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G = C49⋊C8order 392 = 23·72

The semidirect product of C49 and C8 acting via C8/C4=C2

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

Aliases: C49⋊C8, C98.C4, C28.4D7, C4.2D49, C2.Dic49, C196.2C2, C14.1Dic7, C7.(C7⋊C8), SmallGroup(392,1)

Series: Derived Chief Lower central Upper central

C1C49 — C49⋊C8
C1C7C49C98C196 — C49⋊C8
C49 — C49⋊C8
C1C4

Generators and relations for C49⋊C8
 G = < a,b | a49=b8=1, bab-1=a-1 >

49C8
7C7⋊C8

Smallest permutation representation of C49⋊C8
Regular action on 392 points
Generators in S392
(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)(344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392)
(1 367 189 292 59 335 111 237)(2 366 190 291 60 334 112 236)(3 365 191 290 61 333 113 235)(4 364 192 289 62 332 114 234)(5 363 193 288 63 331 115 233)(6 362 194 287 64 330 116 232)(7 361 195 286 65 329 117 231)(8 360 196 285 66 328 118 230)(9 359 148 284 67 327 119 229)(10 358 149 283 68 326 120 228)(11 357 150 282 69 325 121 227)(12 356 151 281 70 324 122 226)(13 355 152 280 71 323 123 225)(14 354 153 279 72 322 124 224)(15 353 154 278 73 321 125 223)(16 352 155 277 74 320 126 222)(17 351 156 276 75 319 127 221)(18 350 157 275 76 318 128 220)(19 349 158 274 77 317 129 219)(20 348 159 273 78 316 130 218)(21 347 160 272 79 315 131 217)(22 346 161 271 80 314 132 216)(23 345 162 270 81 313 133 215)(24 344 163 269 82 312 134 214)(25 392 164 268 83 311 135 213)(26 391 165 267 84 310 136 212)(27 390 166 266 85 309 137 211)(28 389 167 265 86 308 138 210)(29 388 168 264 87 307 139 209)(30 387 169 263 88 306 140 208)(31 386 170 262 89 305 141 207)(32 385 171 261 90 304 142 206)(33 384 172 260 91 303 143 205)(34 383 173 259 92 302 144 204)(35 382 174 258 93 301 145 203)(36 381 175 257 94 300 146 202)(37 380 176 256 95 299 147 201)(38 379 177 255 96 298 99 200)(39 378 178 254 97 297 100 199)(40 377 179 253 98 296 101 198)(41 376 180 252 50 295 102 197)(42 375 181 251 51 343 103 245)(43 374 182 250 52 342 104 244)(44 373 183 249 53 341 105 243)(45 372 184 248 54 340 106 242)(46 371 185 247 55 339 107 241)(47 370 186 246 56 338 108 240)(48 369 187 294 57 337 109 239)(49 368 188 293 58 336 110 238)

G:=sub<Sym(392)| (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)(344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,367,189,292,59,335,111,237)(2,366,190,291,60,334,112,236)(3,365,191,290,61,333,113,235)(4,364,192,289,62,332,114,234)(5,363,193,288,63,331,115,233)(6,362,194,287,64,330,116,232)(7,361,195,286,65,329,117,231)(8,360,196,285,66,328,118,230)(9,359,148,284,67,327,119,229)(10,358,149,283,68,326,120,228)(11,357,150,282,69,325,121,227)(12,356,151,281,70,324,122,226)(13,355,152,280,71,323,123,225)(14,354,153,279,72,322,124,224)(15,353,154,278,73,321,125,223)(16,352,155,277,74,320,126,222)(17,351,156,276,75,319,127,221)(18,350,157,275,76,318,128,220)(19,349,158,274,77,317,129,219)(20,348,159,273,78,316,130,218)(21,347,160,272,79,315,131,217)(22,346,161,271,80,314,132,216)(23,345,162,270,81,313,133,215)(24,344,163,269,82,312,134,214)(25,392,164,268,83,311,135,213)(26,391,165,267,84,310,136,212)(27,390,166,266,85,309,137,211)(28,389,167,265,86,308,138,210)(29,388,168,264,87,307,139,209)(30,387,169,263,88,306,140,208)(31,386,170,262,89,305,141,207)(32,385,171,261,90,304,142,206)(33,384,172,260,91,303,143,205)(34,383,173,259,92,302,144,204)(35,382,174,258,93,301,145,203)(36,381,175,257,94,300,146,202)(37,380,176,256,95,299,147,201)(38,379,177,255,96,298,99,200)(39,378,178,254,97,297,100,199)(40,377,179,253,98,296,101,198)(41,376,180,252,50,295,102,197)(42,375,181,251,51,343,103,245)(43,374,182,250,52,342,104,244)(44,373,183,249,53,341,105,243)(45,372,184,248,54,340,106,242)(46,371,185,247,55,339,107,241)(47,370,186,246,56,338,108,240)(48,369,187,294,57,337,109,239)(49,368,188,293,58,336,110,238)>;

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)(344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,367,189,292,59,335,111,237)(2,366,190,291,60,334,112,236)(3,365,191,290,61,333,113,235)(4,364,192,289,62,332,114,234)(5,363,193,288,63,331,115,233)(6,362,194,287,64,330,116,232)(7,361,195,286,65,329,117,231)(8,360,196,285,66,328,118,230)(9,359,148,284,67,327,119,229)(10,358,149,283,68,326,120,228)(11,357,150,282,69,325,121,227)(12,356,151,281,70,324,122,226)(13,355,152,280,71,323,123,225)(14,354,153,279,72,322,124,224)(15,353,154,278,73,321,125,223)(16,352,155,277,74,320,126,222)(17,351,156,276,75,319,127,221)(18,350,157,275,76,318,128,220)(19,349,158,274,77,317,129,219)(20,348,159,273,78,316,130,218)(21,347,160,272,79,315,131,217)(22,346,161,271,80,314,132,216)(23,345,162,270,81,313,133,215)(24,344,163,269,82,312,134,214)(25,392,164,268,83,311,135,213)(26,391,165,267,84,310,136,212)(27,390,166,266,85,309,137,211)(28,389,167,265,86,308,138,210)(29,388,168,264,87,307,139,209)(30,387,169,263,88,306,140,208)(31,386,170,262,89,305,141,207)(32,385,171,261,90,304,142,206)(33,384,172,260,91,303,143,205)(34,383,173,259,92,302,144,204)(35,382,174,258,93,301,145,203)(36,381,175,257,94,300,146,202)(37,380,176,256,95,299,147,201)(38,379,177,255,96,298,99,200)(39,378,178,254,97,297,100,199)(40,377,179,253,98,296,101,198)(41,376,180,252,50,295,102,197)(42,375,181,251,51,343,103,245)(43,374,182,250,52,342,104,244)(44,373,183,249,53,341,105,243)(45,372,184,248,54,340,106,242)(46,371,185,247,55,339,107,241)(47,370,186,246,56,338,108,240)(48,369,187,294,57,337,109,239)(49,368,188,293,58,336,110,238) );

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),(344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392)], [(1,367,189,292,59,335,111,237),(2,366,190,291,60,334,112,236),(3,365,191,290,61,333,113,235),(4,364,192,289,62,332,114,234),(5,363,193,288,63,331,115,233),(6,362,194,287,64,330,116,232),(7,361,195,286,65,329,117,231),(8,360,196,285,66,328,118,230),(9,359,148,284,67,327,119,229),(10,358,149,283,68,326,120,228),(11,357,150,282,69,325,121,227),(12,356,151,281,70,324,122,226),(13,355,152,280,71,323,123,225),(14,354,153,279,72,322,124,224),(15,353,154,278,73,321,125,223),(16,352,155,277,74,320,126,222),(17,351,156,276,75,319,127,221),(18,350,157,275,76,318,128,220),(19,349,158,274,77,317,129,219),(20,348,159,273,78,316,130,218),(21,347,160,272,79,315,131,217),(22,346,161,271,80,314,132,216),(23,345,162,270,81,313,133,215),(24,344,163,269,82,312,134,214),(25,392,164,268,83,311,135,213),(26,391,165,267,84,310,136,212),(27,390,166,266,85,309,137,211),(28,389,167,265,86,308,138,210),(29,388,168,264,87,307,139,209),(30,387,169,263,88,306,140,208),(31,386,170,262,89,305,141,207),(32,385,171,261,90,304,142,206),(33,384,172,260,91,303,143,205),(34,383,173,259,92,302,144,204),(35,382,174,258,93,301,145,203),(36,381,175,257,94,300,146,202),(37,380,176,256,95,299,147,201),(38,379,177,255,96,298,99,200),(39,378,178,254,97,297,100,199),(40,377,179,253,98,296,101,198),(41,376,180,252,50,295,102,197),(42,375,181,251,51,343,103,245),(43,374,182,250,52,342,104,244),(44,373,183,249,53,341,105,243),(45,372,184,248,54,340,106,242),(46,371,185,247,55,339,107,241),(47,370,186,246,56,338,108,240),(48,369,187,294,57,337,109,239),(49,368,188,293,58,336,110,238)])

104 conjugacy classes

class 1  2 4A4B7A7B7C8A8B8C8D14A14B14C28A···28F49A···49U98A···98U196A···196AP
order1244777888814141428···2849···4998···98196···196
size1111222494949492222···22···22···22···2

104 irreducible representations

dim1111222222
type+++-+-
imageC1C2C4C8D7Dic7C7⋊C8D49Dic49C49⋊C8
kernelC49⋊C8C196C98C49C28C14C7C4C2C1
# reps1124336212142

Matrix representation of C49⋊C8 in GL3(𝔽3137) generated by

100
02422797
012272152
,
194100
01724293
014941413
G:=sub<GL(3,GF(3137))| [1,0,0,0,242,1227,0,2797,2152],[1941,0,0,0,1724,1494,0,293,1413] >;

C49⋊C8 in GAP, Magma, Sage, TeX

C_{49}\rtimes C_8
% in TeX

G:=Group("C49:C8");
// GroupNames label

G:=SmallGroup(392,1);
// by ID

G=gap.SmallGroup(392,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,10,26,2083,858,8404]);
// Polycyclic

G:=Group<a,b|a^49=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C49⋊C8 in TeX

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