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## G = Dic49order 196 = 22·72

### Dicyclic group

Aliases: Dic49, C49⋊C4, C98.C2, C2.D49, C7.Dic7, C14.1D7, SmallGroup(196,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C49 — Dic49
 Chief series C1 — C7 — C49 — C98 — Dic49
 Lower central C49 — Dic49
 Upper central C1 — C2

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)
(1 127 50 176)(2 126 51 175)(3 125 52 174)(4 124 53 173)(5 123 54 172)(6 122 55 171)(7 121 56 170)(8 120 57 169)(9 119 58 168)(10 118 59 167)(11 117 60 166)(12 116 61 165)(13 115 62 164)(14 114 63 163)(15 113 64 162)(16 112 65 161)(17 111 66 160)(18 110 67 159)(19 109 68 158)(20 108 69 157)(21 107 70 156)(22 106 71 155)(23 105 72 154)(24 104 73 153)(25 103 74 152)(26 102 75 151)(27 101 76 150)(28 100 77 149)(29 99 78 148)(30 196 79 147)(31 195 80 146)(32 194 81 145)(33 193 82 144)(34 192 83 143)(35 191 84 142)(36 190 85 141)(37 189 86 140)(38 188 87 139)(39 187 88 138)(40 186 89 137)(41 185 90 136)(42 184 91 135)(43 183 92 134)(44 182 93 133)(45 181 94 132)(46 180 95 131)(47 179 96 130)(48 178 97 129)(49 177 98 128)

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), (1,127,50,176)(2,126,51,175)(3,125,52,174)(4,124,53,173)(5,123,54,172)(6,122,55,171)(7,121,56,170)(8,120,57,169)(9,119,58,168)(10,118,59,167)(11,117,60,166)(12,116,61,165)(13,115,62,164)(14,114,63,163)(15,113,64,162)(16,112,65,161)(17,111,66,160)(18,110,67,159)(19,109,68,158)(20,108,69,157)(21,107,70,156)(22,106,71,155)(23,105,72,154)(24,104,73,153)(25,103,74,152)(26,102,75,151)(27,101,76,150)(28,100,77,149)(29,99,78,148)(30,196,79,147)(31,195,80,146)(32,194,81,145)(33,193,82,144)(34,192,83,143)(35,191,84,142)(36,190,85,141)(37,189,86,140)(38,188,87,139)(39,187,88,138)(40,186,89,137)(41,185,90,136)(42,184,91,135)(43,183,92,134)(44,182,93,133)(45,181,94,132)(46,180,95,131)(47,179,96,130)(48,178,97,129)(49,177,98,128)>;

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), (1,127,50,176)(2,126,51,175)(3,125,52,174)(4,124,53,173)(5,123,54,172)(6,122,55,171)(7,121,56,170)(8,120,57,169)(9,119,58,168)(10,118,59,167)(11,117,60,166)(12,116,61,165)(13,115,62,164)(14,114,63,163)(15,113,64,162)(16,112,65,161)(17,111,66,160)(18,110,67,159)(19,109,68,158)(20,108,69,157)(21,107,70,156)(22,106,71,155)(23,105,72,154)(24,104,73,153)(25,103,74,152)(26,102,75,151)(27,101,76,150)(28,100,77,149)(29,99,78,148)(30,196,79,147)(31,195,80,146)(32,194,81,145)(33,193,82,144)(34,192,83,143)(35,191,84,142)(36,190,85,141)(37,189,86,140)(38,188,87,139)(39,187,88,138)(40,186,89,137)(41,185,90,136)(42,184,91,135)(43,183,92,134)(44,182,93,133)(45,181,94,132)(46,180,95,131)(47,179,96,130)(48,178,97,129)(49,177,98,128) );

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)], [(1,127,50,176),(2,126,51,175),(3,125,52,174),(4,124,53,173),(5,123,54,172),(6,122,55,171),(7,121,56,170),(8,120,57,169),(9,119,58,168),(10,118,59,167),(11,117,60,166),(12,116,61,165),(13,115,62,164),(14,114,63,163),(15,113,64,162),(16,112,65,161),(17,111,66,160),(18,110,67,159),(19,109,68,158),(20,108,69,157),(21,107,70,156),(22,106,71,155),(23,105,72,154),(24,104,73,153),(25,103,74,152),(26,102,75,151),(27,101,76,150),(28,100,77,149),(29,99,78,148),(30,196,79,147),(31,195,80,146),(32,194,81,145),(33,193,82,144),(34,192,83,143),(35,191,84,142),(36,190,85,141),(37,189,86,140),(38,188,87,139),(39,187,88,138),(40,186,89,137),(41,185,90,136),(42,184,91,135),(43,183,92,134),(44,182,93,133),(45,181,94,132),(46,180,95,131),(47,179,96,130),(48,178,97,129),(49,177,98,128)])

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

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