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G = C47⋊C8order 376 = 23·47

The semidirect product of C47 and C8 acting via C8/C4=C2

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

Aliases: C47⋊C8, C94.C4, C4.2D47, C2.Dic47, C188.2C2, SmallGroup(376,1)

Series: Derived Chief Lower central Upper central

C1C47 — C47⋊C8
C1C47C94C188 — C47⋊C8
C47 — C47⋊C8
C1C4

Generators and relations for C47⋊C8
 G = < a,b | a47=b8=1, bab-1=a-1 >

47C8

Smallest permutation representation of C47⋊C8
Regular action on 376 points
Generators in S376
(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)(236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282)(283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329)(330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376)
(1 330 186 236 69 283 101 189)(2 376 187 282 70 329 102 235)(3 375 188 281 71 328 103 234)(4 374 142 280 72 327 104 233)(5 373 143 279 73 326 105 232)(6 372 144 278 74 325 106 231)(7 371 145 277 75 324 107 230)(8 370 146 276 76 323 108 229)(9 369 147 275 77 322 109 228)(10 368 148 274 78 321 110 227)(11 367 149 273 79 320 111 226)(12 366 150 272 80 319 112 225)(13 365 151 271 81 318 113 224)(14 364 152 270 82 317 114 223)(15 363 153 269 83 316 115 222)(16 362 154 268 84 315 116 221)(17 361 155 267 85 314 117 220)(18 360 156 266 86 313 118 219)(19 359 157 265 87 312 119 218)(20 358 158 264 88 311 120 217)(21 357 159 263 89 310 121 216)(22 356 160 262 90 309 122 215)(23 355 161 261 91 308 123 214)(24 354 162 260 92 307 124 213)(25 353 163 259 93 306 125 212)(26 352 164 258 94 305 126 211)(27 351 165 257 48 304 127 210)(28 350 166 256 49 303 128 209)(29 349 167 255 50 302 129 208)(30 348 168 254 51 301 130 207)(31 347 169 253 52 300 131 206)(32 346 170 252 53 299 132 205)(33 345 171 251 54 298 133 204)(34 344 172 250 55 297 134 203)(35 343 173 249 56 296 135 202)(36 342 174 248 57 295 136 201)(37 341 175 247 58 294 137 200)(38 340 176 246 59 293 138 199)(39 339 177 245 60 292 139 198)(40 338 178 244 61 291 140 197)(41 337 179 243 62 290 141 196)(42 336 180 242 63 289 95 195)(43 335 181 241 64 288 96 194)(44 334 182 240 65 287 97 193)(45 333 183 239 66 286 98 192)(46 332 184 238 67 285 99 191)(47 331 185 237 68 284 100 190)

G:=sub<Sym(376)| (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)(236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282)(283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329)(330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376), (1,330,186,236,69,283,101,189)(2,376,187,282,70,329,102,235)(3,375,188,281,71,328,103,234)(4,374,142,280,72,327,104,233)(5,373,143,279,73,326,105,232)(6,372,144,278,74,325,106,231)(7,371,145,277,75,324,107,230)(8,370,146,276,76,323,108,229)(9,369,147,275,77,322,109,228)(10,368,148,274,78,321,110,227)(11,367,149,273,79,320,111,226)(12,366,150,272,80,319,112,225)(13,365,151,271,81,318,113,224)(14,364,152,270,82,317,114,223)(15,363,153,269,83,316,115,222)(16,362,154,268,84,315,116,221)(17,361,155,267,85,314,117,220)(18,360,156,266,86,313,118,219)(19,359,157,265,87,312,119,218)(20,358,158,264,88,311,120,217)(21,357,159,263,89,310,121,216)(22,356,160,262,90,309,122,215)(23,355,161,261,91,308,123,214)(24,354,162,260,92,307,124,213)(25,353,163,259,93,306,125,212)(26,352,164,258,94,305,126,211)(27,351,165,257,48,304,127,210)(28,350,166,256,49,303,128,209)(29,349,167,255,50,302,129,208)(30,348,168,254,51,301,130,207)(31,347,169,253,52,300,131,206)(32,346,170,252,53,299,132,205)(33,345,171,251,54,298,133,204)(34,344,172,250,55,297,134,203)(35,343,173,249,56,296,135,202)(36,342,174,248,57,295,136,201)(37,341,175,247,58,294,137,200)(38,340,176,246,59,293,138,199)(39,339,177,245,60,292,139,198)(40,338,178,244,61,291,140,197)(41,337,179,243,62,290,141,196)(42,336,180,242,63,289,95,195)(43,335,181,241,64,288,96,194)(44,334,182,240,65,287,97,193)(45,333,183,239,66,286,98,192)(46,332,184,238,67,285,99,191)(47,331,185,237,68,284,100,190)>;

G:=Group( (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)(236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282)(283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329)(330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376), (1,330,186,236,69,283,101,189)(2,376,187,282,70,329,102,235)(3,375,188,281,71,328,103,234)(4,374,142,280,72,327,104,233)(5,373,143,279,73,326,105,232)(6,372,144,278,74,325,106,231)(7,371,145,277,75,324,107,230)(8,370,146,276,76,323,108,229)(9,369,147,275,77,322,109,228)(10,368,148,274,78,321,110,227)(11,367,149,273,79,320,111,226)(12,366,150,272,80,319,112,225)(13,365,151,271,81,318,113,224)(14,364,152,270,82,317,114,223)(15,363,153,269,83,316,115,222)(16,362,154,268,84,315,116,221)(17,361,155,267,85,314,117,220)(18,360,156,266,86,313,118,219)(19,359,157,265,87,312,119,218)(20,358,158,264,88,311,120,217)(21,357,159,263,89,310,121,216)(22,356,160,262,90,309,122,215)(23,355,161,261,91,308,123,214)(24,354,162,260,92,307,124,213)(25,353,163,259,93,306,125,212)(26,352,164,258,94,305,126,211)(27,351,165,257,48,304,127,210)(28,350,166,256,49,303,128,209)(29,349,167,255,50,302,129,208)(30,348,168,254,51,301,130,207)(31,347,169,253,52,300,131,206)(32,346,170,252,53,299,132,205)(33,345,171,251,54,298,133,204)(34,344,172,250,55,297,134,203)(35,343,173,249,56,296,135,202)(36,342,174,248,57,295,136,201)(37,341,175,247,58,294,137,200)(38,340,176,246,59,293,138,199)(39,339,177,245,60,292,139,198)(40,338,178,244,61,291,140,197)(41,337,179,243,62,290,141,196)(42,336,180,242,63,289,95,195)(43,335,181,241,64,288,96,194)(44,334,182,240,65,287,97,193)(45,333,183,239,66,286,98,192)(46,332,184,238,67,285,99,191)(47,331,185,237,68,284,100,190) );

G=PermutationGroup([(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),(236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282),(283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329),(330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376)], [(1,330,186,236,69,283,101,189),(2,376,187,282,70,329,102,235),(3,375,188,281,71,328,103,234),(4,374,142,280,72,327,104,233),(5,373,143,279,73,326,105,232),(6,372,144,278,74,325,106,231),(7,371,145,277,75,324,107,230),(8,370,146,276,76,323,108,229),(9,369,147,275,77,322,109,228),(10,368,148,274,78,321,110,227),(11,367,149,273,79,320,111,226),(12,366,150,272,80,319,112,225),(13,365,151,271,81,318,113,224),(14,364,152,270,82,317,114,223),(15,363,153,269,83,316,115,222),(16,362,154,268,84,315,116,221),(17,361,155,267,85,314,117,220),(18,360,156,266,86,313,118,219),(19,359,157,265,87,312,119,218),(20,358,158,264,88,311,120,217),(21,357,159,263,89,310,121,216),(22,356,160,262,90,309,122,215),(23,355,161,261,91,308,123,214),(24,354,162,260,92,307,124,213),(25,353,163,259,93,306,125,212),(26,352,164,258,94,305,126,211),(27,351,165,257,48,304,127,210),(28,350,166,256,49,303,128,209),(29,349,167,255,50,302,129,208),(30,348,168,254,51,301,130,207),(31,347,169,253,52,300,131,206),(32,346,170,252,53,299,132,205),(33,345,171,251,54,298,133,204),(34,344,172,250,55,297,134,203),(35,343,173,249,56,296,135,202),(36,342,174,248,57,295,136,201),(37,341,175,247,58,294,137,200),(38,340,176,246,59,293,138,199),(39,339,177,245,60,292,139,198),(40,338,178,244,61,291,140,197),(41,337,179,243,62,290,141,196),(42,336,180,242,63,289,95,195),(43,335,181,241,64,288,96,194),(44,334,182,240,65,287,97,193),(45,333,183,239,66,286,98,192),(46,332,184,238,67,285,99,191),(47,331,185,237,68,284,100,190)])

100 conjugacy classes

class 1  2 4A4B8A8B8C8D47A···47W94A···94W188A···188AT
order1244888847···4794···94188···188
size1111474747472···22···22···2

100 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D47Dic47C47⋊C8
kernelC47⋊C8C188C94C47C4C2C1
# reps1124232346

Matrix representation of C47⋊C8 in GL3(𝔽1129) generated by

100
001
01128873
,
109800
0618523
0375511
G:=sub<GL(3,GF(1129))| [1,0,0,0,0,1128,0,1,873],[1098,0,0,0,618,375,0,523,511] >;

C47⋊C8 in GAP, Magma, Sage, TeX

C_{47}\rtimes C_8
% in TeX

G:=Group("C47:C8");
// GroupNames label

G:=SmallGroup(376,1);
// by ID

G=gap.SmallGroup(376,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-47,8,21,5891]);
// Polycyclic

G:=Group<a,b|a^47=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C47⋊C8 in TeX

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