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## G = C7×C5⋊C8order 280 = 23·5·7

### Direct product of C7 and C5⋊C8

Aliases: C7×C5⋊C8, C5⋊C56, C352C8, C10.C28, C70.2C4, C14.2F5, Dic5.2C14, C2.(C7×F5), (C7×Dic5).4C2, SmallGroup(280,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C7×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C7×Dic5 — C7×C5⋊C8
 Lower central C5 — C7×C5⋊C8
 Upper central C1 — C14

Generators and relations for C7×C5⋊C8
G = < a,b,c | a7=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C7×C5⋊C8
Regular action on 280 points
Generators in S280
(1 220 140 59 261 180 100)(2 221 141 60 262 181 101)(3 222 142 61 263 182 102)(4 223 143 62 264 183 103)(5 224 144 63 257 184 104)(6 217 137 64 258 177 97)(7 218 138 57 259 178 98)(8 219 139 58 260 179 99)(9 206 136 55 247 166 96)(10 207 129 56 248 167 89)(11 208 130 49 241 168 90)(12 201 131 50 242 161 91)(13 202 132 51 243 162 92)(14 203 133 52 244 163 93)(15 204 134 53 245 164 94)(16 205 135 54 246 165 95)(17 211 276 195 252 171 236)(18 212 277 196 253 172 237)(19 213 278 197 254 173 238)(20 214 279 198 255 174 239)(21 215 280 199 256 175 240)(22 216 273 200 249 176 233)(23 209 274 193 250 169 234)(24 210 275 194 251 170 235)(25 88 146 65 128 47 106)(26 81 147 66 121 48 107)(27 82 148 67 122 41 108)(28 83 149 68 123 42 109)(29 84 150 69 124 43 110)(30 85 151 70 125 44 111)(31 86 152 71 126 45 112)(32 87 145 72 127 46 105)(33 228 154 73 268 187 114)(34 229 155 74 269 188 115)(35 230 156 75 270 189 116)(36 231 157 76 271 190 117)(37 232 158 77 272 191 118)(38 225 159 78 265 192 119)(39 226 160 79 266 185 120)(40 227 153 80 267 186 113)
(1 90 18 86 229)(2 87 91 230 19)(3 231 88 20 92)(4 21 232 93 81)(5 94 22 82 225)(6 83 95 226 23)(7 227 84 24 96)(8 17 228 89 85)(9 218 153 150 210)(10 151 219 211 154)(11 212 152 155 220)(12 156 213 221 145)(13 222 157 146 214)(14 147 223 215 158)(15 216 148 159 224)(16 160 209 217 149)(25 239 162 102 36)(26 103 240 37 163)(27 38 104 164 233)(28 165 39 234 97)(29 235 166 98 40)(30 99 236 33 167)(31 34 100 168 237)(32 161 35 238 101)(41 192 257 53 249)(42 54 185 250 258)(43 251 55 259 186)(44 260 252 187 56)(45 188 261 49 253)(46 50 189 254 262)(47 255 51 263 190)(48 264 256 191 52)(57 267 124 194 136)(58 195 268 129 125)(59 130 196 126 269)(60 127 131 270 197)(61 271 128 198 132)(62 199 272 133 121)(63 134 200 122 265)(64 123 135 266 193)(65 279 202 142 76)(66 143 280 77 203)(67 78 144 204 273)(68 205 79 274 137)(69 275 206 138 80)(70 139 276 73 207)(71 74 140 208 277)(72 201 75 278 141)(105 242 116 173 181)(106 174 243 182 117)(107 183 175 118 244)(108 119 184 245 176)(109 246 120 169 177)(110 170 247 178 113)(111 179 171 114 248)(112 115 180 241 172)
(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)| (1,220,140,59,261,180,100)(2,221,141,60,262,181,101)(3,222,142,61,263,182,102)(4,223,143,62,264,183,103)(5,224,144,63,257,184,104)(6,217,137,64,258,177,97)(7,218,138,57,259,178,98)(8,219,139,58,260,179,99)(9,206,136,55,247,166,96)(10,207,129,56,248,167,89)(11,208,130,49,241,168,90)(12,201,131,50,242,161,91)(13,202,132,51,243,162,92)(14,203,133,52,244,163,93)(15,204,134,53,245,164,94)(16,205,135,54,246,165,95)(17,211,276,195,252,171,236)(18,212,277,196,253,172,237)(19,213,278,197,254,173,238)(20,214,279,198,255,174,239)(21,215,280,199,256,175,240)(22,216,273,200,249,176,233)(23,209,274,193,250,169,234)(24,210,275,194,251,170,235)(25,88,146,65,128,47,106)(26,81,147,66,121,48,107)(27,82,148,67,122,41,108)(28,83,149,68,123,42,109)(29,84,150,69,124,43,110)(30,85,151,70,125,44,111)(31,86,152,71,126,45,112)(32,87,145,72,127,46,105)(33,228,154,73,268,187,114)(34,229,155,74,269,188,115)(35,230,156,75,270,189,116)(36,231,157,76,271,190,117)(37,232,158,77,272,191,118)(38,225,159,78,265,192,119)(39,226,160,79,266,185,120)(40,227,153,80,267,186,113), (1,90,18,86,229)(2,87,91,230,19)(3,231,88,20,92)(4,21,232,93,81)(5,94,22,82,225)(6,83,95,226,23)(7,227,84,24,96)(8,17,228,89,85)(9,218,153,150,210)(10,151,219,211,154)(11,212,152,155,220)(12,156,213,221,145)(13,222,157,146,214)(14,147,223,215,158)(15,216,148,159,224)(16,160,209,217,149)(25,239,162,102,36)(26,103,240,37,163)(27,38,104,164,233)(28,165,39,234,97)(29,235,166,98,40)(30,99,236,33,167)(31,34,100,168,237)(32,161,35,238,101)(41,192,257,53,249)(42,54,185,250,258)(43,251,55,259,186)(44,260,252,187,56)(45,188,261,49,253)(46,50,189,254,262)(47,255,51,263,190)(48,264,256,191,52)(57,267,124,194,136)(58,195,268,129,125)(59,130,196,126,269)(60,127,131,270,197)(61,271,128,198,132)(62,199,272,133,121)(63,134,200,122,265)(64,123,135,266,193)(65,279,202,142,76)(66,143,280,77,203)(67,78,144,204,273)(68,205,79,274,137)(69,275,206,138,80)(70,139,276,73,207)(71,74,140,208,277)(72,201,75,278,141)(105,242,116,173,181)(106,174,243,182,117)(107,183,175,118,244)(108,119,184,245,176)(109,246,120,169,177)(110,170,247,178,113)(111,179,171,114,248)(112,115,180,241,172), (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( (1,220,140,59,261,180,100)(2,221,141,60,262,181,101)(3,222,142,61,263,182,102)(4,223,143,62,264,183,103)(5,224,144,63,257,184,104)(6,217,137,64,258,177,97)(7,218,138,57,259,178,98)(8,219,139,58,260,179,99)(9,206,136,55,247,166,96)(10,207,129,56,248,167,89)(11,208,130,49,241,168,90)(12,201,131,50,242,161,91)(13,202,132,51,243,162,92)(14,203,133,52,244,163,93)(15,204,134,53,245,164,94)(16,205,135,54,246,165,95)(17,211,276,195,252,171,236)(18,212,277,196,253,172,237)(19,213,278,197,254,173,238)(20,214,279,198,255,174,239)(21,215,280,199,256,175,240)(22,216,273,200,249,176,233)(23,209,274,193,250,169,234)(24,210,275,194,251,170,235)(25,88,146,65,128,47,106)(26,81,147,66,121,48,107)(27,82,148,67,122,41,108)(28,83,149,68,123,42,109)(29,84,150,69,124,43,110)(30,85,151,70,125,44,111)(31,86,152,71,126,45,112)(32,87,145,72,127,46,105)(33,228,154,73,268,187,114)(34,229,155,74,269,188,115)(35,230,156,75,270,189,116)(36,231,157,76,271,190,117)(37,232,158,77,272,191,118)(38,225,159,78,265,192,119)(39,226,160,79,266,185,120)(40,227,153,80,267,186,113), (1,90,18,86,229)(2,87,91,230,19)(3,231,88,20,92)(4,21,232,93,81)(5,94,22,82,225)(6,83,95,226,23)(7,227,84,24,96)(8,17,228,89,85)(9,218,153,150,210)(10,151,219,211,154)(11,212,152,155,220)(12,156,213,221,145)(13,222,157,146,214)(14,147,223,215,158)(15,216,148,159,224)(16,160,209,217,149)(25,239,162,102,36)(26,103,240,37,163)(27,38,104,164,233)(28,165,39,234,97)(29,235,166,98,40)(30,99,236,33,167)(31,34,100,168,237)(32,161,35,238,101)(41,192,257,53,249)(42,54,185,250,258)(43,251,55,259,186)(44,260,252,187,56)(45,188,261,49,253)(46,50,189,254,262)(47,255,51,263,190)(48,264,256,191,52)(57,267,124,194,136)(58,195,268,129,125)(59,130,196,126,269)(60,127,131,270,197)(61,271,128,198,132)(62,199,272,133,121)(63,134,200,122,265)(64,123,135,266,193)(65,279,202,142,76)(66,143,280,77,203)(67,78,144,204,273)(68,205,79,274,137)(69,275,206,138,80)(70,139,276,73,207)(71,74,140,208,277)(72,201,75,278,141)(105,242,116,173,181)(106,174,243,182,117)(107,183,175,118,244)(108,119,184,245,176)(109,246,120,169,177)(110,170,247,178,113)(111,179,171,114,248)(112,115,180,241,172), (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([[(1,220,140,59,261,180,100),(2,221,141,60,262,181,101),(3,222,142,61,263,182,102),(4,223,143,62,264,183,103),(5,224,144,63,257,184,104),(6,217,137,64,258,177,97),(7,218,138,57,259,178,98),(8,219,139,58,260,179,99),(9,206,136,55,247,166,96),(10,207,129,56,248,167,89),(11,208,130,49,241,168,90),(12,201,131,50,242,161,91),(13,202,132,51,243,162,92),(14,203,133,52,244,163,93),(15,204,134,53,245,164,94),(16,205,135,54,246,165,95),(17,211,276,195,252,171,236),(18,212,277,196,253,172,237),(19,213,278,197,254,173,238),(20,214,279,198,255,174,239),(21,215,280,199,256,175,240),(22,216,273,200,249,176,233),(23,209,274,193,250,169,234),(24,210,275,194,251,170,235),(25,88,146,65,128,47,106),(26,81,147,66,121,48,107),(27,82,148,67,122,41,108),(28,83,149,68,123,42,109),(29,84,150,69,124,43,110),(30,85,151,70,125,44,111),(31,86,152,71,126,45,112),(32,87,145,72,127,46,105),(33,228,154,73,268,187,114),(34,229,155,74,269,188,115),(35,230,156,75,270,189,116),(36,231,157,76,271,190,117),(37,232,158,77,272,191,118),(38,225,159,78,265,192,119),(39,226,160,79,266,185,120),(40,227,153,80,267,186,113)], [(1,90,18,86,229),(2,87,91,230,19),(3,231,88,20,92),(4,21,232,93,81),(5,94,22,82,225),(6,83,95,226,23),(7,227,84,24,96),(8,17,228,89,85),(9,218,153,150,210),(10,151,219,211,154),(11,212,152,155,220),(12,156,213,221,145),(13,222,157,146,214),(14,147,223,215,158),(15,216,148,159,224),(16,160,209,217,149),(25,239,162,102,36),(26,103,240,37,163),(27,38,104,164,233),(28,165,39,234,97),(29,235,166,98,40),(30,99,236,33,167),(31,34,100,168,237),(32,161,35,238,101),(41,192,257,53,249),(42,54,185,250,258),(43,251,55,259,186),(44,260,252,187,56),(45,188,261,49,253),(46,50,189,254,262),(47,255,51,263,190),(48,264,256,191,52),(57,267,124,194,136),(58,195,268,129,125),(59,130,196,126,269),(60,127,131,270,197),(61,271,128,198,132),(62,199,272,133,121),(63,134,200,122,265),(64,123,135,266,193),(65,279,202,142,76),(66,143,280,77,203),(67,78,144,204,273),(68,205,79,274,137),(69,275,206,138,80),(70,139,276,73,207),(71,74,140,208,277),(72,201,75,278,141),(105,242,116,173,181),(106,174,243,182,117),(107,183,175,118,244),(108,119,184,245,176),(109,246,120,169,177),(110,170,247,178,113),(111,179,171,114,248),(112,115,180,241,172)], [(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)]])

70 conjugacy classes

 class 1 2 4A 4B 5 7A ··· 7F 8A 8B 8C 8D 10 14A ··· 14F 28A ··· 28L 35A ··· 35F 56A ··· 56X 70A ··· 70F order 1 2 4 4 5 7 ··· 7 8 8 8 8 10 14 ··· 14 28 ··· 28 35 ··· 35 56 ··· 56 70 ··· 70 size 1 1 5 5 4 1 ··· 1 5 5 5 5 4 1 ··· 1 5 ··· 5 4 ··· 4 5 ··· 5 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + - image C1 C2 C4 C7 C8 C14 C28 C56 F5 C5⋊C8 C7×F5 C7×C5⋊C8 kernel C7×C5⋊C8 C7×Dic5 C70 C5⋊C8 C35 Dic5 C10 C5 C14 C7 C2 C1 # reps 1 1 2 6 4 6 12 24 1 1 6 6

Matrix representation of C7×C5⋊C8 in GL5(𝔽281)

 109 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 280 280 280 280 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 106 83 184 259 0 101 176 198 23 0 22 128 105 206 0 258 78 153 175

G:=sub<GL(5,GF(281))| [109,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,280,1,0,0,0,280,0,1,0,0,280,0,0,1,0,280,0,0,0],[1,0,0,0,0,0,106,101,22,258,0,83,176,128,78,0,184,198,105,153,0,259,23,206,175] >;

C7×C5⋊C8 in GAP, Magma, Sage, TeX

C_7\times C_5\rtimes C_8
% in TeX

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

G:=SmallGroup(280,5);
// by ID

G=gap.SmallGroup(280,5);
# by ID

G:=PCGroup([5,-2,-7,-2,-2,-5,70,42,2804,414]);
// 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^3>;
// generators/relations

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