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G = C7×C5⋊C8order 280 = 23·5·7

Direct product of C7 and C5⋊C8

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

C1C5 — C7×C5⋊C8
C1C5C10Dic5C7×Dic5 — C7×C5⋊C8
C5 — C7×C5⋊C8
C1C14

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

5C4
5C8
5C28
5C56

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)(9 206 131 50 247 166 91)(10 207 132 51 248 167 92)(11 208 133 52 241 168 93)(12 201 134 53 242 161 94)(13 202 135 54 243 162 95)(14 203 136 55 244 163 96)(15 204 129 56 245 164 89)(16 205 130 49 246 165 90)(17 215 280 199 256 175 240)(18 216 273 200 249 176 233)(19 209 274 193 250 169 234)(20 210 275 194 251 170 235)(21 211 276 195 252 171 236)(22 212 277 196 253 172 237)(23 213 278 197 254 173 238)(24 214 279 198 255 174 239)(25 85 146 65 125 44 106)(26 86 147 66 126 45 107)(27 87 148 67 127 46 108)(28 88 149 68 128 47 109)(29 81 150 69 121 48 110)(30 82 151 70 122 41 111)(31 83 152 71 123 42 112)(32 84 145 72 124 43 105)(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 93 22 86 229)(2 87 94 230 23)(3 231 88 24 95)(4 17 232 96 81)(5 89 18 82 225)(6 83 90 226 19)(7 227 84 20 91)(8 21 228 92 85)(9 218 153 145 210)(10 146 219 211 154)(11 212 147 155 220)(12 156 213 221 148)(13 222 157 149 214)(14 150 223 215 158)(15 216 151 159 224)(16 160 209 217 152)(25 99 236 33 167)(26 34 100 168 237)(27 161 35 238 101)(28 239 162 102 36)(29 103 240 37 163)(30 38 104 164 233)(31 165 39 234 97)(32 235 166 98 40)(41 192 257 56 249)(42 49 185 250 258)(43 251 50 259 186)(44 260 252 187 51)(45 188 261 52 253)(46 53 189 254 262)(47 255 54 263 190)(48 264 256 191 55)(57 267 124 194 131)(58 195 268 132 125)(59 133 196 126 269)(60 127 134 270 197)(61 271 128 198 135)(62 199 272 136 121)(63 129 200 122 265)(64 123 130 266 193)(65 139 276 73 207)(66 74 140 208 277)(67 201 75 278 141)(68 279 202 142 76)(69 143 280 77 203)(70 78 144 204 273)(71 205 79 274 137)(72 275 206 138 80)(105 170 247 178 113)(106 179 171 114 248)(107 115 180 241 172)(108 242 116 173 181)(109 174 243 182 117)(110 183 175 118 244)(111 119 184 245 176)(112 246 120 169 177)
(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)(9,206,131,50,247,166,91)(10,207,132,51,248,167,92)(11,208,133,52,241,168,93)(12,201,134,53,242,161,94)(13,202,135,54,243,162,95)(14,203,136,55,244,163,96)(15,204,129,56,245,164,89)(16,205,130,49,246,165,90)(17,215,280,199,256,175,240)(18,216,273,200,249,176,233)(19,209,274,193,250,169,234)(20,210,275,194,251,170,235)(21,211,276,195,252,171,236)(22,212,277,196,253,172,237)(23,213,278,197,254,173,238)(24,214,279,198,255,174,239)(25,85,146,65,125,44,106)(26,86,147,66,126,45,107)(27,87,148,67,127,46,108)(28,88,149,68,128,47,109)(29,81,150,69,121,48,110)(30,82,151,70,122,41,111)(31,83,152,71,123,42,112)(32,84,145,72,124,43,105)(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), (1,93,22,86,229)(2,87,94,230,23)(3,231,88,24,95)(4,17,232,96,81)(5,89,18,82,225)(6,83,90,226,19)(7,227,84,20,91)(8,21,228,92,85)(9,218,153,145,210)(10,146,219,211,154)(11,212,147,155,220)(12,156,213,221,148)(13,222,157,149,214)(14,150,223,215,158)(15,216,151,159,224)(16,160,209,217,152)(25,99,236,33,167)(26,34,100,168,237)(27,161,35,238,101)(28,239,162,102,36)(29,103,240,37,163)(30,38,104,164,233)(31,165,39,234,97)(32,235,166,98,40)(41,192,257,56,249)(42,49,185,250,258)(43,251,50,259,186)(44,260,252,187,51)(45,188,261,52,253)(46,53,189,254,262)(47,255,54,263,190)(48,264,256,191,55)(57,267,124,194,131)(58,195,268,132,125)(59,133,196,126,269)(60,127,134,270,197)(61,271,128,198,135)(62,199,272,136,121)(63,129,200,122,265)(64,123,130,266,193)(65,139,276,73,207)(66,74,140,208,277)(67,201,75,278,141)(68,279,202,142,76)(69,143,280,77,203)(70,78,144,204,273)(71,205,79,274,137)(72,275,206,138,80)(105,170,247,178,113)(106,179,171,114,248)(107,115,180,241,172)(108,242,116,173,181)(109,174,243,182,117)(110,183,175,118,244)(111,119,184,245,176)(112,246,120,169,177), (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)(9,206,131,50,247,166,91)(10,207,132,51,248,167,92)(11,208,133,52,241,168,93)(12,201,134,53,242,161,94)(13,202,135,54,243,162,95)(14,203,136,55,244,163,96)(15,204,129,56,245,164,89)(16,205,130,49,246,165,90)(17,215,280,199,256,175,240)(18,216,273,200,249,176,233)(19,209,274,193,250,169,234)(20,210,275,194,251,170,235)(21,211,276,195,252,171,236)(22,212,277,196,253,172,237)(23,213,278,197,254,173,238)(24,214,279,198,255,174,239)(25,85,146,65,125,44,106)(26,86,147,66,126,45,107)(27,87,148,67,127,46,108)(28,88,149,68,128,47,109)(29,81,150,69,121,48,110)(30,82,151,70,122,41,111)(31,83,152,71,123,42,112)(32,84,145,72,124,43,105)(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), (1,93,22,86,229)(2,87,94,230,23)(3,231,88,24,95)(4,17,232,96,81)(5,89,18,82,225)(6,83,90,226,19)(7,227,84,20,91)(8,21,228,92,85)(9,218,153,145,210)(10,146,219,211,154)(11,212,147,155,220)(12,156,213,221,148)(13,222,157,149,214)(14,150,223,215,158)(15,216,151,159,224)(16,160,209,217,152)(25,99,236,33,167)(26,34,100,168,237)(27,161,35,238,101)(28,239,162,102,36)(29,103,240,37,163)(30,38,104,164,233)(31,165,39,234,97)(32,235,166,98,40)(41,192,257,56,249)(42,49,185,250,258)(43,251,50,259,186)(44,260,252,187,51)(45,188,261,52,253)(46,53,189,254,262)(47,255,54,263,190)(48,264,256,191,55)(57,267,124,194,131)(58,195,268,132,125)(59,133,196,126,269)(60,127,134,270,197)(61,271,128,198,135)(62,199,272,136,121)(63,129,200,122,265)(64,123,130,266,193)(65,139,276,73,207)(66,74,140,208,277)(67,201,75,278,141)(68,279,202,142,76)(69,143,280,77,203)(70,78,144,204,273)(71,205,79,274,137)(72,275,206,138,80)(105,170,247,178,113)(106,179,171,114,248)(107,115,180,241,172)(108,242,116,173,181)(109,174,243,182,117)(110,183,175,118,244)(111,119,184,245,176)(112,246,120,169,177), (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),(9,206,131,50,247,166,91),(10,207,132,51,248,167,92),(11,208,133,52,241,168,93),(12,201,134,53,242,161,94),(13,202,135,54,243,162,95),(14,203,136,55,244,163,96),(15,204,129,56,245,164,89),(16,205,130,49,246,165,90),(17,215,280,199,256,175,240),(18,216,273,200,249,176,233),(19,209,274,193,250,169,234),(20,210,275,194,251,170,235),(21,211,276,195,252,171,236),(22,212,277,196,253,172,237),(23,213,278,197,254,173,238),(24,214,279,198,255,174,239),(25,85,146,65,125,44,106),(26,86,147,66,126,45,107),(27,87,148,67,127,46,108),(28,88,149,68,128,47,109),(29,81,150,69,121,48,110),(30,82,151,70,122,41,111),(31,83,152,71,123,42,112),(32,84,145,72,124,43,105),(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)], [(1,93,22,86,229),(2,87,94,230,23),(3,231,88,24,95),(4,17,232,96,81),(5,89,18,82,225),(6,83,90,226,19),(7,227,84,20,91),(8,21,228,92,85),(9,218,153,145,210),(10,146,219,211,154),(11,212,147,155,220),(12,156,213,221,148),(13,222,157,149,214),(14,150,223,215,158),(15,216,151,159,224),(16,160,209,217,152),(25,99,236,33,167),(26,34,100,168,237),(27,161,35,238,101),(28,239,162,102,36),(29,103,240,37,163),(30,38,104,164,233),(31,165,39,234,97),(32,235,166,98,40),(41,192,257,56,249),(42,49,185,250,258),(43,251,50,259,186),(44,260,252,187,51),(45,188,261,52,253),(46,53,189,254,262),(47,255,54,263,190),(48,264,256,191,55),(57,267,124,194,131),(58,195,268,132,125),(59,133,196,126,269),(60,127,134,270,197),(61,271,128,198,135),(62,199,272,136,121),(63,129,200,122,265),(64,123,130,266,193),(65,139,276,73,207),(66,74,140,208,277),(67,201,75,278,141),(68,279,202,142,76),(69,143,280,77,203),(70,78,144,204,273),(71,205,79,274,137),(72,275,206,138,80),(105,170,247,178,113),(106,179,171,114,248),(107,115,180,241,172),(108,242,116,173,181),(109,174,243,182,117),(110,183,175,118,244),(111,119,184,245,176),(112,246,120,169,177)], [(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 4A4B 5 7A···7F8A8B8C8D 10 14A···14F28A···28L35A···35F56A···56X70A···70F
order124457···788881014···1428···2835···3556···5670···70
size115541···1555541···15···54···45···54···4

70 irreducible representations

dim111111114444
type+++-
imageC1C2C4C7C8C14C28C56F5C5⋊C8C7×F5C7×C5⋊C8
kernelC7×C5⋊C8C7×Dic5C70C5⋊C8C35Dic5C10C5C14C7C2C1
# reps11264612241166

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

1090000
01000
00100
00010
00001
,
10000
0280280280280
01000
00100
00010
,
10000
010683184259
010117619823
022128105206
025878153175

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

Export

Subgroup lattice of C7×C5⋊C8 in TeX

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