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G = Dic59order 236 = 22·59

Dicyclic group

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

Aliases: Dic59, C59⋊C4, C2.D59, C118.C2, SmallGroup(236,1)

Series: Derived Chief Lower central Upper central

C1C59 — Dic59
C1C59C118 — Dic59
C59 — Dic59
C1C2

Generators and relations for Dic59
 G = < a,b | a118=1, b2=a59, bab-1=a-1 >

59C4

Smallest permutation representation of Dic59
Regular action on 236 points
Generators in S236
(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)(119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236)
(1 193 60 134)(2 192 61 133)(3 191 62 132)(4 190 63 131)(5 189 64 130)(6 188 65 129)(7 187 66 128)(8 186 67 127)(9 185 68 126)(10 184 69 125)(11 183 70 124)(12 182 71 123)(13 181 72 122)(14 180 73 121)(15 179 74 120)(16 178 75 119)(17 177 76 236)(18 176 77 235)(19 175 78 234)(20 174 79 233)(21 173 80 232)(22 172 81 231)(23 171 82 230)(24 170 83 229)(25 169 84 228)(26 168 85 227)(27 167 86 226)(28 166 87 225)(29 165 88 224)(30 164 89 223)(31 163 90 222)(32 162 91 221)(33 161 92 220)(34 160 93 219)(35 159 94 218)(36 158 95 217)(37 157 96 216)(38 156 97 215)(39 155 98 214)(40 154 99 213)(41 153 100 212)(42 152 101 211)(43 151 102 210)(44 150 103 209)(45 149 104 208)(46 148 105 207)(47 147 106 206)(48 146 107 205)(49 145 108 204)(50 144 109 203)(51 143 110 202)(52 142 111 201)(53 141 112 200)(54 140 113 199)(55 139 114 198)(56 138 115 197)(57 137 116 196)(58 136 117 195)(59 135 118 194)

G:=sub<Sym(236)| (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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236), (1,193,60,134)(2,192,61,133)(3,191,62,132)(4,190,63,131)(5,189,64,130)(6,188,65,129)(7,187,66,128)(8,186,67,127)(9,185,68,126)(10,184,69,125)(11,183,70,124)(12,182,71,123)(13,181,72,122)(14,180,73,121)(15,179,74,120)(16,178,75,119)(17,177,76,236)(18,176,77,235)(19,175,78,234)(20,174,79,233)(21,173,80,232)(22,172,81,231)(23,171,82,230)(24,170,83,229)(25,169,84,228)(26,168,85,227)(27,167,86,226)(28,166,87,225)(29,165,88,224)(30,164,89,223)(31,163,90,222)(32,162,91,221)(33,161,92,220)(34,160,93,219)(35,159,94,218)(36,158,95,217)(37,157,96,216)(38,156,97,215)(39,155,98,214)(40,154,99,213)(41,153,100,212)(42,152,101,211)(43,151,102,210)(44,150,103,209)(45,149,104,208)(46,148,105,207)(47,147,106,206)(48,146,107,205)(49,145,108,204)(50,144,109,203)(51,143,110,202)(52,142,111,201)(53,141,112,200)(54,140,113,199)(55,139,114,198)(56,138,115,197)(57,137,116,196)(58,136,117,195)(59,135,118,194)>;

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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236), (1,193,60,134)(2,192,61,133)(3,191,62,132)(4,190,63,131)(5,189,64,130)(6,188,65,129)(7,187,66,128)(8,186,67,127)(9,185,68,126)(10,184,69,125)(11,183,70,124)(12,182,71,123)(13,181,72,122)(14,180,73,121)(15,179,74,120)(16,178,75,119)(17,177,76,236)(18,176,77,235)(19,175,78,234)(20,174,79,233)(21,173,80,232)(22,172,81,231)(23,171,82,230)(24,170,83,229)(25,169,84,228)(26,168,85,227)(27,167,86,226)(28,166,87,225)(29,165,88,224)(30,164,89,223)(31,163,90,222)(32,162,91,221)(33,161,92,220)(34,160,93,219)(35,159,94,218)(36,158,95,217)(37,157,96,216)(38,156,97,215)(39,155,98,214)(40,154,99,213)(41,153,100,212)(42,152,101,211)(43,151,102,210)(44,150,103,209)(45,149,104,208)(46,148,105,207)(47,147,106,206)(48,146,107,205)(49,145,108,204)(50,144,109,203)(51,143,110,202)(52,142,111,201)(53,141,112,200)(54,140,113,199)(55,139,114,198)(56,138,115,197)(57,137,116,196)(58,136,117,195)(59,135,118,194) );

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),(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236)], [(1,193,60,134),(2,192,61,133),(3,191,62,132),(4,190,63,131),(5,189,64,130),(6,188,65,129),(7,187,66,128),(8,186,67,127),(9,185,68,126),(10,184,69,125),(11,183,70,124),(12,182,71,123),(13,181,72,122),(14,180,73,121),(15,179,74,120),(16,178,75,119),(17,177,76,236),(18,176,77,235),(19,175,78,234),(20,174,79,233),(21,173,80,232),(22,172,81,231),(23,171,82,230),(24,170,83,229),(25,169,84,228),(26,168,85,227),(27,167,86,226),(28,166,87,225),(29,165,88,224),(30,164,89,223),(31,163,90,222),(32,162,91,221),(33,161,92,220),(34,160,93,219),(35,159,94,218),(36,158,95,217),(37,157,96,216),(38,156,97,215),(39,155,98,214),(40,154,99,213),(41,153,100,212),(42,152,101,211),(43,151,102,210),(44,150,103,209),(45,149,104,208),(46,148,105,207),(47,147,106,206),(48,146,107,205),(49,145,108,204),(50,144,109,203),(51,143,110,202),(52,142,111,201),(53,141,112,200),(54,140,113,199),(55,139,114,198),(56,138,115,197),(57,137,116,196),(58,136,117,195),(59,135,118,194)])

Dic59 is a maximal subgroup of   Dic118  C4×D59  C59⋊D4
Dic59 is a maximal quotient of   C59⋊C8

62 conjugacy classes

class 1  2 4A4B59A···59AC118A···118AC
order124459···59118···118
size1159592···22···2

62 irreducible representations

dim11122
type+++-
imageC1C2C4D59Dic59
kernelDic59C118C59C2C1
# reps1122929

Matrix representation of Dic59 in GL3(𝔽709) generated by

70800
0605708
010
,
9600
0244282
0430465
G:=sub<GL(3,GF(709))| [708,0,0,0,605,1,0,708,0],[96,0,0,0,244,430,0,282,465] >;

Dic59 in GAP, Magma, Sage, TeX

{\rm Dic}_{59}
% in TeX

G:=Group("Dic59");
// GroupNames label

G:=SmallGroup(236,1);
// by ID

G=gap.SmallGroup(236,1);
# by ID

G:=PCGroup([3,-2,-2,-59,6,2090]);
// Polycyclic

G:=Group<a,b|a^118=1,b^2=a^59,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic59 in TeX

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