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## G = C10×Dic7order 280 = 23·5·7

### Direct product of C10 and Dic7

Aliases: C10×Dic7, C14⋊C20, C704C4, C10.16D14, C70.16C22, C72(C2×C20), C3511(C2×C4), (C2×C14).C10, C22.(C5×D7), (C2×C70).2C2, (C2×C10).2D7, C2.2(C10×D7), C14.4(C2×C10), SmallGroup(280,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C10×Dic7
 Chief series C1 — C7 — C14 — C70 — C5×Dic7 — C10×Dic7
 Lower central C7 — C10×Dic7
 Upper central C1 — C2×C10

Generators and relations for C10×Dic7
G = < a,b,c | a10=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C10×Dic7
Regular action on 280 points
Generators in S280
(1 111 69 94 47 84 42 131 15 126)(2 112 70 95 48 71 29 132 16 113)(3 99 57 96 49 72 30 133 17 114)(4 100 58 97 50 73 31 134 18 115)(5 101 59 98 51 74 32 135 19 116)(6 102 60 85 52 75 33 136 20 117)(7 103 61 86 53 76 34 137 21 118)(8 104 62 87 54 77 35 138 22 119)(9 105 63 88 55 78 36 139 23 120)(10 106 64 89 56 79 37 140 24 121)(11 107 65 90 43 80 38 127 25 122)(12 108 66 91 44 81 39 128 26 123)(13 109 67 92 45 82 40 129 27 124)(14 110 68 93 46 83 41 130 28 125)(141 246 204 225 190 218 169 267 155 253)(142 247 205 226 191 219 170 268 156 254)(143 248 206 227 192 220 171 269 157 255)(144 249 207 228 193 221 172 270 158 256)(145 250 208 229 194 222 173 271 159 257)(146 251 209 230 195 223 174 272 160 258)(147 252 210 231 196 224 175 273 161 259)(148 239 197 232 183 211 176 274 162 260)(149 240 198 233 184 212 177 275 163 261)(150 241 199 234 185 213 178 276 164 262)(151 242 200 235 186 214 179 277 165 263)(152 243 201 236 187 215 180 278 166 264)(153 244 202 237 188 216 181 279 167 265)(154 245 203 238 189 217 182 280 168 266)
(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)
(1 148 8 141)(2 147 9 154)(3 146 10 153)(4 145 11 152)(5 144 12 151)(6 143 13 150)(7 142 14 149)(15 162 22 155)(16 161 23 168)(17 160 24 167)(18 159 25 166)(19 158 26 165)(20 157 27 164)(21 156 28 163)(29 175 36 182)(30 174 37 181)(31 173 38 180)(32 172 39 179)(33 171 40 178)(34 170 41 177)(35 169 42 176)(43 187 50 194)(44 186 51 193)(45 185 52 192)(46 184 53 191)(47 183 54 190)(48 196 55 189)(49 195 56 188)(57 209 64 202)(58 208 65 201)(59 207 66 200)(60 206 67 199)(61 205 68 198)(62 204 69 197)(63 203 70 210)(71 224 78 217)(72 223 79 216)(73 222 80 215)(74 221 81 214)(75 220 82 213)(76 219 83 212)(77 218 84 211)(85 227 92 234)(86 226 93 233)(87 225 94 232)(88 238 95 231)(89 237 96 230)(90 236 97 229)(91 235 98 228)(99 251 106 244)(100 250 107 243)(101 249 108 242)(102 248 109 241)(103 247 110 240)(104 246 111 239)(105 245 112 252)(113 259 120 266)(114 258 121 265)(115 257 122 264)(116 256 123 263)(117 255 124 262)(118 254 125 261)(119 253 126 260)(127 278 134 271)(128 277 135 270)(129 276 136 269)(130 275 137 268)(131 274 138 267)(132 273 139 280)(133 272 140 279)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 7A 7B 7C 10A ··· 10L 14A ··· 14I 20A ··· 20P 35A ··· 35L 70A ··· 70AJ order 1 2 2 2 4 4 4 4 5 5 5 5 7 7 7 10 ··· 10 14 ··· 14 20 ··· 20 35 ··· 35 70 ··· 70 size 1 1 1 1 7 7 7 7 1 1 1 1 2 2 2 1 ··· 1 2 ··· 2 7 ··· 7 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 D7 Dic7 D14 C5×D7 C5×Dic7 C10×D7 kernel C10×Dic7 C5×Dic7 C2×C70 C70 C2×Dic7 Dic7 C2×C14 C14 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 3 6 3 12 24 12

Matrix representation of C10×Dic7 in GL3(𝔽281) generated by

 280 0 0 0 232 0 0 0 232
,
 1 0 0 0 1 280 0 49 233
,
 1 0 0 0 43 219 0 193 238
G:=sub<GL(3,GF(281))| [280,0,0,0,232,0,0,0,232],[1,0,0,0,1,49,0,280,233],[1,0,0,0,43,193,0,219,238] >;

C10×Dic7 in GAP, Magma, Sage, TeX

C_{10}\times {\rm Dic}_7
% in TeX

G:=Group("C10xDic7");
// GroupNames label

G:=SmallGroup(280,17);
// by ID

G=gap.SmallGroup(280,17);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-7,100,6004]);
// Polycyclic

G:=Group<a,b,c|a^10=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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