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G = Dic57order 228 = 22·3·19

Dicyclic group

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

Aliases: Dic57, C571C4, C38.S3, C6.D19, C2.D57, C19⋊Dic3, C3⋊Dic19, C114.1C2, SmallGroup(228,5)

Series: Derived Chief Lower central Upper central

C1C57 — Dic57
C1C19C57C114 — Dic57
C57 — Dic57
C1C2

Generators and relations for Dic57
 G = < a,b | a114=1, b2=a57, bab-1=a-1 >

57C4
19Dic3
3Dic19

Smallest permutation representation of Dic57
Regular action on 228 points
Generators in S228
(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)(115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228)
(1 206 58 149)(2 205 59 148)(3 204 60 147)(4 203 61 146)(5 202 62 145)(6 201 63 144)(7 200 64 143)(8 199 65 142)(9 198 66 141)(10 197 67 140)(11 196 68 139)(12 195 69 138)(13 194 70 137)(14 193 71 136)(15 192 72 135)(16 191 73 134)(17 190 74 133)(18 189 75 132)(19 188 76 131)(20 187 77 130)(21 186 78 129)(22 185 79 128)(23 184 80 127)(24 183 81 126)(25 182 82 125)(26 181 83 124)(27 180 84 123)(28 179 85 122)(29 178 86 121)(30 177 87 120)(31 176 88 119)(32 175 89 118)(33 174 90 117)(34 173 91 116)(35 172 92 115)(36 171 93 228)(37 170 94 227)(38 169 95 226)(39 168 96 225)(40 167 97 224)(41 166 98 223)(42 165 99 222)(43 164 100 221)(44 163 101 220)(45 162 102 219)(46 161 103 218)(47 160 104 217)(48 159 105 216)(49 158 106 215)(50 157 107 214)(51 156 108 213)(52 155 109 212)(53 154 110 211)(54 153 111 210)(55 152 112 209)(56 151 113 208)(57 150 114 207)

G:=sub<Sym(228)| (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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228), (1,206,58,149)(2,205,59,148)(3,204,60,147)(4,203,61,146)(5,202,62,145)(6,201,63,144)(7,200,64,143)(8,199,65,142)(9,198,66,141)(10,197,67,140)(11,196,68,139)(12,195,69,138)(13,194,70,137)(14,193,71,136)(15,192,72,135)(16,191,73,134)(17,190,74,133)(18,189,75,132)(19,188,76,131)(20,187,77,130)(21,186,78,129)(22,185,79,128)(23,184,80,127)(24,183,81,126)(25,182,82,125)(26,181,83,124)(27,180,84,123)(28,179,85,122)(29,178,86,121)(30,177,87,120)(31,176,88,119)(32,175,89,118)(33,174,90,117)(34,173,91,116)(35,172,92,115)(36,171,93,228)(37,170,94,227)(38,169,95,226)(39,168,96,225)(40,167,97,224)(41,166,98,223)(42,165,99,222)(43,164,100,221)(44,163,101,220)(45,162,102,219)(46,161,103,218)(47,160,104,217)(48,159,105,216)(49,158,106,215)(50,157,107,214)(51,156,108,213)(52,155,109,212)(53,154,110,211)(54,153,111,210)(55,152,112,209)(56,151,113,208)(57,150,114,207)>;

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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228), (1,206,58,149)(2,205,59,148)(3,204,60,147)(4,203,61,146)(5,202,62,145)(6,201,63,144)(7,200,64,143)(8,199,65,142)(9,198,66,141)(10,197,67,140)(11,196,68,139)(12,195,69,138)(13,194,70,137)(14,193,71,136)(15,192,72,135)(16,191,73,134)(17,190,74,133)(18,189,75,132)(19,188,76,131)(20,187,77,130)(21,186,78,129)(22,185,79,128)(23,184,80,127)(24,183,81,126)(25,182,82,125)(26,181,83,124)(27,180,84,123)(28,179,85,122)(29,178,86,121)(30,177,87,120)(31,176,88,119)(32,175,89,118)(33,174,90,117)(34,173,91,116)(35,172,92,115)(36,171,93,228)(37,170,94,227)(38,169,95,226)(39,168,96,225)(40,167,97,224)(41,166,98,223)(42,165,99,222)(43,164,100,221)(44,163,101,220)(45,162,102,219)(46,161,103,218)(47,160,104,217)(48,159,105,216)(49,158,106,215)(50,157,107,214)(51,156,108,213)(52,155,109,212)(53,154,110,211)(54,153,111,210)(55,152,112,209)(56,151,113,208)(57,150,114,207) );

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),(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)], [(1,206,58,149),(2,205,59,148),(3,204,60,147),(4,203,61,146),(5,202,62,145),(6,201,63,144),(7,200,64,143),(8,199,65,142),(9,198,66,141),(10,197,67,140),(11,196,68,139),(12,195,69,138),(13,194,70,137),(14,193,71,136),(15,192,72,135),(16,191,73,134),(17,190,74,133),(18,189,75,132),(19,188,76,131),(20,187,77,130),(21,186,78,129),(22,185,79,128),(23,184,80,127),(24,183,81,126),(25,182,82,125),(26,181,83,124),(27,180,84,123),(28,179,85,122),(29,178,86,121),(30,177,87,120),(31,176,88,119),(32,175,89,118),(33,174,90,117),(34,173,91,116),(35,172,92,115),(36,171,93,228),(37,170,94,227),(38,169,95,226),(39,168,96,225),(40,167,97,224),(41,166,98,223),(42,165,99,222),(43,164,100,221),(44,163,101,220),(45,162,102,219),(46,161,103,218),(47,160,104,217),(48,159,105,216),(49,158,106,215),(50,157,107,214),(51,156,108,213),(52,155,109,212),(53,154,110,211),(54,153,111,210),(55,152,112,209),(56,151,113,208),(57,150,114,207)])

Dic57 is a maximal subgroup of   Dic3×D19  S3×Dic19  C57⋊D4  C57⋊Q8  Dic114  C4×D57  C577D4
Dic57 is a maximal quotient of   C57⋊C8

60 conjugacy classes

class 1  2  3 4A4B 6 19A···19I38A···38I57A···57R114A···114R
order12344619···1938···3857···57114···114
size112575722···22···22···22···2

60 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D19Dic19D57Dic57
kernelDic57C114C57C38C19C6C3C2C1
# reps11211991818

Matrix representation of Dic57 in GL4(𝔽229) generated by

170100
228000
0019227
0087215
,
11710400
13711200
00190137
0013639
G:=sub<GL(4,GF(229))| [170,228,0,0,1,0,0,0,0,0,192,87,0,0,27,215],[117,137,0,0,104,112,0,0,0,0,190,136,0,0,137,39] >;

Dic57 in GAP, Magma, Sage, TeX

{\rm Dic}_{57}
% in TeX

G:=Group("Dic57");
// GroupNames label

G:=SmallGroup(228,5);
// by ID

G=gap.SmallGroup(228,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-19,8,98,3459]);
// Polycyclic

G:=Group<a,b|a^114=1,b^2=a^57,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic57 in TeX

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