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## G = C5×C7⋊C8order 280 = 23·5·7

### Direct product of C5 and C7⋊C8

Aliases: C5×C7⋊C8, C7⋊C40, C354C8, C14.C20, C70.4C4, C20.4D7, C140.5C2, C28.2C10, C10.3Dic7, C4.2(C5×D7), C2.(C5×Dic7), SmallGroup(280,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C5×C7⋊C8
 Chief series C1 — C7 — C14 — C28 — C140 — C5×C7⋊C8
 Lower central C7 — C5×C7⋊C8
 Upper central C1 — C20

Generators and relations for C5×C7⋊C8
G = < a,b,c | a5=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

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

100 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 7A 7B 7C 8A 8B 8C 8D 10A 10B 10C 10D 14A 14B 14C 20A ··· 20H 28A ··· 28F 35A ··· 35L 40A ··· 40P 70A ··· 70L 140A ··· 140X order 1 2 4 4 5 5 5 5 7 7 7 8 8 8 8 10 10 10 10 14 14 14 20 ··· 20 28 ··· 28 35 ··· 35 40 ··· 40 70 ··· 70 140 ··· 140 size 1 1 1 1 1 1 1 1 2 2 2 7 7 7 7 1 1 1 1 2 2 2 1 ··· 1 2 ··· 2 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 C4 C5 C8 C10 C20 C40 D7 Dic7 C7⋊C8 C5×D7 C5×Dic7 C5×C7⋊C8 kernel C5×C7⋊C8 C140 C70 C7⋊C8 C35 C28 C14 C7 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 4 4 8 16 3 3 6 12 12 24

Matrix representation of C5×C7⋊C8 in GL2(𝔽41) generated by

 16 0 0 16
,
 4 2 4 33
,
 0 36 10 0
G:=sub<GL(2,GF(41))| [16,0,0,16],[4,4,2,33],[0,10,36,0] >;

C5×C7⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_7\rtimes C_8
% in TeX

G:=Group("C5xC7:C8");
// GroupNames label

G:=SmallGroup(280,2);
// by ID

G=gap.SmallGroup(280,2);
# by ID

G:=PCGroup([5,-2,-5,-2,-2,-7,50,42,6004]);
// Polycyclic

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

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