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G = Dic69order 276 = 22·3·23

Dicyclic group

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

Aliases: Dic69, C691C4, C46.S3, C6.D23, C2.D69, C3⋊Dic23, C23⋊Dic3, C138.1C2, SmallGroup(276,3)

Series: Derived Chief Lower central Upper central

C1C69 — Dic69
C1C23C69C138 — Dic69
C69 — Dic69
C1C2

Generators and relations for Dic69
 G = < a,b | a138=1, b2=a69, bab-1=a-1 >

69C4
23Dic3
3Dic23

Smallest permutation representation of Dic69
Regular action on 276 points
Generators in S276
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276)
(1 176 70 245)(2 175 71 244)(3 174 72 243)(4 173 73 242)(5 172 74 241)(6 171 75 240)(7 170 76 239)(8 169 77 238)(9 168 78 237)(10 167 79 236)(11 166 80 235)(12 165 81 234)(13 164 82 233)(14 163 83 232)(15 162 84 231)(16 161 85 230)(17 160 86 229)(18 159 87 228)(19 158 88 227)(20 157 89 226)(21 156 90 225)(22 155 91 224)(23 154 92 223)(24 153 93 222)(25 152 94 221)(26 151 95 220)(27 150 96 219)(28 149 97 218)(29 148 98 217)(30 147 99 216)(31 146 100 215)(32 145 101 214)(33 144 102 213)(34 143 103 212)(35 142 104 211)(36 141 105 210)(37 140 106 209)(38 139 107 208)(39 276 108 207)(40 275 109 206)(41 274 110 205)(42 273 111 204)(43 272 112 203)(44 271 113 202)(45 270 114 201)(46 269 115 200)(47 268 116 199)(48 267 117 198)(49 266 118 197)(50 265 119 196)(51 264 120 195)(52 263 121 194)(53 262 122 193)(54 261 123 192)(55 260 124 191)(56 259 125 190)(57 258 126 189)(58 257 127 188)(59 256 128 187)(60 255 129 186)(61 254 130 185)(62 253 131 184)(63 252 132 183)(64 251 133 182)(65 250 134 181)(66 249 135 180)(67 248 136 179)(68 247 137 178)(69 246 138 177)

G:=sub<Sym(276)| (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276), (1,176,70,245)(2,175,71,244)(3,174,72,243)(4,173,73,242)(5,172,74,241)(6,171,75,240)(7,170,76,239)(8,169,77,238)(9,168,78,237)(10,167,79,236)(11,166,80,235)(12,165,81,234)(13,164,82,233)(14,163,83,232)(15,162,84,231)(16,161,85,230)(17,160,86,229)(18,159,87,228)(19,158,88,227)(20,157,89,226)(21,156,90,225)(22,155,91,224)(23,154,92,223)(24,153,93,222)(25,152,94,221)(26,151,95,220)(27,150,96,219)(28,149,97,218)(29,148,98,217)(30,147,99,216)(31,146,100,215)(32,145,101,214)(33,144,102,213)(34,143,103,212)(35,142,104,211)(36,141,105,210)(37,140,106,209)(38,139,107,208)(39,276,108,207)(40,275,109,206)(41,274,110,205)(42,273,111,204)(43,272,112,203)(44,271,113,202)(45,270,114,201)(46,269,115,200)(47,268,116,199)(48,267,117,198)(49,266,118,197)(50,265,119,196)(51,264,120,195)(52,263,121,194)(53,262,122,193)(54,261,123,192)(55,260,124,191)(56,259,125,190)(57,258,126,189)(58,257,127,188)(59,256,128,187)(60,255,129,186)(61,254,130,185)(62,253,131,184)(63,252,132,183)(64,251,133,182)(65,250,134,181)(66,249,135,180)(67,248,136,179)(68,247,137,178)(69,246,138,177)>;

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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276), (1,176,70,245)(2,175,71,244)(3,174,72,243)(4,173,73,242)(5,172,74,241)(6,171,75,240)(7,170,76,239)(8,169,77,238)(9,168,78,237)(10,167,79,236)(11,166,80,235)(12,165,81,234)(13,164,82,233)(14,163,83,232)(15,162,84,231)(16,161,85,230)(17,160,86,229)(18,159,87,228)(19,158,88,227)(20,157,89,226)(21,156,90,225)(22,155,91,224)(23,154,92,223)(24,153,93,222)(25,152,94,221)(26,151,95,220)(27,150,96,219)(28,149,97,218)(29,148,98,217)(30,147,99,216)(31,146,100,215)(32,145,101,214)(33,144,102,213)(34,143,103,212)(35,142,104,211)(36,141,105,210)(37,140,106,209)(38,139,107,208)(39,276,108,207)(40,275,109,206)(41,274,110,205)(42,273,111,204)(43,272,112,203)(44,271,113,202)(45,270,114,201)(46,269,115,200)(47,268,116,199)(48,267,117,198)(49,266,118,197)(50,265,119,196)(51,264,120,195)(52,263,121,194)(53,262,122,193)(54,261,123,192)(55,260,124,191)(56,259,125,190)(57,258,126,189)(58,257,127,188)(59,256,128,187)(60,255,129,186)(61,254,130,185)(62,253,131,184)(63,252,132,183)(64,251,133,182)(65,250,134,181)(66,249,135,180)(67,248,136,179)(68,247,137,178)(69,246,138,177) );

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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276)], [(1,176,70,245),(2,175,71,244),(3,174,72,243),(4,173,73,242),(5,172,74,241),(6,171,75,240),(7,170,76,239),(8,169,77,238),(9,168,78,237),(10,167,79,236),(11,166,80,235),(12,165,81,234),(13,164,82,233),(14,163,83,232),(15,162,84,231),(16,161,85,230),(17,160,86,229),(18,159,87,228),(19,158,88,227),(20,157,89,226),(21,156,90,225),(22,155,91,224),(23,154,92,223),(24,153,93,222),(25,152,94,221),(26,151,95,220),(27,150,96,219),(28,149,97,218),(29,148,98,217),(30,147,99,216),(31,146,100,215),(32,145,101,214),(33,144,102,213),(34,143,103,212),(35,142,104,211),(36,141,105,210),(37,140,106,209),(38,139,107,208),(39,276,108,207),(40,275,109,206),(41,274,110,205),(42,273,111,204),(43,272,112,203),(44,271,113,202),(45,270,114,201),(46,269,115,200),(47,268,116,199),(48,267,117,198),(49,266,118,197),(50,265,119,196),(51,264,120,195),(52,263,121,194),(53,262,122,193),(54,261,123,192),(55,260,124,191),(56,259,125,190),(57,258,126,189),(58,257,127,188),(59,256,128,187),(60,255,129,186),(61,254,130,185),(62,253,131,184),(63,252,132,183),(64,251,133,182),(65,250,134,181),(66,249,135,180),(67,248,136,179),(68,247,137,178),(69,246,138,177)])

72 conjugacy classes

class 1  2  3 4A4B 6 23A···23K46A···46K69A···69V138A···138V
order12344623···2346···4669···69138···138
size112696922···22···22···22···2

72 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D23Dic23D69Dic69
kernelDic69C138C69C46C23C6C3C2C1
# reps1121111112222

Matrix representation of Dic69 in GL3(𝔽277) generated by

27600
01715
026252
,
6000
022283
04755
G:=sub<GL(3,GF(277))| [276,0,0,0,17,262,0,15,52],[60,0,0,0,222,47,0,83,55] >;

Dic69 in GAP, Magma, Sage, TeX

{\rm Dic}_{69}
% in TeX

G:=Group("Dic69");
// GroupNames label

G:=SmallGroup(276,3);
// by ID

G=gap.SmallGroup(276,3);
# by ID

G:=PCGroup([4,-2,-2,-3,-23,8,98,4227]);
// Polycyclic

G:=Group<a,b|a^138=1,b^2=a^69,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic69 in TeX

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