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G = Dic51order 204 = 22·3·17

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic51, C513C4, C34.S3, C6.D17, C2.D51, C3⋊Dic17, C172Dic3, C102.1C2, SmallGroup(204,3)

Series: Derived Chief Lower central Upper central

C1C51 — Dic51
C1C17C51C102 — Dic51
C51 — Dic51
C1C2

Generators and relations for Dic51
 G = < a,b | a102=1, b2=a51, bab-1=a-1 >

51C4
17Dic3
3Dic17

Smallest permutation representation of Dic51
Regular action on 204 points
Generators in S204
(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)
(1 133 52 184)(2 132 53 183)(3 131 54 182)(4 130 55 181)(5 129 56 180)(6 128 57 179)(7 127 58 178)(8 126 59 177)(9 125 60 176)(10 124 61 175)(11 123 62 174)(12 122 63 173)(13 121 64 172)(14 120 65 171)(15 119 66 170)(16 118 67 169)(17 117 68 168)(18 116 69 167)(19 115 70 166)(20 114 71 165)(21 113 72 164)(22 112 73 163)(23 111 74 162)(24 110 75 161)(25 109 76 160)(26 108 77 159)(27 107 78 158)(28 106 79 157)(29 105 80 156)(30 104 81 155)(31 103 82 154)(32 204 83 153)(33 203 84 152)(34 202 85 151)(35 201 86 150)(36 200 87 149)(37 199 88 148)(38 198 89 147)(39 197 90 146)(40 196 91 145)(41 195 92 144)(42 194 93 143)(43 193 94 142)(44 192 95 141)(45 191 96 140)(46 190 97 139)(47 189 98 138)(48 188 99 137)(49 187 100 136)(50 186 101 135)(51 185 102 134)

G:=sub<Sym(204)| (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), (1,133,52,184)(2,132,53,183)(3,131,54,182)(4,130,55,181)(5,129,56,180)(6,128,57,179)(7,127,58,178)(8,126,59,177)(9,125,60,176)(10,124,61,175)(11,123,62,174)(12,122,63,173)(13,121,64,172)(14,120,65,171)(15,119,66,170)(16,118,67,169)(17,117,68,168)(18,116,69,167)(19,115,70,166)(20,114,71,165)(21,113,72,164)(22,112,73,163)(23,111,74,162)(24,110,75,161)(25,109,76,160)(26,108,77,159)(27,107,78,158)(28,106,79,157)(29,105,80,156)(30,104,81,155)(31,103,82,154)(32,204,83,153)(33,203,84,152)(34,202,85,151)(35,201,86,150)(36,200,87,149)(37,199,88,148)(38,198,89,147)(39,197,90,146)(40,196,91,145)(41,195,92,144)(42,194,93,143)(43,193,94,142)(44,192,95,141)(45,191,96,140)(46,190,97,139)(47,189,98,138)(48,188,99,137)(49,187,100,136)(50,186,101,135)(51,185,102,134)>;

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), (1,133,52,184)(2,132,53,183)(3,131,54,182)(4,130,55,181)(5,129,56,180)(6,128,57,179)(7,127,58,178)(8,126,59,177)(9,125,60,176)(10,124,61,175)(11,123,62,174)(12,122,63,173)(13,121,64,172)(14,120,65,171)(15,119,66,170)(16,118,67,169)(17,117,68,168)(18,116,69,167)(19,115,70,166)(20,114,71,165)(21,113,72,164)(22,112,73,163)(23,111,74,162)(24,110,75,161)(25,109,76,160)(26,108,77,159)(27,107,78,158)(28,106,79,157)(29,105,80,156)(30,104,81,155)(31,103,82,154)(32,204,83,153)(33,203,84,152)(34,202,85,151)(35,201,86,150)(36,200,87,149)(37,199,88,148)(38,198,89,147)(39,197,90,146)(40,196,91,145)(41,195,92,144)(42,194,93,143)(43,193,94,142)(44,192,95,141)(45,191,96,140)(46,190,97,139)(47,189,98,138)(48,188,99,137)(49,187,100,136)(50,186,101,135)(51,185,102,134) );

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)], [(1,133,52,184),(2,132,53,183),(3,131,54,182),(4,130,55,181),(5,129,56,180),(6,128,57,179),(7,127,58,178),(8,126,59,177),(9,125,60,176),(10,124,61,175),(11,123,62,174),(12,122,63,173),(13,121,64,172),(14,120,65,171),(15,119,66,170),(16,118,67,169),(17,117,68,168),(18,116,69,167),(19,115,70,166),(20,114,71,165),(21,113,72,164),(22,112,73,163),(23,111,74,162),(24,110,75,161),(25,109,76,160),(26,108,77,159),(27,107,78,158),(28,106,79,157),(29,105,80,156),(30,104,81,155),(31,103,82,154),(32,204,83,153),(33,203,84,152),(34,202,85,151),(35,201,86,150),(36,200,87,149),(37,199,88,148),(38,198,89,147),(39,197,90,146),(40,196,91,145),(41,195,92,144),(42,194,93,143),(43,193,94,142),(44,192,95,141),(45,191,96,140),(46,190,97,139),(47,189,98,138),(48,188,99,137),(49,187,100,136),(50,186,101,135),(51,185,102,134)])

Dic51 is a maximal subgroup of   Dic3×D17  S3×Dic17  C51⋊D4  C51⋊Q8  Dic102  C4×D51  C517D4
Dic51 is a maximal quotient of   C515C8

54 conjugacy classes

class 1  2  3 4A4B 6 17A···17H34A···34H51A···51P102A···102P
order12344617···1734···3451···51102···102
size112515122···22···22···22···2

54 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D17Dic17D51Dic51
kernelDic51C102C51C34C17C6C3C2C1
# reps11211881616

Matrix representation of Dic51 in GL3(𝔽409) generated by

40800
0143239
017021
,
26600
0388239
017121
G:=sub<GL(3,GF(409))| [408,0,0,0,143,170,0,239,21],[266,0,0,0,388,171,0,239,21] >;

Dic51 in GAP, Magma, Sage, TeX

{\rm Dic}_{51}
% in TeX

G:=Group("Dic51");
// GroupNames label

G:=SmallGroup(204,3);
// by ID

G=gap.SmallGroup(204,3);
# by ID

G:=PCGroup([4,-2,-2,-3,-17,8,98,3075]);
// Polycyclic

G:=Group<a,b|a^102=1,b^2=a^51,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic51 in TeX

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