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G = Dic70order 280 = 23·5·7

Dicyclic group

Aliases: Dic70, C4.D35, C352Q8, C20.1D7, C28.1D5, C2.3D70, C72Dic10, C52Dic14, C140.1C2, C10.8D14, C14.8D10, C70.8C22, Dic35.1C2, SmallGroup(280,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C70 — Dic70
 Chief series C1 — C7 — C35 — C70 — Dic35 — Dic70
 Lower central C35 — C70 — Dic70
 Upper central C1 — C2 — C4

Generators and relations for Dic70
G = < a,b | a140=1, b2=a70, bab-1=a-1 >

Smallest permutation representation of Dic70
Regular action on 280 points
Generators in S280
(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)
(1 278 71 208)(2 277 72 207)(3 276 73 206)(4 275 74 205)(5 274 75 204)(6 273 76 203)(7 272 77 202)(8 271 78 201)(9 270 79 200)(10 269 80 199)(11 268 81 198)(12 267 82 197)(13 266 83 196)(14 265 84 195)(15 264 85 194)(16 263 86 193)(17 262 87 192)(18 261 88 191)(19 260 89 190)(20 259 90 189)(21 258 91 188)(22 257 92 187)(23 256 93 186)(24 255 94 185)(25 254 95 184)(26 253 96 183)(27 252 97 182)(28 251 98 181)(29 250 99 180)(30 249 100 179)(31 248 101 178)(32 247 102 177)(33 246 103 176)(34 245 104 175)(35 244 105 174)(36 243 106 173)(37 242 107 172)(38 241 108 171)(39 240 109 170)(40 239 110 169)(41 238 111 168)(42 237 112 167)(43 236 113 166)(44 235 114 165)(45 234 115 164)(46 233 116 163)(47 232 117 162)(48 231 118 161)(49 230 119 160)(50 229 120 159)(51 228 121 158)(52 227 122 157)(53 226 123 156)(54 225 124 155)(55 224 125 154)(56 223 126 153)(57 222 127 152)(58 221 128 151)(59 220 129 150)(60 219 130 149)(61 218 131 148)(62 217 132 147)(63 216 133 146)(64 215 134 145)(65 214 135 144)(66 213 136 143)(67 212 137 142)(68 211 138 141)(69 210 139 280)(70 209 140 279)

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

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

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

73 conjugacy classes

 class 1 2 4A 4B 4C 5A 5B 7A 7B 7C 10A 10B 14A 14B 14C 20A 20B 20C 20D 28A ··· 28F 35A ··· 35L 70A ··· 70L 140A ··· 140X order 1 2 4 4 4 5 5 7 7 7 10 10 14 14 14 20 20 20 20 28 ··· 28 35 ··· 35 70 ··· 70 140 ··· 140 size 1 1 2 70 70 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

73 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + - + + + + - - + + - image C1 C2 C2 Q8 D5 D7 D10 D14 Dic10 Dic14 D35 D70 Dic70 kernel Dic70 Dic35 C140 C35 C28 C20 C14 C10 C7 C5 C4 C2 C1 # reps 1 2 1 1 2 3 2 3 4 6 12 12 24

Matrix representation of Dic70 in GL4(𝔽281) generated by

 38 1 0 0 280 0 0 0 0 0 265 201 0 0 252 13
,
 1 0 0 0 243 280 0 0 0 0 239 93 0 0 259 42
G:=sub<GL(4,GF(281))| [38,280,0,0,1,0,0,0,0,0,265,252,0,0,201,13],[1,243,0,0,0,280,0,0,0,0,239,259,0,0,93,42] >;

Dic70 in GAP, Magma, Sage, TeX

{\rm Dic}_{70}
% in TeX

G:=Group("Dic70");
// GroupNames label

G:=SmallGroup(280,24);
// by ID

G=gap.SmallGroup(280,24);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,20,61,26,643,6004]);
// Polycyclic

G:=Group<a,b|a^140=1,b^2=a^70,b*a*b^-1=a^-1>;
// generators/relations

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