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G = C5×C7⋊C8order 280 = 23·5·7

Direct product of C5 and C7⋊C8

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

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

C1C7 — C5×C7⋊C8
C1C7C14C28C140 — C5×C7⋊C8
C7 — C5×C7⋊C8
C1C20

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

7C8
7C40

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 78 135 172 229)(10 79 136 173 230)(11 80 129 174 231)(12 73 130 175 232)(13 74 131 176 225)(14 75 132 169 226)(15 76 133 170 227)(16 77 134 171 228)(17 205 262 180 237)(18 206 263 181 238)(19 207 264 182 239)(20 208 257 183 240)(21 201 258 184 233)(22 202 259 177 234)(23 203 260 178 235)(24 204 261 179 236)(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 90 152 213 270)(34 91 145 214 271)(35 92 146 215 272)(36 93 147 216 265)(37 94 148 209 266)(38 95 149 210 267)(39 96 150 211 268)(40 89 151 212 269)(41 97 160 59 116)(42 98 153 60 117)(43 99 154 61 118)(44 100 155 62 119)(45 101 156 63 120)(46 102 157 64 113)(47 103 158 57 114)(48 104 159 58 115)(49 106 168 67 124)(50 107 161 68 125)(51 108 162 69 126)(52 109 163 70 127)(53 110 164 71 128)(54 111 165 72 121)(55 112 166 65 122)(56 105 167 66 123)
(1 25 43 228 239 89 107)(2 108 90 240 229 44 26)(3 27 45 230 233 91 109)(4 110 92 234 231 46 28)(5 29 47 232 235 93 111)(6 112 94 236 225 48 30)(7 31 41 226 237 95 105)(8 106 96 238 227 42 32)(9 100 82 221 162 152 20)(10 21 145 163 222 83 101)(11 102 84 223 164 146 22)(12 23 147 165 224 85 103)(13 104 86 217 166 148 24)(14 17 149 167 218 87 97)(15 98 88 219 168 150 18)(16 19 151 161 220 81 99)(33 183 172 119 252 246 51)(34 52 247 253 120 173 184)(35 177 174 113 254 248 53)(36 54 241 255 114 175 178)(37 179 176 115 256 242 55)(38 56 243 249 116 169 180)(39 181 170 117 250 244 49)(40 50 245 251 118 171 182)(57 130 260 265 121 192 198)(58 199 185 122 266 261 131)(59 132 262 267 123 186 200)(60 193 187 124 268 263 133)(61 134 264 269 125 188 194)(62 195 189 126 270 257 135)(63 136 258 271 127 190 196)(64 197 191 128 272 259 129)(65 209 204 74 159 141 274)(66 275 142 160 75 205 210)(67 211 206 76 153 143 276)(68 277 144 154 77 207 212)(69 213 208 78 155 137 278)(70 279 138 156 79 201 214)(71 215 202 80 157 139 280)(72 273 140 158 73 203 216)
(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)(9,78,135,172,229)(10,79,136,173,230)(11,80,129,174,231)(12,73,130,175,232)(13,74,131,176,225)(14,75,132,169,226)(15,76,133,170,227)(16,77,134,171,228)(17,205,262,180,237)(18,206,263,181,238)(19,207,264,182,239)(20,208,257,183,240)(21,201,258,184,233)(22,202,259,177,234)(23,203,260,178,235)(24,204,261,179,236)(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)(33,90,152,213,270)(34,91,145,214,271)(35,92,146,215,272)(36,93,147,216,265)(37,94,148,209,266)(38,95,149,210,267)(39,96,150,211,268)(40,89,151,212,269)(41,97,160,59,116)(42,98,153,60,117)(43,99,154,61,118)(44,100,155,62,119)(45,101,156,63,120)(46,102,157,64,113)(47,103,158,57,114)(48,104,159,58,115)(49,106,168,67,124)(50,107,161,68,125)(51,108,162,69,126)(52,109,163,70,127)(53,110,164,71,128)(54,111,165,72,121)(55,112,166,65,122)(56,105,167,66,123), (1,25,43,228,239,89,107)(2,108,90,240,229,44,26)(3,27,45,230,233,91,109)(4,110,92,234,231,46,28)(5,29,47,232,235,93,111)(6,112,94,236,225,48,30)(7,31,41,226,237,95,105)(8,106,96,238,227,42,32)(9,100,82,221,162,152,20)(10,21,145,163,222,83,101)(11,102,84,223,164,146,22)(12,23,147,165,224,85,103)(13,104,86,217,166,148,24)(14,17,149,167,218,87,97)(15,98,88,219,168,150,18)(16,19,151,161,220,81,99)(33,183,172,119,252,246,51)(34,52,247,253,120,173,184)(35,177,174,113,254,248,53)(36,54,241,255,114,175,178)(37,179,176,115,256,242,55)(38,56,243,249,116,169,180)(39,181,170,117,250,244,49)(40,50,245,251,118,171,182)(57,130,260,265,121,192,198)(58,199,185,122,266,261,131)(59,132,262,267,123,186,200)(60,193,187,124,268,263,133)(61,134,264,269,125,188,194)(62,195,189,126,270,257,135)(63,136,258,271,127,190,196)(64,197,191,128,272,259,129)(65,209,204,74,159,141,274)(66,275,142,160,75,205,210)(67,211,206,76,153,143,276)(68,277,144,154,77,207,212)(69,213,208,78,155,137,278)(70,279,138,156,79,201,214)(71,215,202,80,157,139,280)(72,273,140,158,73,203,216), (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)(9,78,135,172,229)(10,79,136,173,230)(11,80,129,174,231)(12,73,130,175,232)(13,74,131,176,225)(14,75,132,169,226)(15,76,133,170,227)(16,77,134,171,228)(17,205,262,180,237)(18,206,263,181,238)(19,207,264,182,239)(20,208,257,183,240)(21,201,258,184,233)(22,202,259,177,234)(23,203,260,178,235)(24,204,261,179,236)(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)(33,90,152,213,270)(34,91,145,214,271)(35,92,146,215,272)(36,93,147,216,265)(37,94,148,209,266)(38,95,149,210,267)(39,96,150,211,268)(40,89,151,212,269)(41,97,160,59,116)(42,98,153,60,117)(43,99,154,61,118)(44,100,155,62,119)(45,101,156,63,120)(46,102,157,64,113)(47,103,158,57,114)(48,104,159,58,115)(49,106,168,67,124)(50,107,161,68,125)(51,108,162,69,126)(52,109,163,70,127)(53,110,164,71,128)(54,111,165,72,121)(55,112,166,65,122)(56,105,167,66,123), (1,25,43,228,239,89,107)(2,108,90,240,229,44,26)(3,27,45,230,233,91,109)(4,110,92,234,231,46,28)(5,29,47,232,235,93,111)(6,112,94,236,225,48,30)(7,31,41,226,237,95,105)(8,106,96,238,227,42,32)(9,100,82,221,162,152,20)(10,21,145,163,222,83,101)(11,102,84,223,164,146,22)(12,23,147,165,224,85,103)(13,104,86,217,166,148,24)(14,17,149,167,218,87,97)(15,98,88,219,168,150,18)(16,19,151,161,220,81,99)(33,183,172,119,252,246,51)(34,52,247,253,120,173,184)(35,177,174,113,254,248,53)(36,54,241,255,114,175,178)(37,179,176,115,256,242,55)(38,56,243,249,116,169,180)(39,181,170,117,250,244,49)(40,50,245,251,118,171,182)(57,130,260,265,121,192,198)(58,199,185,122,266,261,131)(59,132,262,267,123,186,200)(60,193,187,124,268,263,133)(61,134,264,269,125,188,194)(62,195,189,126,270,257,135)(63,136,258,271,127,190,196)(64,197,191,128,272,259,129)(65,209,204,74,159,141,274)(66,275,142,160,75,205,210)(67,211,206,76,153,143,276)(68,277,144,154,77,207,212)(69,213,208,78,155,137,278)(70,279,138,156,79,201,214)(71,215,202,80,157,139,280)(72,273,140,158,73,203,216), (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),(9,78,135,172,229),(10,79,136,173,230),(11,80,129,174,231),(12,73,130,175,232),(13,74,131,176,225),(14,75,132,169,226),(15,76,133,170,227),(16,77,134,171,228),(17,205,262,180,237),(18,206,263,181,238),(19,207,264,182,239),(20,208,257,183,240),(21,201,258,184,233),(22,202,259,177,234),(23,203,260,178,235),(24,204,261,179,236),(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),(33,90,152,213,270),(34,91,145,214,271),(35,92,146,215,272),(36,93,147,216,265),(37,94,148,209,266),(38,95,149,210,267),(39,96,150,211,268),(40,89,151,212,269),(41,97,160,59,116),(42,98,153,60,117),(43,99,154,61,118),(44,100,155,62,119),(45,101,156,63,120),(46,102,157,64,113),(47,103,158,57,114),(48,104,159,58,115),(49,106,168,67,124),(50,107,161,68,125),(51,108,162,69,126),(52,109,163,70,127),(53,110,164,71,128),(54,111,165,72,121),(55,112,166,65,122),(56,105,167,66,123)], [(1,25,43,228,239,89,107),(2,108,90,240,229,44,26),(3,27,45,230,233,91,109),(4,110,92,234,231,46,28),(5,29,47,232,235,93,111),(6,112,94,236,225,48,30),(7,31,41,226,237,95,105),(8,106,96,238,227,42,32),(9,100,82,221,162,152,20),(10,21,145,163,222,83,101),(11,102,84,223,164,146,22),(12,23,147,165,224,85,103),(13,104,86,217,166,148,24),(14,17,149,167,218,87,97),(15,98,88,219,168,150,18),(16,19,151,161,220,81,99),(33,183,172,119,252,246,51),(34,52,247,253,120,173,184),(35,177,174,113,254,248,53),(36,54,241,255,114,175,178),(37,179,176,115,256,242,55),(38,56,243,249,116,169,180),(39,181,170,117,250,244,49),(40,50,245,251,118,171,182),(57,130,260,265,121,192,198),(58,199,185,122,266,261,131),(59,132,262,267,123,186,200),(60,193,187,124,268,263,133),(61,134,264,269,125,188,194),(62,195,189,126,270,257,135),(63,136,258,271,127,190,196),(64,197,191,128,272,259,129),(65,209,204,74,159,141,274),(66,275,142,160,75,205,210),(67,211,206,76,153,143,276),(68,277,144,154,77,207,212),(69,213,208,78,155,137,278),(70,279,138,156,79,201,214),(71,215,202,80,157,139,280),(72,273,140,158,73,203,216)], [(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 4A4B5A5B5C5D7A7B7C8A8B8C8D10A10B10C10D14A14B14C20A···20H28A···28F35A···35L40A···40P70A···70L140A···140X
order1244555577788881010101014141420···2028···2835···3540···4070···70140···140
size11111111222777711112221···12···22···27···72···22···2

100 irreducible representations

dim11111111222222
type+++-
imageC1C2C4C5C8C10C20C40D7Dic7C7⋊C8C5×D7C5×Dic7C5×C7⋊C8
kernelC5×C7⋊C8C140C70C7⋊C8C35C28C14C7C20C10C5C4C2C1
# reps112444816336121224

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

160
016
,
42
433
,
036
100
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

Export

Subgroup lattice of C5×C7⋊C8 in TeX

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