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G = Dic70order 280 = 23·5·7

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

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

C1C70 — Dic70
C1C7C35C70Dic35 — Dic70
C35C70 — Dic70
C1C2C4

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

35C4
35C4
35Q8
7Dic5
7Dic5
5Dic7
5Dic7
7Dic10
5Dic14

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

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

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

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

73 conjugacy classes

class 1  2 4A4B4C5A5B7A7B7C10A10B14A14B14C20A20B20C20D28A···28F35A···35L70A···70L140A···140X
order124445577710101414142020202028···2835···3570···70140···140
size1127070222222222222222···22···22···22···2

73 irreducible representations

dim1112222222222
type+++-++++--++-
imageC1C2C2Q8D5D7D10D14Dic10Dic14D35D70Dic70
kernelDic70Dic35C140C35C28C20C14C10C7C5C4C2C1
# reps1211232346121224

Matrix representation of Dic70 in GL4(𝔽281) generated by

38100
280000
00265201
0025213
,
1000
24328000
0023993
0025942
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

Export

Subgroup lattice of Dic70 in TeX

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