Copied to
clipboard

G = Dic65order 260 = 22·5·13

Dicyclic group

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

Aliases: Dic65, C657C4, C26.D5, C2.D65, C10.D13, C52Dic13, C132Dic5, C130.1C2, SmallGroup(260,3)

Series: Derived Chief Lower central Upper central

C1C65 — Dic65
C1C13C65C130 — Dic65
C65 — Dic65
C1C2

Generators and relations for Dic65
 G = < a,b | a130=1, b2=a65, bab-1=a-1 >

65C4
13Dic5
5Dic13

Smallest permutation representation of Dic65
Regular action on 260 points
Generators in S260
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)
(1 176 66 241)(2 175 67 240)(3 174 68 239)(4 173 69 238)(5 172 70 237)(6 171 71 236)(7 170 72 235)(8 169 73 234)(9 168 74 233)(10 167 75 232)(11 166 76 231)(12 165 77 230)(13 164 78 229)(14 163 79 228)(15 162 80 227)(16 161 81 226)(17 160 82 225)(18 159 83 224)(19 158 84 223)(20 157 85 222)(21 156 86 221)(22 155 87 220)(23 154 88 219)(24 153 89 218)(25 152 90 217)(26 151 91 216)(27 150 92 215)(28 149 93 214)(29 148 94 213)(30 147 95 212)(31 146 96 211)(32 145 97 210)(33 144 98 209)(34 143 99 208)(35 142 100 207)(36 141 101 206)(37 140 102 205)(38 139 103 204)(39 138 104 203)(40 137 105 202)(41 136 106 201)(42 135 107 200)(43 134 108 199)(44 133 109 198)(45 132 110 197)(46 131 111 196)(47 260 112 195)(48 259 113 194)(49 258 114 193)(50 257 115 192)(51 256 116 191)(52 255 117 190)(53 254 118 189)(54 253 119 188)(55 252 120 187)(56 251 121 186)(57 250 122 185)(58 249 123 184)(59 248 124 183)(60 247 125 182)(61 246 126 181)(62 245 127 180)(63 244 128 179)(64 243 129 178)(65 242 130 177)

G:=sub<Sym(260)| (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260), (1,176,66,241)(2,175,67,240)(3,174,68,239)(4,173,69,238)(5,172,70,237)(6,171,71,236)(7,170,72,235)(8,169,73,234)(9,168,74,233)(10,167,75,232)(11,166,76,231)(12,165,77,230)(13,164,78,229)(14,163,79,228)(15,162,80,227)(16,161,81,226)(17,160,82,225)(18,159,83,224)(19,158,84,223)(20,157,85,222)(21,156,86,221)(22,155,87,220)(23,154,88,219)(24,153,89,218)(25,152,90,217)(26,151,91,216)(27,150,92,215)(28,149,93,214)(29,148,94,213)(30,147,95,212)(31,146,96,211)(32,145,97,210)(33,144,98,209)(34,143,99,208)(35,142,100,207)(36,141,101,206)(37,140,102,205)(38,139,103,204)(39,138,104,203)(40,137,105,202)(41,136,106,201)(42,135,107,200)(43,134,108,199)(44,133,109,198)(45,132,110,197)(46,131,111,196)(47,260,112,195)(48,259,113,194)(49,258,114,193)(50,257,115,192)(51,256,116,191)(52,255,117,190)(53,254,118,189)(54,253,119,188)(55,252,120,187)(56,251,121,186)(57,250,122,185)(58,249,123,184)(59,248,124,183)(60,247,125,182)(61,246,126,181)(62,245,127,180)(63,244,128,179)(64,243,129,178)(65,242,130,177)>;

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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260), (1,176,66,241)(2,175,67,240)(3,174,68,239)(4,173,69,238)(5,172,70,237)(6,171,71,236)(7,170,72,235)(8,169,73,234)(9,168,74,233)(10,167,75,232)(11,166,76,231)(12,165,77,230)(13,164,78,229)(14,163,79,228)(15,162,80,227)(16,161,81,226)(17,160,82,225)(18,159,83,224)(19,158,84,223)(20,157,85,222)(21,156,86,221)(22,155,87,220)(23,154,88,219)(24,153,89,218)(25,152,90,217)(26,151,91,216)(27,150,92,215)(28,149,93,214)(29,148,94,213)(30,147,95,212)(31,146,96,211)(32,145,97,210)(33,144,98,209)(34,143,99,208)(35,142,100,207)(36,141,101,206)(37,140,102,205)(38,139,103,204)(39,138,104,203)(40,137,105,202)(41,136,106,201)(42,135,107,200)(43,134,108,199)(44,133,109,198)(45,132,110,197)(46,131,111,196)(47,260,112,195)(48,259,113,194)(49,258,114,193)(50,257,115,192)(51,256,116,191)(52,255,117,190)(53,254,118,189)(54,253,119,188)(55,252,120,187)(56,251,121,186)(57,250,122,185)(58,249,123,184)(59,248,124,183)(60,247,125,182)(61,246,126,181)(62,245,127,180)(63,244,128,179)(64,243,129,178)(65,242,130,177) );

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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)], [(1,176,66,241),(2,175,67,240),(3,174,68,239),(4,173,69,238),(5,172,70,237),(6,171,71,236),(7,170,72,235),(8,169,73,234),(9,168,74,233),(10,167,75,232),(11,166,76,231),(12,165,77,230),(13,164,78,229),(14,163,79,228),(15,162,80,227),(16,161,81,226),(17,160,82,225),(18,159,83,224),(19,158,84,223),(20,157,85,222),(21,156,86,221),(22,155,87,220),(23,154,88,219),(24,153,89,218),(25,152,90,217),(26,151,91,216),(27,150,92,215),(28,149,93,214),(29,148,94,213),(30,147,95,212),(31,146,96,211),(32,145,97,210),(33,144,98,209),(34,143,99,208),(35,142,100,207),(36,141,101,206),(37,140,102,205),(38,139,103,204),(39,138,104,203),(40,137,105,202),(41,136,106,201),(42,135,107,200),(43,134,108,199),(44,133,109,198),(45,132,110,197),(46,131,111,196),(47,260,112,195),(48,259,113,194),(49,258,114,193),(50,257,115,192),(51,256,116,191),(52,255,117,190),(53,254,118,189),(54,253,119,188),(55,252,120,187),(56,251,121,186),(57,250,122,185),(58,249,123,184),(59,248,124,183),(60,247,125,182),(61,246,126,181),(62,245,127,180),(63,244,128,179),(64,243,129,178),(65,242,130,177)])

68 conjugacy classes

class 1  2 4A4B5A5B10A10B13A···13F26A···26F65A···65X130A···130X
order124455101013···1326···2665···65130···130
size11656522222···22···22···22···2

68 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4D5Dic5D13Dic13D65Dic65
kernelDic65C130C65C26C13C10C5C2C1
# reps11222662424

Matrix representation of Dic65 in GL3(𝔽521) generated by

52000
040646
0475109
,
28600
012192
0409509
G:=sub<GL(3,GF(521))| [520,0,0,0,406,475,0,46,109],[286,0,0,0,12,409,0,192,509] >;

Dic65 in GAP, Magma, Sage, TeX

{\rm Dic}_{65}
% in TeX

G:=Group("Dic65");
// GroupNames label

G:=SmallGroup(260,3);
// by ID

G=gap.SmallGroup(260,3);
# by ID

G:=PCGroup([4,-2,-2,-5,-13,8,194,3843]);
// Polycyclic

G:=Group<a,b|a^130=1,b^2=a^65,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic65 in TeX

׿
×
𝔽