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G = Dic71order 284 = 22·71

Dicyclic group

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

Aliases: Dic71, C71⋊C4, C2.D71, C142.C2, SmallGroup(284,1)

Series: Derived Chief Lower central Upper central

C1C71 — Dic71
C1C71C142 — Dic71
C71 — Dic71
C1C2

Generators and relations for Dic71
 G = < a,b | a142=1, b2=a71, bab-1=a-1 >

71C4

Smallest permutation representation of Dic71
Regular action on 284 points
Generators in S284
(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)(143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284)
(1 181 72 252)(2 180 73 251)(3 179 74 250)(4 178 75 249)(5 177 76 248)(6 176 77 247)(7 175 78 246)(8 174 79 245)(9 173 80 244)(10 172 81 243)(11 171 82 242)(12 170 83 241)(13 169 84 240)(14 168 85 239)(15 167 86 238)(16 166 87 237)(17 165 88 236)(18 164 89 235)(19 163 90 234)(20 162 91 233)(21 161 92 232)(22 160 93 231)(23 159 94 230)(24 158 95 229)(25 157 96 228)(26 156 97 227)(27 155 98 226)(28 154 99 225)(29 153 100 224)(30 152 101 223)(31 151 102 222)(32 150 103 221)(33 149 104 220)(34 148 105 219)(35 147 106 218)(36 146 107 217)(37 145 108 216)(38 144 109 215)(39 143 110 214)(40 284 111 213)(41 283 112 212)(42 282 113 211)(43 281 114 210)(44 280 115 209)(45 279 116 208)(46 278 117 207)(47 277 118 206)(48 276 119 205)(49 275 120 204)(50 274 121 203)(51 273 122 202)(52 272 123 201)(53 271 124 200)(54 270 125 199)(55 269 126 198)(56 268 127 197)(57 267 128 196)(58 266 129 195)(59 265 130 194)(60 264 131 193)(61 263 132 192)(62 262 133 191)(63 261 134 190)(64 260 135 189)(65 259 136 188)(66 258 137 187)(67 257 138 186)(68 256 139 185)(69 255 140 184)(70 254 141 183)(71 253 142 182)

G:=sub<Sym(284)| (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)(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284), (1,181,72,252)(2,180,73,251)(3,179,74,250)(4,178,75,249)(5,177,76,248)(6,176,77,247)(7,175,78,246)(8,174,79,245)(9,173,80,244)(10,172,81,243)(11,171,82,242)(12,170,83,241)(13,169,84,240)(14,168,85,239)(15,167,86,238)(16,166,87,237)(17,165,88,236)(18,164,89,235)(19,163,90,234)(20,162,91,233)(21,161,92,232)(22,160,93,231)(23,159,94,230)(24,158,95,229)(25,157,96,228)(26,156,97,227)(27,155,98,226)(28,154,99,225)(29,153,100,224)(30,152,101,223)(31,151,102,222)(32,150,103,221)(33,149,104,220)(34,148,105,219)(35,147,106,218)(36,146,107,217)(37,145,108,216)(38,144,109,215)(39,143,110,214)(40,284,111,213)(41,283,112,212)(42,282,113,211)(43,281,114,210)(44,280,115,209)(45,279,116,208)(46,278,117,207)(47,277,118,206)(48,276,119,205)(49,275,120,204)(50,274,121,203)(51,273,122,202)(52,272,123,201)(53,271,124,200)(54,270,125,199)(55,269,126,198)(56,268,127,197)(57,267,128,196)(58,266,129,195)(59,265,130,194)(60,264,131,193)(61,263,132,192)(62,262,133,191)(63,261,134,190)(64,260,135,189)(65,259,136,188)(66,258,137,187)(67,257,138,186)(68,256,139,185)(69,255,140,184)(70,254,141,183)(71,253,142,182)>;

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)(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284), (1,181,72,252)(2,180,73,251)(3,179,74,250)(4,178,75,249)(5,177,76,248)(6,176,77,247)(7,175,78,246)(8,174,79,245)(9,173,80,244)(10,172,81,243)(11,171,82,242)(12,170,83,241)(13,169,84,240)(14,168,85,239)(15,167,86,238)(16,166,87,237)(17,165,88,236)(18,164,89,235)(19,163,90,234)(20,162,91,233)(21,161,92,232)(22,160,93,231)(23,159,94,230)(24,158,95,229)(25,157,96,228)(26,156,97,227)(27,155,98,226)(28,154,99,225)(29,153,100,224)(30,152,101,223)(31,151,102,222)(32,150,103,221)(33,149,104,220)(34,148,105,219)(35,147,106,218)(36,146,107,217)(37,145,108,216)(38,144,109,215)(39,143,110,214)(40,284,111,213)(41,283,112,212)(42,282,113,211)(43,281,114,210)(44,280,115,209)(45,279,116,208)(46,278,117,207)(47,277,118,206)(48,276,119,205)(49,275,120,204)(50,274,121,203)(51,273,122,202)(52,272,123,201)(53,271,124,200)(54,270,125,199)(55,269,126,198)(56,268,127,197)(57,267,128,196)(58,266,129,195)(59,265,130,194)(60,264,131,193)(61,263,132,192)(62,262,133,191)(63,261,134,190)(64,260,135,189)(65,259,136,188)(66,258,137,187)(67,257,138,186)(68,256,139,185)(69,255,140,184)(70,254,141,183)(71,253,142,182) );

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),(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284)], [(1,181,72,252),(2,180,73,251),(3,179,74,250),(4,178,75,249),(5,177,76,248),(6,176,77,247),(7,175,78,246),(8,174,79,245),(9,173,80,244),(10,172,81,243),(11,171,82,242),(12,170,83,241),(13,169,84,240),(14,168,85,239),(15,167,86,238),(16,166,87,237),(17,165,88,236),(18,164,89,235),(19,163,90,234),(20,162,91,233),(21,161,92,232),(22,160,93,231),(23,159,94,230),(24,158,95,229),(25,157,96,228),(26,156,97,227),(27,155,98,226),(28,154,99,225),(29,153,100,224),(30,152,101,223),(31,151,102,222),(32,150,103,221),(33,149,104,220),(34,148,105,219),(35,147,106,218),(36,146,107,217),(37,145,108,216),(38,144,109,215),(39,143,110,214),(40,284,111,213),(41,283,112,212),(42,282,113,211),(43,281,114,210),(44,280,115,209),(45,279,116,208),(46,278,117,207),(47,277,118,206),(48,276,119,205),(49,275,120,204),(50,274,121,203),(51,273,122,202),(52,272,123,201),(53,271,124,200),(54,270,125,199),(55,269,126,198),(56,268,127,197),(57,267,128,196),(58,266,129,195),(59,265,130,194),(60,264,131,193),(61,263,132,192),(62,262,133,191),(63,261,134,190),(64,260,135,189),(65,259,136,188),(66,258,137,187),(67,257,138,186),(68,256,139,185),(69,255,140,184),(70,254,141,183),(71,253,142,182)])

74 conjugacy classes

class 1  2 4A4B71A···71AI142A···142AI
order124471···71142···142
size1171712···22···2

74 irreducible representations

dim11122
type+++-
imageC1C2C4D71Dic71
kernelDic71C142C71C2C1
# reps1123535

Matrix representation of Dic71 in GL2(𝔽569) generated by

5041
5680
,
354493
174215
G:=sub<GL(2,GF(569))| [504,568,1,0],[354,174,493,215] >;

Dic71 in GAP, Magma, Sage, TeX

{\rm Dic}_{71}
% in TeX

G:=Group("Dic71");
// GroupNames label

G:=SmallGroup(284,1);
// by ID

G=gap.SmallGroup(284,1);
# by ID

G:=PCGroup([3,-2,-2,-71,6,2522]);
// Polycyclic

G:=Group<a,b|a^142=1,b^2=a^71,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic71 in TeX

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