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## G = Q8×C72order 392 = 23·72

### Direct product of C72 and Q8

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

Aliases: Q8×C72, C28.5C14, C2.2C142, C4.(C7×C14), (C7×C28).5C2, C14.9(C2×C14), (C7×C14).17C22, SmallGroup(392,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C72
 Chief series C1 — C2 — C14 — C7×C14 — C7×C28 — Q8×C72
 Lower central C1 — C2 — Q8×C72
 Upper central C1 — C7×C14 — Q8×C72

Generators and relations for Q8×C72
G = < a,b,c,d | a7=b7=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 60, all normal (6 characteristic)
C1, C2, C4, C7, Q8, C14, C28, C72, C7×Q8, C7×C14, C7×C28, Q8×C72
Quotients: C1, C2, C22, C7, Q8, C14, C2×C14, C72, C7×Q8, C7×C14, C142, Q8×C72

Smallest permutation representation of Q8×C72
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)
(1 111 121 138 13 26 36)(2 112 122 139 14 27 37)(3 106 123 140 8 28 38)(4 107 124 134 9 22 39)(5 108 125 135 10 23 40)(6 109 126 136 11 24 41)(7 110 120 137 12 25 42)(15 32 49 379 346 363 373)(16 33 43 380 347 364 374)(17 34 44 381 348 358 375)(18 35 45 382 349 359 376)(19 29 46 383 350 360 377)(20 30 47 384 344 361 378)(21 31 48 385 345 362 372)(50 142 159 169 87 57 74)(51 143 160 170 88 58 75)(52 144 161 171 89 59 76)(53 145 155 172 90 60 77)(54 146 156 173 91 61 71)(55 147 157 174 85 62 72)(56 141 158 175 86 63 73)(64 81 98 298 315 325 390)(65 82 92 299 309 326 391)(66 83 93 300 310 327 392)(67 84 94 301 311 328 386)(68 78 95 295 312 329 387)(69 79 96 296 313 323 388)(70 80 97 297 314 324 389)(99 116 133 191 242 259 269)(100 117 127 192 243 253 270)(101 118 128 193 244 254 271)(102 119 129 194 245 255 272)(103 113 130 195 239 256 273)(104 114 131 196 240 257 267)(105 115 132 190 241 258 268)(148 165 182 208 218 235 188)(149 166 176 209 219 236 189)(150 167 177 210 220 237 183)(151 168 178 204 221 238 184)(152 162 179 205 222 232 185)(153 163 180 206 223 233 186)(154 164 181 207 224 234 187)(197 214 231 289 340 357 367)(198 215 225 290 341 351 368)(199 216 226 291 342 352 369)(200 217 227 292 343 353 370)(201 211 228 293 337 354 371)(202 212 229 294 338 355 365)(203 213 230 288 339 356 366)(246 263 280 306 316 333 286)(247 264 274 307 317 334 287)(248 265 275 308 318 335 281)(249 266 276 302 319 336 282)(250 260 277 303 320 330 283)(251 261 278 304 321 331 284)(252 262 279 305 322 332 285)
(1 270 172 176)(2 271 173 177)(3 272 174 178)(4 273 175 179)(5 267 169 180)(6 268 170 181)(7 269 171 182)(8 194 55 184)(9 195 56 185)(10 196 50 186)(11 190 51 187)(12 191 52 188)(13 192 53 189)(14 193 54 183)(15 199 297 308)(16 200 298 302)(17 201 299 303)(18 202 300 304)(19 203 301 305)(20 197 295 306)(21 198 296 307)(22 239 141 152)(23 240 142 153)(24 241 143 154)(25 242 144 148)(26 243 145 149)(27 244 146 150)(28 245 147 151)(29 213 311 322)(30 214 312 316)(31 215 313 317)(32 216 314 318)(33 217 315 319)(34 211 309 320)(35 212 310 321)(36 253 155 166)(37 254 156 167)(38 255 157 168)(39 256 158 162)(40 257 159 163)(41 258 160 164)(42 259 161 165)(43 227 325 336)(44 228 326 330)(45 229 327 331)(46 230 328 332)(47 231 329 333)(48 225 323 334)(49 226 324 335)(57 223 125 114)(58 224 126 115)(59 218 120 116)(60 219 121 117)(61 220 122 118)(62 221 123 119)(63 222 124 113)(64 249 347 343)(65 250 348 337)(66 251 349 338)(67 252 350 339)(68 246 344 340)(69 247 345 341)(70 248 346 342)(71 237 139 128)(72 238 140 129)(73 232 134 130)(74 233 135 131)(75 234 136 132)(76 235 137 133)(77 236 138 127)(78 263 361 357)(79 264 362 351)(80 265 363 352)(81 266 364 353)(82 260 358 354)(83 261 359 355)(84 262 360 356)(85 204 106 102)(86 205 107 103)(87 206 108 104)(88 207 109 105)(89 208 110 99)(90 209 111 100)(91 210 112 101)(92 277 375 371)(93 278 376 365)(94 279 377 366)(95 280 378 367)(96 274 372 368)(97 275 373 369)(98 276 374 370)(281 379 291 389)(282 380 292 390)(283 381 293 391)(284 382 294 392)(285 383 288 386)(286 384 289 387)(287 385 290 388)
(1 368 172 274)(2 369 173 275)(3 370 174 276)(4 371 175 277)(5 365 169 278)(6 366 170 279)(7 367 171 280)(8 292 55 282)(9 293 56 283)(10 294 50 284)(11 288 51 285)(12 289 52 286)(13 290 53 287)(14 291 54 281)(15 210 297 101)(16 204 298 102)(17 205 299 103)(18 206 300 104)(19 207 301 105)(20 208 295 99)(21 209 296 100)(22 337 141 250)(23 338 142 251)(24 339 143 252)(25 340 144 246)(26 341 145 247)(27 342 146 248)(28 343 147 249)(29 224 311 115)(30 218 312 116)(31 219 313 117)(32 220 314 118)(33 221 315 119)(34 222 309 113)(35 223 310 114)(36 351 155 264)(37 352 156 265)(38 353 157 266)(39 354 158 260)(40 355 159 261)(41 356 160 262)(42 357 161 263)(43 238 325 129)(44 232 326 130)(45 233 327 131)(46 234 328 132)(47 235 329 133)(48 236 323 127)(49 237 324 128)(57 321 125 212)(58 322 126 213)(59 316 120 214)(60 317 121 215)(61 318 122 216)(62 319 123 217)(63 320 124 211)(64 245 347 151)(65 239 348 152)(66 240 349 153)(67 241 350 154)(68 242 344 148)(69 243 345 149)(70 244 346 150)(71 335 139 226)(72 336 140 227)(73 330 134 228)(74 331 135 229)(75 332 136 230)(76 333 137 231)(77 334 138 225)(78 259 361 165)(79 253 362 166)(80 254 363 167)(81 255 364 168)(82 256 358 162)(83 257 359 163)(84 258 360 164)(85 302 106 200)(86 303 107 201)(87 304 108 202)(88 305 109 203)(89 306 110 197)(90 307 111 198)(91 308 112 199)(92 273 375 179)(93 267 376 180)(94 268 377 181)(95 269 378 182)(96 270 372 176)(97 271 373 177)(98 272 374 178)(183 389 193 379)(184 390 194 380)(185 391 195 381)(186 392 196 382)(187 386 190 383)(188 387 191 384)(189 388 192 385)

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

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

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

245 conjugacy classes

 class 1 2 4A 4B 4C 7A ··· 7AV 14A ··· 14AV 28A ··· 28EN order 1 2 4 4 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2

245 irreducible representations

 dim 1 1 1 1 2 2 type + + - image C1 C2 C7 C14 Q8 C7×Q8 kernel Q8×C72 C7×C28 C7×Q8 C28 C72 C7 # reps 1 3 48 144 1 48

Matrix representation of Q8×C72 in GL3(𝔽29) generated by

 16 0 0 0 20 0 0 0 20
,
 24 0 0 0 24 0 0 0 24
,
 1 0 0 0 0 1 0 28 0
,
 1 0 0 0 11 20 0 20 18
G:=sub<GL(3,GF(29))| [16,0,0,0,20,0,0,0,20],[24,0,0,0,24,0,0,0,24],[1,0,0,0,0,28,0,1,0],[1,0,0,0,11,20,0,20,18] >;

Q8×C72 in GAP, Magma, Sage, TeX

Q_8\times C_7^2
% in TeX

G:=Group("Q8xC7^2");
// GroupNames label

G:=SmallGroup(392,35);
// by ID

G=gap.SmallGroup(392,35);
# by ID

G:=PCGroup([5,-2,-2,-7,-7,-2,980,1981,986]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^7=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽