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## G = C7×Dic10order 280 = 23·5·7

### Direct product of C7 and Dic10

Aliases: C7×Dic10, C354Q8, C28.3D5, C140.4C2, C20.1C14, C14.13D10, C70.18C22, Dic5.1C14, C5⋊(C7×Q8), C4.(C7×D5), C2.3(D5×C14), C10.1(C2×C14), (C7×Dic5).3C2, SmallGroup(280,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C7×Dic10
 Chief series C1 — C5 — C10 — C70 — C7×Dic5 — C7×Dic10
 Lower central C5 — C10 — C7×Dic10
 Upper central C1 — C14 — C28

Generators and relations for C7×Dic10
G = < a,b,c | a7=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×Dic10
Regular action on 280 points
Generators in S280
(1 28 231 42 160 220 124)(2 29 232 43 141 201 125)(3 30 233 44 142 202 126)(4 31 234 45 143 203 127)(5 32 235 46 144 204 128)(6 33 236 47 145 205 129)(7 34 237 48 146 206 130)(8 35 238 49 147 207 131)(9 36 239 50 148 208 132)(10 37 240 51 149 209 133)(11 38 221 52 150 210 134)(12 39 222 53 151 211 135)(13 40 223 54 152 212 136)(14 21 224 55 153 213 137)(15 22 225 56 154 214 138)(16 23 226 57 155 215 139)(17 24 227 58 156 216 140)(18 25 228 59 157 217 121)(19 26 229 60 158 218 122)(20 27 230 41 159 219 123)(61 256 176 109 262 188 90)(62 257 177 110 263 189 91)(63 258 178 111 264 190 92)(64 259 179 112 265 191 93)(65 260 180 113 266 192 94)(66 241 161 114 267 193 95)(67 242 162 115 268 194 96)(68 243 163 116 269 195 97)(69 244 164 117 270 196 98)(70 245 165 118 271 197 99)(71 246 166 119 272 198 100)(72 247 167 120 273 199 81)(73 248 168 101 274 200 82)(74 249 169 102 275 181 83)(75 250 170 103 276 182 84)(76 251 171 104 277 183 85)(77 252 172 105 278 184 86)(78 253 173 106 279 185 87)(79 254 174 107 280 186 88)(80 255 175 108 261 187 89)
(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)
(1 266 11 276)(2 265 12 275)(3 264 13 274)(4 263 14 273)(5 262 15 272)(6 261 16 271)(7 280 17 270)(8 279 18 269)(9 278 19 268)(10 277 20 267)(21 199 31 189)(22 198 32 188)(23 197 33 187)(24 196 34 186)(25 195 35 185)(26 194 36 184)(27 193 37 183)(28 192 38 182)(29 191 39 181)(30 190 40 200)(41 66 51 76)(42 65 52 75)(43 64 53 74)(44 63 54 73)(45 62 55 72)(46 61 56 71)(47 80 57 70)(48 79 58 69)(49 78 59 68)(50 77 60 67)(81 234 91 224)(82 233 92 223)(83 232 93 222)(84 231 94 221)(85 230 95 240)(86 229 96 239)(87 228 97 238)(88 227 98 237)(89 226 99 236)(90 225 100 235)(101 126 111 136)(102 125 112 135)(103 124 113 134)(104 123 114 133)(105 122 115 132)(106 121 116 131)(107 140 117 130)(108 139 118 129)(109 138 119 128)(110 137 120 127)(141 259 151 249)(142 258 152 248)(143 257 153 247)(144 256 154 246)(145 255 155 245)(146 254 156 244)(147 253 157 243)(148 252 158 242)(149 251 159 241)(150 250 160 260)(161 209 171 219)(162 208 172 218)(163 207 173 217)(164 206 174 216)(165 205 175 215)(166 204 176 214)(167 203 177 213)(168 202 178 212)(169 201 179 211)(170 220 180 210)

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

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

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

91 conjugacy classes

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

91 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + + - image C1 C2 C2 C7 C14 C14 Q8 D5 D10 Dic10 C7×Q8 C7×D5 D5×C14 C7×Dic10 kernel C7×Dic10 C7×Dic5 C140 Dic10 Dic5 C20 C35 C28 C14 C7 C5 C4 C2 C1 # reps 1 2 1 6 12 6 1 2 2 4 6 12 12 24

Matrix representation of C7×Dic10 in GL2(𝔽281) generated by

 165 0 0 165
,
 231 84 214 17
,
 126 142 187 155
G:=sub<GL(2,GF(281))| [165,0,0,165],[231,214,84,17],[126,187,142,155] >;

C7×Dic10 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_{10}
% in TeX

G:=Group("C7xDic10");
// GroupNames label

G:=SmallGroup(280,19);
// by ID

G=gap.SmallGroup(280,19);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-5,140,301,146,5604]);
// Polycyclic

G:=Group<a,b,c|a^7=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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