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## G = Dic57order 228 = 22·3·19

### Dicyclic group

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

 Derived series C1 — C57 — Dic57
 Chief series C1 — C19 — C57 — C114 — Dic57
 Lower central C57 — Dic57
 Upper central C1 — C2

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

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

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

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

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

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 4A 4B 6 19A ··· 19I 38A ··· 38I 57A ··· 57R 114A ··· 114R order 1 2 3 4 4 6 19 ··· 19 38 ··· 38 57 ··· 57 114 ··· 114 size 1 1 2 57 57 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + - + - + - image C1 C2 C4 S3 Dic3 D19 Dic19 D57 Dic57 kernel Dic57 C114 C57 C38 C19 C6 C3 C2 C1 # reps 1 1 2 1 1 9 9 18 18

Matrix representation of Dic57 in GL4(𝔽229) generated by

 170 1 0 0 228 0 0 0 0 0 192 27 0 0 87 215
,
 117 104 0 0 137 112 0 0 0 0 190 137 0 0 136 39
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

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