// generators/relations
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