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## G = C14×Dic5order 280 = 23·5·7

### Direct product of C14 and Dic5

Aliases: C14×Dic5, C705C4, C102C28, C14.16D10, C70.21C22, C53(C2×C28), C3512(C2×C4), (C2×C10).C14, C22.(C7×D5), (C2×C70).3C2, (C2×C14).2D5, C2.2(D5×C14), C10.4(C2×C14), SmallGroup(280,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C14×Dic5
 Chief series C1 — C5 — C10 — C70 — C7×Dic5 — C14×Dic5
 Lower central C5 — C14×Dic5
 Upper central C1 — C2×C14

Generators and relations for C14×Dic5
G = < a,b,c | a14=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C14×Dic5
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)
(1 249 137 230 223 80 116 155 254 25)(2 250 138 231 224 81 117 156 255 26)(3 251 139 232 211 82 118 157 256 27)(4 252 140 233 212 83 119 158 257 28)(5 239 127 234 213 84 120 159 258 15)(6 240 128 235 214 71 121 160 259 16)(7 241 129 236 215 72 122 161 260 17)(8 242 130 237 216 73 123 162 261 18)(9 243 131 238 217 74 124 163 262 19)(10 244 132 225 218 75 125 164 263 20)(11 245 133 226 219 76 126 165 264 21)(12 246 134 227 220 77 113 166 265 22)(13 247 135 228 221 78 114 167 266 23)(14 248 136 229 222 79 115 168 253 24)(29 206 106 92 48 59 194 180 279 142)(30 207 107 93 49 60 195 181 280 143)(31 208 108 94 50 61 196 182 267 144)(32 209 109 95 51 62 183 169 268 145)(33 210 110 96 52 63 184 170 269 146)(34 197 111 97 53 64 185 171 270 147)(35 198 112 98 54 65 186 172 271 148)(36 199 99 85 55 66 187 173 272 149)(37 200 100 86 56 67 188 174 273 150)(38 201 101 87 43 68 189 175 274 151)(39 202 102 88 44 69 190 176 275 152)(40 203 103 89 45 70 191 177 276 153)(41 204 104 90 46 57 192 178 277 154)(42 205 105 91 47 58 193 179 278 141)
(1 51 80 145)(2 52 81 146)(3 53 82 147)(4 54 83 148)(5 55 84 149)(6 56 71 150)(7 43 72 151)(8 44 73 152)(9 45 74 153)(10 46 75 154)(11 47 76 141)(12 48 77 142)(13 49 78 143)(14 50 79 144)(15 66 213 36)(16 67 214 37)(17 68 215 38)(18 69 216 39)(19 70 217 40)(20 57 218 41)(21 58 219 42)(22 59 220 29)(23 60 221 30)(24 61 222 31)(25 62 223 32)(26 63 224 33)(27 64 211 34)(28 65 212 35)(85 120 272 239)(86 121 273 240)(87 122 274 241)(88 123 275 242)(89 124 276 243)(90 125 277 244)(91 126 278 245)(92 113 279 246)(93 114 280 247)(94 115 267 248)(95 116 268 249)(96 117 269 250)(97 118 270 251)(98 119 271 252)(99 159 173 127)(100 160 174 128)(101 161 175 129)(102 162 176 130)(103 163 177 131)(104 164 178 132)(105 165 179 133)(106 166 180 134)(107 167 181 135)(108 168 182 136)(109 155 169 137)(110 156 170 138)(111 157 171 139)(112 158 172 140)(183 230 209 254)(184 231 210 255)(185 232 197 256)(186 233 198 257)(187 234 199 258)(188 235 200 259)(189 236 201 260)(190 237 202 261)(191 238 203 262)(192 225 204 263)(193 226 205 264)(194 227 206 265)(195 228 207 266)(196 229 208 253)

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

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

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

112 conjugacy classes

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

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C7 C14 C14 C28 D5 Dic5 D10 C7×D5 C7×Dic5 D5×C14 kernel C14×Dic5 C7×Dic5 C2×C70 C70 C2×Dic5 Dic5 C2×C10 C10 C2×C14 C14 C14 C22 C2 C2 # reps 1 2 1 4 6 12 6 24 2 4 2 12 24 12

Matrix representation of C14×Dic5 in GL3(𝔽281) generated by

 1 0 0 0 172 0 0 0 172
,
 280 0 0 0 0 1 0 280 243
,
 228 0 0 0 124 252 0 36 157
G:=sub<GL(3,GF(281))| [1,0,0,0,172,0,0,0,172],[280,0,0,0,0,280,0,1,243],[228,0,0,0,124,36,0,252,157] >;

C14×Dic5 in GAP, Magma, Sage, TeX

C_{14}\times {\rm Dic}_5
% in TeX

G:=Group("C14xDic5");
// GroupNames label

G:=SmallGroup(280,22);
// by ID

G=gap.SmallGroup(280,22);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-5,140,5604]);
// Polycyclic

G:=Group<a,b,c|a^14=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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