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## G = Dic60order 240 = 24·3·5

### Dicyclic group

Aliases: Dic60, C8.D15, C154Q16, C40.1S3, C2.5D60, C6.3D20, C24.1D5, C51Dic12, C31Dic20, C120.1C2, C10.3D12, C4.10D30, C30.21D4, C20.45D6, C12.45D10, C60.52C22, Dic30.1C2, SmallGroup(240,69)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic60
 Chief series C1 — C5 — C15 — C30 — C60 — Dic30 — Dic60
 Lower central C15 — C30 — C60 — Dic60
 Upper central C1 — C2 — C4 — C8

Generators and relations for Dic60
G = < a,b | a120=1, b2=a60, bab-1=a-1 >

Smallest permutation representation of Dic60
Regular action on 240 points
Generators in S240
(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)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 188 61 128)(2 187 62 127)(3 186 63 126)(4 185 64 125)(5 184 65 124)(6 183 66 123)(7 182 67 122)(8 181 68 121)(9 180 69 240)(10 179 70 239)(11 178 71 238)(12 177 72 237)(13 176 73 236)(14 175 74 235)(15 174 75 234)(16 173 76 233)(17 172 77 232)(18 171 78 231)(19 170 79 230)(20 169 80 229)(21 168 81 228)(22 167 82 227)(23 166 83 226)(24 165 84 225)(25 164 85 224)(26 163 86 223)(27 162 87 222)(28 161 88 221)(29 160 89 220)(30 159 90 219)(31 158 91 218)(32 157 92 217)(33 156 93 216)(34 155 94 215)(35 154 95 214)(36 153 96 213)(37 152 97 212)(38 151 98 211)(39 150 99 210)(40 149 100 209)(41 148 101 208)(42 147 102 207)(43 146 103 206)(44 145 104 205)(45 144 105 204)(46 143 106 203)(47 142 107 202)(48 141 108 201)(49 140 109 200)(50 139 110 199)(51 138 111 198)(52 137 112 197)(53 136 113 196)(54 135 114 195)(55 134 115 194)(56 133 116 193)(57 132 117 192)(58 131 118 191)(59 130 119 190)(60 129 120 189)

G:=sub<Sym(240)| (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,188,61,128)(2,187,62,127)(3,186,63,126)(4,185,64,125)(5,184,65,124)(6,183,66,123)(7,182,67,122)(8,181,68,121)(9,180,69,240)(10,179,70,239)(11,178,71,238)(12,177,72,237)(13,176,73,236)(14,175,74,235)(15,174,75,234)(16,173,76,233)(17,172,77,232)(18,171,78,231)(19,170,79,230)(20,169,80,229)(21,168,81,228)(22,167,82,227)(23,166,83,226)(24,165,84,225)(25,164,85,224)(26,163,86,223)(27,162,87,222)(28,161,88,221)(29,160,89,220)(30,159,90,219)(31,158,91,218)(32,157,92,217)(33,156,93,216)(34,155,94,215)(35,154,95,214)(36,153,96,213)(37,152,97,212)(38,151,98,211)(39,150,99,210)(40,149,100,209)(41,148,101,208)(42,147,102,207)(43,146,103,206)(44,145,104,205)(45,144,105,204)(46,143,106,203)(47,142,107,202)(48,141,108,201)(49,140,109,200)(50,139,110,199)(51,138,111,198)(52,137,112,197)(53,136,113,196)(54,135,114,195)(55,134,115,194)(56,133,116,193)(57,132,117,192)(58,131,118,191)(59,130,119,190)(60,129,120,189)>;

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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,188,61,128)(2,187,62,127)(3,186,63,126)(4,185,64,125)(5,184,65,124)(6,183,66,123)(7,182,67,122)(8,181,68,121)(9,180,69,240)(10,179,70,239)(11,178,71,238)(12,177,72,237)(13,176,73,236)(14,175,74,235)(15,174,75,234)(16,173,76,233)(17,172,77,232)(18,171,78,231)(19,170,79,230)(20,169,80,229)(21,168,81,228)(22,167,82,227)(23,166,83,226)(24,165,84,225)(25,164,85,224)(26,163,86,223)(27,162,87,222)(28,161,88,221)(29,160,89,220)(30,159,90,219)(31,158,91,218)(32,157,92,217)(33,156,93,216)(34,155,94,215)(35,154,95,214)(36,153,96,213)(37,152,97,212)(38,151,98,211)(39,150,99,210)(40,149,100,209)(41,148,101,208)(42,147,102,207)(43,146,103,206)(44,145,104,205)(45,144,105,204)(46,143,106,203)(47,142,107,202)(48,141,108,201)(49,140,109,200)(50,139,110,199)(51,138,111,198)(52,137,112,197)(53,136,113,196)(54,135,114,195)(55,134,115,194)(56,133,116,193)(57,132,117,192)(58,131,118,191)(59,130,119,190)(60,129,120,189) );

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),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,188,61,128),(2,187,62,127),(3,186,63,126),(4,185,64,125),(5,184,65,124),(6,183,66,123),(7,182,67,122),(8,181,68,121),(9,180,69,240),(10,179,70,239),(11,178,71,238),(12,177,72,237),(13,176,73,236),(14,175,74,235),(15,174,75,234),(16,173,76,233),(17,172,77,232),(18,171,78,231),(19,170,79,230),(20,169,80,229),(21,168,81,228),(22,167,82,227),(23,166,83,226),(24,165,84,225),(25,164,85,224),(26,163,86,223),(27,162,87,222),(28,161,88,221),(29,160,89,220),(30,159,90,219),(31,158,91,218),(32,157,92,217),(33,156,93,216),(34,155,94,215),(35,154,95,214),(36,153,96,213),(37,152,97,212),(38,151,98,211),(39,150,99,210),(40,149,100,209),(41,148,101,208),(42,147,102,207),(43,146,103,206),(44,145,104,205),(45,144,105,204),(46,143,106,203),(47,142,107,202),(48,141,108,201),(49,140,109,200),(50,139,110,199),(51,138,111,198),(52,137,112,197),(53,136,113,196),(54,135,114,195),(55,134,115,194),(56,133,116,193),(57,132,117,192),(58,131,118,191),(59,130,119,190),(60,129,120,189)]])

Dic60 is a maximal subgroup of
D40.S3  D24.D5  C3⋊Dic40  C5⋊Dic24  C48⋊D5  Dic120  D8.D15  C157Q32  D5×Dic12  Dic60⋊C2  S3×Dic20  D247D5  D407S3  C40.2D6  C40.69D6  C8.D30  D83D15  SD16⋊D15  Q16×D15
Dic60 is a maximal quotient of
Dic308C4  C1209C4

63 conjugacy classes

 class 1 2 3 4A 4B 4C 5A 5B 6 8A 8B 10A 10B 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 60A ··· 60H 120A ··· 120P order 1 2 3 4 4 4 5 5 6 8 8 10 10 12 12 15 15 15 15 20 20 20 20 24 24 24 24 30 30 30 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 2 2 60 60 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

63 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + + - + - + - image C1 C2 C2 S3 D4 D5 D6 Q16 D10 D12 D15 D20 Dic12 D30 Dic20 D60 Dic60 kernel Dic60 C120 Dic30 C40 C30 C24 C20 C15 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 2 1 1 2 1 2 2 2 4 4 4 4 8 8 16

Matrix representation of Dic60 in GL2(𝔽241) generated by

 187 202 39 46
,
 86 52 159 155
G:=sub<GL(2,GF(241))| [187,39,202,46],[86,159,52,155] >;

Dic60 in GAP, Magma, Sage, TeX

{\rm Dic}_{60}
% in TeX

G:=Group("Dic60");
// GroupNames label

G:=SmallGroup(240,69);
// by ID

G=gap.SmallGroup(240,69);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,79,218,50,964,6917]);
// Polycyclic

G:=Group<a,b|a^120=1,b^2=a^60,b*a*b^-1=a^-1>;
// generators/relations

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