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G = Q8×C49order 392 = 23·72

Direct product of C49 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C49, C4.C98, C196.3C2, C28.4C14, C98.7C22, C7.(C7×Q8), (C7×Q8).C7, C2.2(C2×C98), C14.7(C2×C14), SmallGroup(392,10)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C49
C1C7C14C98C196 — Q8×C49
C1C2 — Q8×C49
C1C98 — Q8×C49

Generators and relations for Q8×C49
 G = < a,b,c | a49=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C49
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)
(1 223 124 360)(2 224 125 361)(3 225 126 362)(4 226 127 363)(5 227 128 364)(6 228 129 365)(7 229 130 366)(8 230 131 367)(9 231 132 368)(10 232 133 369)(11 233 134 370)(12 234 135 371)(13 235 136 372)(14 236 137 373)(15 237 138 374)(16 238 139 375)(17 239 140 376)(18 240 141 377)(19 241 142 378)(20 242 143 379)(21 243 144 380)(22 244 145 381)(23 245 146 382)(24 197 147 383)(25 198 99 384)(26 199 100 385)(27 200 101 386)(28 201 102 387)(29 202 103 388)(30 203 104 389)(31 204 105 390)(32 205 106 391)(33 206 107 392)(34 207 108 344)(35 208 109 345)(36 209 110 346)(37 210 111 347)(38 211 112 348)(39 212 113 349)(40 213 114 350)(41 214 115 351)(42 215 116 352)(43 216 117 353)(44 217 118 354)(45 218 119 355)(46 219 120 356)(47 220 121 357)(48 221 122 358)(49 222 123 359)(50 330 249 161)(51 331 250 162)(52 332 251 163)(53 333 252 164)(54 334 253 165)(55 335 254 166)(56 336 255 167)(57 337 256 168)(58 338 257 169)(59 339 258 170)(60 340 259 171)(61 341 260 172)(62 342 261 173)(63 343 262 174)(64 295 263 175)(65 296 264 176)(66 297 265 177)(67 298 266 178)(68 299 267 179)(69 300 268 180)(70 301 269 181)(71 302 270 182)(72 303 271 183)(73 304 272 184)(74 305 273 185)(75 306 274 186)(76 307 275 187)(77 308 276 188)(78 309 277 189)(79 310 278 190)(80 311 279 191)(81 312 280 192)(82 313 281 193)(83 314 282 194)(84 315 283 195)(85 316 284 196)(86 317 285 148)(87 318 286 149)(88 319 287 150)(89 320 288 151)(90 321 289 152)(91 322 290 153)(92 323 291 154)(93 324 292 155)(94 325 293 156)(95 326 294 157)(96 327 246 158)(97 328 247 159)(98 329 248 160)
(1 330 124 161)(2 331 125 162)(3 332 126 163)(4 333 127 164)(5 334 128 165)(6 335 129 166)(7 336 130 167)(8 337 131 168)(9 338 132 169)(10 339 133 170)(11 340 134 171)(12 341 135 172)(13 342 136 173)(14 343 137 174)(15 295 138 175)(16 296 139 176)(17 297 140 177)(18 298 141 178)(19 299 142 179)(20 300 143 180)(21 301 144 181)(22 302 145 182)(23 303 146 183)(24 304 147 184)(25 305 99 185)(26 306 100 186)(27 307 101 187)(28 308 102 188)(29 309 103 189)(30 310 104 190)(31 311 105 191)(32 312 106 192)(33 313 107 193)(34 314 108 194)(35 315 109 195)(36 316 110 196)(37 317 111 148)(38 318 112 149)(39 319 113 150)(40 320 114 151)(41 321 115 152)(42 322 116 153)(43 323 117 154)(44 324 118 155)(45 325 119 156)(46 326 120 157)(47 327 121 158)(48 328 122 159)(49 329 123 160)(50 360 249 223)(51 361 250 224)(52 362 251 225)(53 363 252 226)(54 364 253 227)(55 365 254 228)(56 366 255 229)(57 367 256 230)(58 368 257 231)(59 369 258 232)(60 370 259 233)(61 371 260 234)(62 372 261 235)(63 373 262 236)(64 374 263 237)(65 375 264 238)(66 376 265 239)(67 377 266 240)(68 378 267 241)(69 379 268 242)(70 380 269 243)(71 381 270 244)(72 382 271 245)(73 383 272 197)(74 384 273 198)(75 385 274 199)(76 386 275 200)(77 387 276 201)(78 388 277 202)(79 389 278 203)(80 390 279 204)(81 391 280 205)(82 392 281 206)(83 344 282 207)(84 345 283 208)(85 346 284 209)(86 347 285 210)(87 348 286 211)(88 349 287 212)(89 350 288 213)(90 351 289 214)(91 352 290 215)(92 353 291 216)(93 354 292 217)(94 355 293 218)(95 356 294 219)(96 357 246 220)(97 358 247 221)(98 359 248 222)

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

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

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

245 conjugacy classes

class 1  2 4A4B4C7A···7F14A···14F28A···28R49A···49AP98A···98AP196A···196DV
order124447···714···1428···2849···4998···98196···196
size112221···11···12···21···11···12···2

245 irreducible representations

dim111111222
type++-
imageC1C2C7C14C49C98Q8C7×Q8Q8×C49
kernelQ8×C49C196C7×Q8C28Q8C4C49C7C1
# reps13618421261642

Matrix representation of Q8×C49 in GL2(𝔽197) generated by

850
085
,
1195
1196
,
281
141195
G:=sub<GL(2,GF(197))| [85,0,0,85],[1,1,195,196],[2,141,81,195] >;

Q8×C49 in GAP, Magma, Sage, TeX

Q_8\times C_{49}
% in TeX

G:=Group("Q8xC49");
// GroupNames label

G:=SmallGroup(392,10);
// by ID

G=gap.SmallGroup(392,10);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-7,140,301,146,222]);
// Polycyclic

G:=Group<a,b,c|a^49=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C49 in TeX

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