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G = Dic52order 208 = 24·13

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic52, C8.D13, C131Q16, C2.5D52, C26.3D4, C104.1C2, C4.10D26, C52.10C22, Dic26.1C2, SmallGroup(208,8)

Series: Derived Chief Lower central Upper central

C1C52 — Dic52
C1C13C26C52Dic26 — Dic52
C13C26C52 — Dic52
C1C2C4C8

Generators and relations for Dic52
 G = < a,b | a104=1, b2=a52, bab-1=a-1 >

26C4
26C4
13Q8
13Q8
2Dic13
2Dic13
13Q16

Smallest permutation representation of Dic52
Regular action on 208 points
Generators in S208
(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)
(1 203 53 151)(2 202 54 150)(3 201 55 149)(4 200 56 148)(5 199 57 147)(6 198 58 146)(7 197 59 145)(8 196 60 144)(9 195 61 143)(10 194 62 142)(11 193 63 141)(12 192 64 140)(13 191 65 139)(14 190 66 138)(15 189 67 137)(16 188 68 136)(17 187 69 135)(18 186 70 134)(19 185 71 133)(20 184 72 132)(21 183 73 131)(22 182 74 130)(23 181 75 129)(24 180 76 128)(25 179 77 127)(26 178 78 126)(27 177 79 125)(28 176 80 124)(29 175 81 123)(30 174 82 122)(31 173 83 121)(32 172 84 120)(33 171 85 119)(34 170 86 118)(35 169 87 117)(36 168 88 116)(37 167 89 115)(38 166 90 114)(39 165 91 113)(40 164 92 112)(41 163 93 111)(42 162 94 110)(43 161 95 109)(44 160 96 108)(45 159 97 107)(46 158 98 106)(47 157 99 105)(48 156 100 208)(49 155 101 207)(50 154 102 206)(51 153 103 205)(52 152 104 204)

G:=sub<Sym(208)| (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), (1,203,53,151)(2,202,54,150)(3,201,55,149)(4,200,56,148)(5,199,57,147)(6,198,58,146)(7,197,59,145)(8,196,60,144)(9,195,61,143)(10,194,62,142)(11,193,63,141)(12,192,64,140)(13,191,65,139)(14,190,66,138)(15,189,67,137)(16,188,68,136)(17,187,69,135)(18,186,70,134)(19,185,71,133)(20,184,72,132)(21,183,73,131)(22,182,74,130)(23,181,75,129)(24,180,76,128)(25,179,77,127)(26,178,78,126)(27,177,79,125)(28,176,80,124)(29,175,81,123)(30,174,82,122)(31,173,83,121)(32,172,84,120)(33,171,85,119)(34,170,86,118)(35,169,87,117)(36,168,88,116)(37,167,89,115)(38,166,90,114)(39,165,91,113)(40,164,92,112)(41,163,93,111)(42,162,94,110)(43,161,95,109)(44,160,96,108)(45,159,97,107)(46,158,98,106)(47,157,99,105)(48,156,100,208)(49,155,101,207)(50,154,102,206)(51,153,103,205)(52,152,104,204)>;

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), (1,203,53,151)(2,202,54,150)(3,201,55,149)(4,200,56,148)(5,199,57,147)(6,198,58,146)(7,197,59,145)(8,196,60,144)(9,195,61,143)(10,194,62,142)(11,193,63,141)(12,192,64,140)(13,191,65,139)(14,190,66,138)(15,189,67,137)(16,188,68,136)(17,187,69,135)(18,186,70,134)(19,185,71,133)(20,184,72,132)(21,183,73,131)(22,182,74,130)(23,181,75,129)(24,180,76,128)(25,179,77,127)(26,178,78,126)(27,177,79,125)(28,176,80,124)(29,175,81,123)(30,174,82,122)(31,173,83,121)(32,172,84,120)(33,171,85,119)(34,170,86,118)(35,169,87,117)(36,168,88,116)(37,167,89,115)(38,166,90,114)(39,165,91,113)(40,164,92,112)(41,163,93,111)(42,162,94,110)(43,161,95,109)(44,160,96,108)(45,159,97,107)(46,158,98,106)(47,157,99,105)(48,156,100,208)(49,155,101,207)(50,154,102,206)(51,153,103,205)(52,152,104,204) );

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)], [(1,203,53,151),(2,202,54,150),(3,201,55,149),(4,200,56,148),(5,199,57,147),(6,198,58,146),(7,197,59,145),(8,196,60,144),(9,195,61,143),(10,194,62,142),(11,193,63,141),(12,192,64,140),(13,191,65,139),(14,190,66,138),(15,189,67,137),(16,188,68,136),(17,187,69,135),(18,186,70,134),(19,185,71,133),(20,184,72,132),(21,183,73,131),(22,182,74,130),(23,181,75,129),(24,180,76,128),(25,179,77,127),(26,178,78,126),(27,177,79,125),(28,176,80,124),(29,175,81,123),(30,174,82,122),(31,173,83,121),(32,172,84,120),(33,171,85,119),(34,170,86,118),(35,169,87,117),(36,168,88,116),(37,167,89,115),(38,166,90,114),(39,165,91,113),(40,164,92,112),(41,163,93,111),(42,162,94,110),(43,161,95,109),(44,160,96,108),(45,159,97,107),(46,158,98,106),(47,157,99,105),(48,156,100,208),(49,155,101,207),(50,154,102,206),(51,153,103,205),(52,152,104,204)])

Dic52 is a maximal subgroup of
C16⋊D13  Dic104  D8.D13  C13⋊Q32  D1047C2  C8.D26  D83D13  D4.D26  Q16×D13
Dic52 is a maximal quotient of
C52.44D4  C1045C4

55 conjugacy classes

class 1  2 4A4B4C8A8B13A···13F26A···26F52A···52L104A···104X
order124448813···1326···2652···52104···104
size1125252222···22···22···22···2

55 irreducible representations

dim111222222
type++++-+++-
imageC1C2C2D4Q16D13D26D52Dic52
kernelDic52C104Dic26C26C13C8C4C2C1
# reps11212661224

Matrix representation of Dic52 in GL2(𝔽313) generated by

99130
223296
,
25644
16857
G:=sub<GL(2,GF(313))| [99,223,130,296],[256,168,44,57] >;

Dic52 in GAP, Magma, Sage, TeX

{\rm Dic}_{52}
% in TeX

G:=Group("Dic52");
// GroupNames label

G:=SmallGroup(208,8);
// by ID

G=gap.SmallGroup(208,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,40,61,66,182,42,4804]);
// Polycyclic

G:=Group<a,b|a^104=1,b^2=a^52,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic52 in TeX

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