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
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 57 209 319)(2 58 210 320)(3 59 211 321)(4 60 212 322)(5 61 213 323)(6 62 214 324)(7 63 215 325)(8 64 216 326)(9 65 217 327)(10 66 218 328)(11 67 219 329)(12 68 220 330)(13 69 221 331)(14 70 222 332)(15 71 223 333)(16 72 224 334)(17 73 225 335)(18 74 226 336)(19 75 227 337)(20 76 228 338)(21 77 229 339)(22 78 230 340)(23 79 231 341)(24 80 232 342)(25 81 233 343)(26 82 234 295)(27 83 235 296)(28 84 236 297)(29 85 237 298)(30 86 238 299)(31 87 239 300)(32 88 240 301)(33 89 241 302)(34 90 242 303)(35 91 243 304)(36 92 244 305)(37 93 245 306)(38 94 197 307)(39 95 198 308)(40 96 199 309)(41 97 200 310)(42 98 201 311)(43 50 202 312)(44 51 203 313)(45 52 204 314)(46 53 205 315)(47 54 206 316)(48 55 207 317)(49 56 208 318)(99 272 392 153)(100 273 344 154)(101 274 345 155)(102 275 346 156)(103 276 347 157)(104 277 348 158)(105 278 349 159)(106 279 350 160)(107 280 351 161)(108 281 352 162)(109 282 353 163)(110 283 354 164)(111 284 355 165)(112 285 356 166)(113 286 357 167)(114 287 358 168)(115 288 359 169)(116 289 360 170)(117 290 361 171)(118 291 362 172)(119 292 363 173)(120 293 364 174)(121 294 365 175)(122 246 366 176)(123 247 367 177)(124 248 368 178)(125 249 369 179)(126 250 370 180)(127 251 371 181)(128 252 372 182)(129 253 373 183)(130 254 374 184)(131 255 375 185)(132 256 376 186)(133 257 377 187)(134 258 378 188)(135 259 379 189)(136 260 380 190)(137 261 381 191)(138 262 382 192)(139 263 383 193)(140 264 384 194)(141 265 385 195)(142 266 386 196)(143 267 387 148)(144 268 388 149)(145 269 389 150)(146 270 390 151)(147 271 391 152)
(1 282 209 163)(2 283 210 164)(3 284 211 165)(4 285 212 166)(5 286 213 167)(6 287 214 168)(7 288 215 169)(8 289 216 170)(9 290 217 171)(10 291 218 172)(11 292 219 173)(12 293 220 174)(13 294 221 175)(14 246 222 176)(15 247 223 177)(16 248 224 178)(17 249 225 179)(18 250 226 180)(19 251 227 181)(20 252 228 182)(21 253 229 183)(22 254 230 184)(23 255 231 185)(24 256 232 186)(25 257 233 187)(26 258 234 188)(27 259 235 189)(28 260 236 190)(29 261 237 191)(30 262 238 192)(31 263 239 193)(32 264 240 194)(33 265 241 195)(34 266 242 196)(35 267 243 148)(36 268 244 149)(37 269 245 150)(38 270 197 151)(39 271 198 152)(40 272 199 153)(41 273 200 154)(42 274 201 155)(43 275 202 156)(44 276 203 157)(45 277 204 158)(46 278 205 159)(47 279 206 160)(48 280 207 161)(49 281 208 162)(50 102 312 346)(51 103 313 347)(52 104 314 348)(53 105 315 349)(54 106 316 350)(55 107 317 351)(56 108 318 352)(57 109 319 353)(58 110 320 354)(59 111 321 355)(60 112 322 356)(61 113 323 357)(62 114 324 358)(63 115 325 359)(64 116 326 360)(65 117 327 361)(66 118 328 362)(67 119 329 363)(68 120 330 364)(69 121 331 365)(70 122 332 366)(71 123 333 367)(72 124 334 368)(73 125 335 369)(74 126 336 370)(75 127 337 371)(76 128 338 372)(77 129 339 373)(78 130 340 374)(79 131 341 375)(80 132 342 376)(81 133 343 377)(82 134 295 378)(83 135 296 379)(84 136 297 380)(85 137 298 381)(86 138 299 382)(87 139 300 383)(88 140 301 384)(89 141 302 385)(90 142 303 386)(91 143 304 387)(92 144 305 388)(93 145 306 389)(94 146 307 390)(95 147 308 391)(96 99 309 392)(97 100 310 344)(98 101 311 345)
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), (1,57,209,319)(2,58,210,320)(3,59,211,321)(4,60,212,322)(5,61,213,323)(6,62,214,324)(7,63,215,325)(8,64,216,326)(9,65,217,327)(10,66,218,328)(11,67,219,329)(12,68,220,330)(13,69,221,331)(14,70,222,332)(15,71,223,333)(16,72,224,334)(17,73,225,335)(18,74,226,336)(19,75,227,337)(20,76,228,338)(21,77,229,339)(22,78,230,340)(23,79,231,341)(24,80,232,342)(25,81,233,343)(26,82,234,295)(27,83,235,296)(28,84,236,297)(29,85,237,298)(30,86,238,299)(31,87,239,300)(32,88,240,301)(33,89,241,302)(34,90,242,303)(35,91,243,304)(36,92,244,305)(37,93,245,306)(38,94,197,307)(39,95,198,308)(40,96,199,309)(41,97,200,310)(42,98,201,311)(43,50,202,312)(44,51,203,313)(45,52,204,314)(46,53,205,315)(47,54,206,316)(48,55,207,317)(49,56,208,318)(99,272,392,153)(100,273,344,154)(101,274,345,155)(102,275,346,156)(103,276,347,157)(104,277,348,158)(105,278,349,159)(106,279,350,160)(107,280,351,161)(108,281,352,162)(109,282,353,163)(110,283,354,164)(111,284,355,165)(112,285,356,166)(113,286,357,167)(114,287,358,168)(115,288,359,169)(116,289,360,170)(117,290,361,171)(118,291,362,172)(119,292,363,173)(120,293,364,174)(121,294,365,175)(122,246,366,176)(123,247,367,177)(124,248,368,178)(125,249,369,179)(126,250,370,180)(127,251,371,181)(128,252,372,182)(129,253,373,183)(130,254,374,184)(131,255,375,185)(132,256,376,186)(133,257,377,187)(134,258,378,188)(135,259,379,189)(136,260,380,190)(137,261,381,191)(138,262,382,192)(139,263,383,193)(140,264,384,194)(141,265,385,195)(142,266,386,196)(143,267,387,148)(144,268,388,149)(145,269,389,150)(146,270,390,151)(147,271,391,152), (1,282,209,163)(2,283,210,164)(3,284,211,165)(4,285,212,166)(5,286,213,167)(6,287,214,168)(7,288,215,169)(8,289,216,170)(9,290,217,171)(10,291,218,172)(11,292,219,173)(12,293,220,174)(13,294,221,175)(14,246,222,176)(15,247,223,177)(16,248,224,178)(17,249,225,179)(18,250,226,180)(19,251,227,181)(20,252,228,182)(21,253,229,183)(22,254,230,184)(23,255,231,185)(24,256,232,186)(25,257,233,187)(26,258,234,188)(27,259,235,189)(28,260,236,190)(29,261,237,191)(30,262,238,192)(31,263,239,193)(32,264,240,194)(33,265,241,195)(34,266,242,196)(35,267,243,148)(36,268,244,149)(37,269,245,150)(38,270,197,151)(39,271,198,152)(40,272,199,153)(41,273,200,154)(42,274,201,155)(43,275,202,156)(44,276,203,157)(45,277,204,158)(46,278,205,159)(47,279,206,160)(48,280,207,161)(49,281,208,162)(50,102,312,346)(51,103,313,347)(52,104,314,348)(53,105,315,349)(54,106,316,350)(55,107,317,351)(56,108,318,352)(57,109,319,353)(58,110,320,354)(59,111,321,355)(60,112,322,356)(61,113,323,357)(62,114,324,358)(63,115,325,359)(64,116,326,360)(65,117,327,361)(66,118,328,362)(67,119,329,363)(68,120,330,364)(69,121,331,365)(70,122,332,366)(71,123,333,367)(72,124,334,368)(73,125,335,369)(74,126,336,370)(75,127,337,371)(76,128,338,372)(77,129,339,373)(78,130,340,374)(79,131,341,375)(80,132,342,376)(81,133,343,377)(82,134,295,378)(83,135,296,379)(84,136,297,380)(85,137,298,381)(86,138,299,382)(87,139,300,383)(88,140,301,384)(89,141,302,385)(90,142,303,386)(91,143,304,387)(92,144,305,388)(93,145,306,389)(94,146,307,390)(95,147,308,391)(96,99,309,392)(97,100,310,344)(98,101,311,345)>;
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), (1,57,209,319)(2,58,210,320)(3,59,211,321)(4,60,212,322)(5,61,213,323)(6,62,214,324)(7,63,215,325)(8,64,216,326)(9,65,217,327)(10,66,218,328)(11,67,219,329)(12,68,220,330)(13,69,221,331)(14,70,222,332)(15,71,223,333)(16,72,224,334)(17,73,225,335)(18,74,226,336)(19,75,227,337)(20,76,228,338)(21,77,229,339)(22,78,230,340)(23,79,231,341)(24,80,232,342)(25,81,233,343)(26,82,234,295)(27,83,235,296)(28,84,236,297)(29,85,237,298)(30,86,238,299)(31,87,239,300)(32,88,240,301)(33,89,241,302)(34,90,242,303)(35,91,243,304)(36,92,244,305)(37,93,245,306)(38,94,197,307)(39,95,198,308)(40,96,199,309)(41,97,200,310)(42,98,201,311)(43,50,202,312)(44,51,203,313)(45,52,204,314)(46,53,205,315)(47,54,206,316)(48,55,207,317)(49,56,208,318)(99,272,392,153)(100,273,344,154)(101,274,345,155)(102,275,346,156)(103,276,347,157)(104,277,348,158)(105,278,349,159)(106,279,350,160)(107,280,351,161)(108,281,352,162)(109,282,353,163)(110,283,354,164)(111,284,355,165)(112,285,356,166)(113,286,357,167)(114,287,358,168)(115,288,359,169)(116,289,360,170)(117,290,361,171)(118,291,362,172)(119,292,363,173)(120,293,364,174)(121,294,365,175)(122,246,366,176)(123,247,367,177)(124,248,368,178)(125,249,369,179)(126,250,370,180)(127,251,371,181)(128,252,372,182)(129,253,373,183)(130,254,374,184)(131,255,375,185)(132,256,376,186)(133,257,377,187)(134,258,378,188)(135,259,379,189)(136,260,380,190)(137,261,381,191)(138,262,382,192)(139,263,383,193)(140,264,384,194)(141,265,385,195)(142,266,386,196)(143,267,387,148)(144,268,388,149)(145,269,389,150)(146,270,390,151)(147,271,391,152), (1,282,209,163)(2,283,210,164)(3,284,211,165)(4,285,212,166)(5,286,213,167)(6,287,214,168)(7,288,215,169)(8,289,216,170)(9,290,217,171)(10,291,218,172)(11,292,219,173)(12,293,220,174)(13,294,221,175)(14,246,222,176)(15,247,223,177)(16,248,224,178)(17,249,225,179)(18,250,226,180)(19,251,227,181)(20,252,228,182)(21,253,229,183)(22,254,230,184)(23,255,231,185)(24,256,232,186)(25,257,233,187)(26,258,234,188)(27,259,235,189)(28,260,236,190)(29,261,237,191)(30,262,238,192)(31,263,239,193)(32,264,240,194)(33,265,241,195)(34,266,242,196)(35,267,243,148)(36,268,244,149)(37,269,245,150)(38,270,197,151)(39,271,198,152)(40,272,199,153)(41,273,200,154)(42,274,201,155)(43,275,202,156)(44,276,203,157)(45,277,204,158)(46,278,205,159)(47,279,206,160)(48,280,207,161)(49,281,208,162)(50,102,312,346)(51,103,313,347)(52,104,314,348)(53,105,315,349)(54,106,316,350)(55,107,317,351)(56,108,318,352)(57,109,319,353)(58,110,320,354)(59,111,321,355)(60,112,322,356)(61,113,323,357)(62,114,324,358)(63,115,325,359)(64,116,326,360)(65,117,327,361)(66,118,328,362)(67,119,329,363)(68,120,330,364)(69,121,331,365)(70,122,332,366)(71,123,333,367)(72,124,334,368)(73,125,335,369)(74,126,336,370)(75,127,337,371)(76,128,338,372)(77,129,339,373)(78,130,340,374)(79,131,341,375)(80,132,342,376)(81,133,343,377)(82,134,295,378)(83,135,296,379)(84,136,297,380)(85,137,298,381)(86,138,299,382)(87,139,300,383)(88,140,301,384)(89,141,302,385)(90,142,303,386)(91,143,304,387)(92,144,305,388)(93,145,306,389)(94,146,307,390)(95,147,308,391)(96,99,309,392)(97,100,310,344)(98,101,311,345) );
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)], [(1,57,209,319),(2,58,210,320),(3,59,211,321),(4,60,212,322),(5,61,213,323),(6,62,214,324),(7,63,215,325),(8,64,216,326),(9,65,217,327),(10,66,218,328),(11,67,219,329),(12,68,220,330),(13,69,221,331),(14,70,222,332),(15,71,223,333),(16,72,224,334),(17,73,225,335),(18,74,226,336),(19,75,227,337),(20,76,228,338),(21,77,229,339),(22,78,230,340),(23,79,231,341),(24,80,232,342),(25,81,233,343),(26,82,234,295),(27,83,235,296),(28,84,236,297),(29,85,237,298),(30,86,238,299),(31,87,239,300),(32,88,240,301),(33,89,241,302),(34,90,242,303),(35,91,243,304),(36,92,244,305),(37,93,245,306),(38,94,197,307),(39,95,198,308),(40,96,199,309),(41,97,200,310),(42,98,201,311),(43,50,202,312),(44,51,203,313),(45,52,204,314),(46,53,205,315),(47,54,206,316),(48,55,207,317),(49,56,208,318),(99,272,392,153),(100,273,344,154),(101,274,345,155),(102,275,346,156),(103,276,347,157),(104,277,348,158),(105,278,349,159),(106,279,350,160),(107,280,351,161),(108,281,352,162),(109,282,353,163),(110,283,354,164),(111,284,355,165),(112,285,356,166),(113,286,357,167),(114,287,358,168),(115,288,359,169),(116,289,360,170),(117,290,361,171),(118,291,362,172),(119,292,363,173),(120,293,364,174),(121,294,365,175),(122,246,366,176),(123,247,367,177),(124,248,368,178),(125,249,369,179),(126,250,370,180),(127,251,371,181),(128,252,372,182),(129,253,373,183),(130,254,374,184),(131,255,375,185),(132,256,376,186),(133,257,377,187),(134,258,378,188),(135,259,379,189),(136,260,380,190),(137,261,381,191),(138,262,382,192),(139,263,383,193),(140,264,384,194),(141,265,385,195),(142,266,386,196),(143,267,387,148),(144,268,388,149),(145,269,389,150),(146,270,390,151),(147,271,391,152)], [(1,282,209,163),(2,283,210,164),(3,284,211,165),(4,285,212,166),(5,286,213,167),(6,287,214,168),(7,288,215,169),(8,289,216,170),(9,290,217,171),(10,291,218,172),(11,292,219,173),(12,293,220,174),(13,294,221,175),(14,246,222,176),(15,247,223,177),(16,248,224,178),(17,249,225,179),(18,250,226,180),(19,251,227,181),(20,252,228,182),(21,253,229,183),(22,254,230,184),(23,255,231,185),(24,256,232,186),(25,257,233,187),(26,258,234,188),(27,259,235,189),(28,260,236,190),(29,261,237,191),(30,262,238,192),(31,263,239,193),(32,264,240,194),(33,265,241,195),(34,266,242,196),(35,267,243,148),(36,268,244,149),(37,269,245,150),(38,270,197,151),(39,271,198,152),(40,272,199,153),(41,273,200,154),(42,274,201,155),(43,275,202,156),(44,276,203,157),(45,277,204,158),(46,278,205,159),(47,279,206,160),(48,280,207,161),(49,281,208,162),(50,102,312,346),(51,103,313,347),(52,104,314,348),(53,105,315,349),(54,106,316,350),(55,107,317,351),(56,108,318,352),(57,109,319,353),(58,110,320,354),(59,111,321,355),(60,112,322,356),(61,113,323,357),(62,114,324,358),(63,115,325,359),(64,116,326,360),(65,117,327,361),(66,118,328,362),(67,119,329,363),(68,120,330,364),(69,121,331,365),(70,122,332,366),(71,123,333,367),(72,124,334,368),(73,125,335,369),(74,126,336,370),(75,127,337,371),(76,128,338,372),(77,129,339,373),(78,130,340,374),(79,131,341,375),(80,132,342,376),(81,133,343,377),(82,134,295,378),(83,135,296,379),(84,136,297,380),(85,137,298,381),(86,138,299,382),(87,139,300,383),(88,140,301,384),(89,141,302,385),(90,142,303,386),(91,143,304,387),(92,144,305,388),(93,145,306,389),(94,146,307,390),(95,147,308,391),(96,99,309,392),(97,100,310,344),(98,101,311,345)]])
245 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | ··· | 7F | 14A | ··· | 14F | 28A | ··· | 28R | 49A | ··· | 49AP | 98A | ··· | 98AP | 196A | ··· | 196DV |
| order | 1 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 49 | ··· | 49 | 98 | ··· | 98 | 196 | ··· | 196 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
245 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | - | ||||||
| image | C1 | C2 | C7 | C14 | C49 | C98 | Q8 | C7×Q8 | Q8×C49 |
| kernel | Q8×C49 | C196 | C7×Q8 | C28 | Q8 | C4 | C49 | C7 | C1 |
| # reps | 1 | 3 | 6 | 18 | 42 | 126 | 1 | 6 | 42 |
Matrix representation of Q8×C49 ►in GL2(𝔽197) generated by
| 85 | 0 |
| 0 | 85 |
| 1 | 195 |
| 1 | 196 |
| 2 | 81 |
| 141 | 195 |
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