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## G = C7×C5⋊2C8order 280 = 23·5·7

### Direct product of C7 and C5⋊2C8

Aliases: C7×C52C8, C52C56, C355C8, C70.5C4, C28.4D5, C20.2C14, C140.6C2, C10.2C28, C14.2Dic5, C4.2(C7×D5), C2.(C7×Dic5), SmallGroup(280,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C7×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C140 — C7×C5⋊2C8
 Lower central C5 — C7×C5⋊2C8
 Upper central C1 — C28

Generators and relations for C7×C52C8
G = < a,b,c | a7=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×C52C8
Regular action on 280 points
Generators in S280
(1 220 145 72 261 180 105)(2 221 146 65 262 181 106)(3 222 147 66 263 182 107)(4 223 148 67 264 183 108)(5 224 149 68 257 184 109)(6 217 150 69 258 177 110)(7 218 151 70 259 178 111)(8 219 152 71 260 179 112)(9 206 123 42 247 166 83)(10 207 124 43 248 167 84)(11 208 125 44 241 168 85)(12 201 126 45 242 161 86)(13 202 127 46 243 162 87)(14 203 128 47 244 163 88)(15 204 121 48 245 164 81)(16 205 122 41 246 165 82)(17 95 138 57 135 54 98)(18 96 139 58 136 55 99)(19 89 140 59 129 56 100)(20 90 141 60 130 49 101)(21 91 142 61 131 50 102)(22 92 143 62 132 51 103)(23 93 144 63 133 52 104)(24 94 137 64 134 53 97)(25 216 273 200 249 176 233)(26 209 274 193 250 169 234)(27 210 275 194 251 170 235)(28 211 276 195 252 171 236)(29 212 277 196 253 172 237)(30 213 278 197 254 173 238)(31 214 279 198 255 174 239)(32 215 280 199 256 175 240)(33 231 154 73 271 190 114)(34 232 155 74 272 191 115)(35 225 156 75 265 192 116)(36 226 157 76 266 185 117)(37 227 158 77 267 186 118)(38 228 159 78 268 187 119)(39 229 160 79 269 188 120)(40 230 153 80 270 189 113)
(1 88 29 91 232)(2 225 92 30 81)(3 82 31 93 226)(4 227 94 32 83)(5 84 25 95 228)(6 229 96 26 85)(7 86 27 89 230)(8 231 90 28 87)(9 223 158 137 215)(10 216 138 159 224)(11 217 160 139 209)(12 210 140 153 218)(13 219 154 141 211)(14 212 142 155 220)(15 221 156 143 213)(16 214 144 157 222)(17 38 109 167 233)(18 234 168 110 39)(19 40 111 161 235)(20 236 162 112 33)(21 34 105 163 237)(22 238 164 106 35)(23 36 107 165 239)(24 240 166 108 37)(41 255 52 185 263)(42 264 186 53 256)(43 249 54 187 257)(44 258 188 55 250)(45 251 56 189 259)(46 260 190 49 252)(47 253 50 191 261)(48 262 192 51 254)(57 78 149 207 273)(58 274 208 150 79)(59 80 151 201 275)(60 276 202 152 73)(61 74 145 203 277)(62 278 204 146 75)(63 76 147 205 279)(64 280 206 148 77)(65 265 132 197 121)(66 122 198 133 266)(67 267 134 199 123)(68 124 200 135 268)(69 269 136 193 125)(70 126 194 129 270)(71 271 130 195 127)(72 128 196 131 272)(97 175 247 183 118)(98 119 184 248 176)(99 169 241 177 120)(100 113 178 242 170)(101 171 243 179 114)(102 115 180 244 172)(103 173 245 181 116)(104 117 182 246 174)
(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)

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

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

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

112 conjugacy classes

 class 1 2 4A 4B 5A 5B 7A ··· 7F 8A 8B 8C 8D 10A 10B 14A ··· 14F 20A 20B 20C 20D 28A ··· 28L 35A ··· 35L 56A ··· 56X 70A ··· 70L 140A ··· 140X order 1 2 4 4 5 5 7 ··· 7 8 8 8 8 10 10 14 ··· 14 20 20 20 20 28 ··· 28 35 ··· 35 56 ··· 56 70 ··· 70 140 ··· 140 size 1 1 1 1 2 2 1 ··· 1 5 5 5 5 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 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 C4 C7 C8 C14 C28 C56 D5 Dic5 C5⋊2C8 C7×D5 C7×Dic5 C7×C5⋊2C8 kernel C7×C5⋊2C8 C140 C70 C5⋊2C8 C35 C20 C10 C5 C28 C14 C7 C4 C2 C1 # reps 1 1 2 6 4 6 12 24 2 2 4 12 12 24

Matrix representation of C7×C52C8 in GL2(𝔽29) generated by

 16 0 0 16
,
 1 27 4 22
,
 0 8 16 0
G:=sub<GL(2,GF(29))| [16,0,0,16],[1,4,27,22],[0,16,8,0] >;

C7×C52C8 in GAP, Magma, Sage, TeX

C_7\times C_5\rtimes_2C_8
% in TeX

G:=Group("C7xC5:2C8");
// GroupNames label

G:=SmallGroup(280,1);
// by ID

G=gap.SmallGroup(280,1);
# by ID

G:=PCGroup([5,-2,-7,-2,-2,-5,70,42,5604]);
// Polycyclic

G:=Group<a,b,c|a^7=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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