metabelian, supersoluble, monomial
Aliases: C72⋊4Q8, C28.3D7, C7⋊2Dic14, C14.12D14, C4.(C7⋊D7), (C7×C28).1C2, C7⋊Dic7.3C2, (C7×C14).11C22, C2.3(C2×C7⋊D7), SmallGroup(392,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊4Q8
G = < a,b,c,d | a7=b7=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 348 in 60 conjugacy classes, 33 normal (7 characteristic)
C1, C2, C4, C4, C7, Q8, C14, Dic7, C28, C72, Dic14, C7×C14, C7⋊Dic7, C7×C28, C72⋊4Q8
Quotients: C1, C2, C22, Q8, D7, D14, Dic14, C7⋊D7, C2×C7⋊D7, C72⋊4Q8
►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 90 68 75 82 54 61)(2 91 69 76 83 55 62)(3 85 70 77 84 56 63)(4 86 64 71 78 50 57)(5 87 65 72 79 51 58)(6 88 66 73 80 52 59)(7 89 67 74 81 53 60)(8 15 392 49 22 29 36)(9 16 386 43 23 30 37)(10 17 387 44 24 31 38)(11 18 388 45 25 32 39)(12 19 389 46 26 33 40)(13 20 390 47 27 34 41)(14 21 391 48 28 35 42)(92 135 113 120 127 99 106)(93 136 114 121 128 100 107)(94 137 115 122 129 101 108)(95 138 116 123 130 102 109)(96 139 117 124 131 103 110)(97 140 118 125 132 104 111)(98 134 119 126 133 105 112)(141 184 162 169 176 148 155)(142 185 163 170 177 149 156)(143 186 164 171 178 150 157)(144 187 165 172 179 151 158)(145 188 166 173 180 152 159)(146 189 167 174 181 153 160)(147 183 168 175 182 154 161)(190 198 218 211 204 232 225)(191 199 219 212 205 233 226)(192 200 220 213 206 234 227)(193 201 221 214 207 235 228)(194 202 222 215 208 236 229)(195 203 223 216 209 237 230)(196 197 224 217 210 238 231)(239 252 267 260 253 281 274)(240 246 268 261 254 282 275)(241 247 269 262 255 283 276)(242 248 270 263 256 284 277)(243 249 271 264 257 285 278)(244 250 272 265 258 286 279)(245 251 273 266 259 287 280)(288 301 316 309 302 330 323)(289 295 317 310 303 331 324)(290 296 318 311 304 332 325)(291 297 319 312 305 333 326)(292 298 320 313 306 334 327)(293 299 321 314 307 335 328)(294 300 322 315 308 336 329)(337 350 365 358 351 379 372)(338 344 366 359 352 380 373)(339 345 367 360 353 381 374)(340 346 368 361 354 382 375)(341 347 369 362 355 383 376)(342 348 370 363 356 384 377)(343 349 371 364 357 385 378)
(1 194 96 145)(2 195 97 146)(3 196 98 147)(4 190 92 141)(5 191 93 142)(6 192 94 143)(7 193 95 144)(8 287 385 336)(9 281 379 330)(10 282 380 331)(11 283 381 332)(12 284 382 333)(13 285 383 334)(14 286 384 335)(15 280 378 329)(16 274 372 323)(17 275 373 324)(18 276 374 325)(19 277 375 326)(20 278 376 327)(21 279 377 328)(22 273 371 322)(23 267 365 316)(24 268 366 317)(25 269 367 318)(26 270 368 319)(27 271 369 320)(28 272 370 321)(29 266 364 315)(30 260 358 309)(31 261 359 310)(32 262 360 311)(33 263 361 312)(34 264 362 313)(35 265 363 314)(36 259 357 308)(37 253 351 302)(38 254 352 303)(39 255 353 304)(40 256 354 305)(41 257 355 306)(42 258 356 307)(43 252 350 301)(44 246 344 295)(45 247 345 296)(46 248 346 297)(47 249 347 298)(48 250 348 299)(49 251 349 300)(50 232 99 148)(51 233 100 149)(52 234 101 150)(53 235 102 151)(54 236 103 152)(55 237 104 153)(56 238 105 154)(57 225 106 155)(58 226 107 156)(59 227 108 157)(60 228 109 158)(61 229 110 159)(62 230 111 160)(63 231 112 161)(64 218 113 162)(65 219 114 163)(66 220 115 164)(67 221 116 165)(68 222 117 166)(69 223 118 167)(70 224 119 168)(71 211 120 169)(72 212 121 170)(73 213 122 171)(74 214 123 172)(75 215 124 173)(76 216 125 174)(77 217 126 175)(78 204 127 176)(79 205 128 177)(80 206 129 178)(81 207 130 179)(82 208 131 180)(83 209 132 181)(84 210 133 182)(85 197 134 183)(86 198 135 184)(87 199 136 185)(88 200 137 186)(89 201 138 187)(90 202 139 188)(91 203 140 189)(239 337 288 386)(240 338 289 387)(241 339 290 388)(242 340 291 389)(243 341 292 390)(244 342 293 391)(245 343 294 392)
(1 317 96 268)(2 316 97 267)(3 322 98 273)(4 321 92 272)(5 320 93 271)(6 319 94 270)(7 318 95 269)(8 210 385 182)(9 209 379 181)(10 208 380 180)(11 207 381 179)(12 206 382 178)(13 205 383 177)(14 204 384 176)(15 217 378 175)(16 216 372 174)(17 215 373 173)(18 214 374 172)(19 213 375 171)(20 212 376 170)(21 211 377 169)(22 196 371 147)(23 195 365 146)(24 194 366 145)(25 193 367 144)(26 192 368 143)(27 191 369 142)(28 190 370 141)(29 231 364 161)(30 230 358 160)(31 229 359 159)(32 228 360 158)(33 227 361 157)(34 226 362 156)(35 225 363 155)(36 238 357 154)(37 237 351 153)(38 236 352 152)(39 235 353 151)(40 234 354 150)(41 233 355 149)(42 232 356 148)(43 203 350 189)(44 202 344 188)(45 201 345 187)(46 200 346 186)(47 199 347 185)(48 198 348 184)(49 197 349 183)(50 307 99 258)(51 306 100 257)(52 305 101 256)(53 304 102 255)(54 303 103 254)(55 302 104 253)(56 308 105 259)(57 314 106 265)(58 313 107 264)(59 312 108 263)(60 311 109 262)(61 310 110 261)(62 309 111 260)(63 315 112 266)(64 293 113 244)(65 292 114 243)(66 291 115 242)(67 290 116 241)(68 289 117 240)(69 288 118 239)(70 294 119 245)(71 328 120 279)(72 327 121 278)(73 326 122 277)(74 325 123 276)(75 324 124 275)(76 323 125 274)(77 329 126 280)(78 335 127 286)(79 334 128 285)(80 333 129 284)(81 332 130 283)(82 331 131 282)(83 330 132 281)(84 336 133 287)(85 300 134 251)(86 299 135 250)(87 298 136 249)(88 297 137 248)(89 296 138 247)(90 295 139 246)(91 301 140 252)(162 391 218 342)(163 390 219 341)(164 389 220 340)(165 388 221 339)(166 387 222 338)(167 386 223 337)(168 392 224 343)
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), (1,90,68,75,82,54,61)(2,91,69,76,83,55,62)(3,85,70,77,84,56,63)(4,86,64,71,78,50,57)(5,87,65,72,79,51,58)(6,88,66,73,80,52,59)(7,89,67,74,81,53,60)(8,15,392,49,22,29,36)(9,16,386,43,23,30,37)(10,17,387,44,24,31,38)(11,18,388,45,25,32,39)(12,19,389,46,26,33,40)(13,20,390,47,27,34,41)(14,21,391,48,28,35,42)(92,135,113,120,127,99,106)(93,136,114,121,128,100,107)(94,137,115,122,129,101,108)(95,138,116,123,130,102,109)(96,139,117,124,131,103,110)(97,140,118,125,132,104,111)(98,134,119,126,133,105,112)(141,184,162,169,176,148,155)(142,185,163,170,177,149,156)(143,186,164,171,178,150,157)(144,187,165,172,179,151,158)(145,188,166,173,180,152,159)(146,189,167,174,181,153,160)(147,183,168,175,182,154,161)(190,198,218,211,204,232,225)(191,199,219,212,205,233,226)(192,200,220,213,206,234,227)(193,201,221,214,207,235,228)(194,202,222,215,208,236,229)(195,203,223,216,209,237,230)(196,197,224,217,210,238,231)(239,252,267,260,253,281,274)(240,246,268,261,254,282,275)(241,247,269,262,255,283,276)(242,248,270,263,256,284,277)(243,249,271,264,257,285,278)(244,250,272,265,258,286,279)(245,251,273,266,259,287,280)(288,301,316,309,302,330,323)(289,295,317,310,303,331,324)(290,296,318,311,304,332,325)(291,297,319,312,305,333,326)(292,298,320,313,306,334,327)(293,299,321,314,307,335,328)(294,300,322,315,308,336,329)(337,350,365,358,351,379,372)(338,344,366,359,352,380,373)(339,345,367,360,353,381,374)(340,346,368,361,354,382,375)(341,347,369,362,355,383,376)(342,348,370,363,356,384,377)(343,349,371,364,357,385,378), (1,194,96,145)(2,195,97,146)(3,196,98,147)(4,190,92,141)(5,191,93,142)(6,192,94,143)(7,193,95,144)(8,287,385,336)(9,281,379,330)(10,282,380,331)(11,283,381,332)(12,284,382,333)(13,285,383,334)(14,286,384,335)(15,280,378,329)(16,274,372,323)(17,275,373,324)(18,276,374,325)(19,277,375,326)(20,278,376,327)(21,279,377,328)(22,273,371,322)(23,267,365,316)(24,268,366,317)(25,269,367,318)(26,270,368,319)(27,271,369,320)(28,272,370,321)(29,266,364,315)(30,260,358,309)(31,261,359,310)(32,262,360,311)(33,263,361,312)(34,264,362,313)(35,265,363,314)(36,259,357,308)(37,253,351,302)(38,254,352,303)(39,255,353,304)(40,256,354,305)(41,257,355,306)(42,258,356,307)(43,252,350,301)(44,246,344,295)(45,247,345,296)(46,248,346,297)(47,249,347,298)(48,250,348,299)(49,251,349,300)(50,232,99,148)(51,233,100,149)(52,234,101,150)(53,235,102,151)(54,236,103,152)(55,237,104,153)(56,238,105,154)(57,225,106,155)(58,226,107,156)(59,227,108,157)(60,228,109,158)(61,229,110,159)(62,230,111,160)(63,231,112,161)(64,218,113,162)(65,219,114,163)(66,220,115,164)(67,221,116,165)(68,222,117,166)(69,223,118,167)(70,224,119,168)(71,211,120,169)(72,212,121,170)(73,213,122,171)(74,214,123,172)(75,215,124,173)(76,216,125,174)(77,217,126,175)(78,204,127,176)(79,205,128,177)(80,206,129,178)(81,207,130,179)(82,208,131,180)(83,209,132,181)(84,210,133,182)(85,197,134,183)(86,198,135,184)(87,199,136,185)(88,200,137,186)(89,201,138,187)(90,202,139,188)(91,203,140,189)(239,337,288,386)(240,338,289,387)(241,339,290,388)(242,340,291,389)(243,341,292,390)(244,342,293,391)(245,343,294,392), (1,317,96,268)(2,316,97,267)(3,322,98,273)(4,321,92,272)(5,320,93,271)(6,319,94,270)(7,318,95,269)(8,210,385,182)(9,209,379,181)(10,208,380,180)(11,207,381,179)(12,206,382,178)(13,205,383,177)(14,204,384,176)(15,217,378,175)(16,216,372,174)(17,215,373,173)(18,214,374,172)(19,213,375,171)(20,212,376,170)(21,211,377,169)(22,196,371,147)(23,195,365,146)(24,194,366,145)(25,193,367,144)(26,192,368,143)(27,191,369,142)(28,190,370,141)(29,231,364,161)(30,230,358,160)(31,229,359,159)(32,228,360,158)(33,227,361,157)(34,226,362,156)(35,225,363,155)(36,238,357,154)(37,237,351,153)(38,236,352,152)(39,235,353,151)(40,234,354,150)(41,233,355,149)(42,232,356,148)(43,203,350,189)(44,202,344,188)(45,201,345,187)(46,200,346,186)(47,199,347,185)(48,198,348,184)(49,197,349,183)(50,307,99,258)(51,306,100,257)(52,305,101,256)(53,304,102,255)(54,303,103,254)(55,302,104,253)(56,308,105,259)(57,314,106,265)(58,313,107,264)(59,312,108,263)(60,311,109,262)(61,310,110,261)(62,309,111,260)(63,315,112,266)(64,293,113,244)(65,292,114,243)(66,291,115,242)(67,290,116,241)(68,289,117,240)(69,288,118,239)(70,294,119,245)(71,328,120,279)(72,327,121,278)(73,326,122,277)(74,325,123,276)(75,324,124,275)(76,323,125,274)(77,329,126,280)(78,335,127,286)(79,334,128,285)(80,333,129,284)(81,332,130,283)(82,331,131,282)(83,330,132,281)(84,336,133,287)(85,300,134,251)(86,299,135,250)(87,298,136,249)(88,297,137,248)(89,296,138,247)(90,295,139,246)(91,301,140,252)(162,391,218,342)(163,390,219,341)(164,389,220,340)(165,388,221,339)(166,387,222,338)(167,386,223,337)(168,392,224,343)>;
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), (1,90,68,75,82,54,61)(2,91,69,76,83,55,62)(3,85,70,77,84,56,63)(4,86,64,71,78,50,57)(5,87,65,72,79,51,58)(6,88,66,73,80,52,59)(7,89,67,74,81,53,60)(8,15,392,49,22,29,36)(9,16,386,43,23,30,37)(10,17,387,44,24,31,38)(11,18,388,45,25,32,39)(12,19,389,46,26,33,40)(13,20,390,47,27,34,41)(14,21,391,48,28,35,42)(92,135,113,120,127,99,106)(93,136,114,121,128,100,107)(94,137,115,122,129,101,108)(95,138,116,123,130,102,109)(96,139,117,124,131,103,110)(97,140,118,125,132,104,111)(98,134,119,126,133,105,112)(141,184,162,169,176,148,155)(142,185,163,170,177,149,156)(143,186,164,171,178,150,157)(144,187,165,172,179,151,158)(145,188,166,173,180,152,159)(146,189,167,174,181,153,160)(147,183,168,175,182,154,161)(190,198,218,211,204,232,225)(191,199,219,212,205,233,226)(192,200,220,213,206,234,227)(193,201,221,214,207,235,228)(194,202,222,215,208,236,229)(195,203,223,216,209,237,230)(196,197,224,217,210,238,231)(239,252,267,260,253,281,274)(240,246,268,261,254,282,275)(241,247,269,262,255,283,276)(242,248,270,263,256,284,277)(243,249,271,264,257,285,278)(244,250,272,265,258,286,279)(245,251,273,266,259,287,280)(288,301,316,309,302,330,323)(289,295,317,310,303,331,324)(290,296,318,311,304,332,325)(291,297,319,312,305,333,326)(292,298,320,313,306,334,327)(293,299,321,314,307,335,328)(294,300,322,315,308,336,329)(337,350,365,358,351,379,372)(338,344,366,359,352,380,373)(339,345,367,360,353,381,374)(340,346,368,361,354,382,375)(341,347,369,362,355,383,376)(342,348,370,363,356,384,377)(343,349,371,364,357,385,378), (1,194,96,145)(2,195,97,146)(3,196,98,147)(4,190,92,141)(5,191,93,142)(6,192,94,143)(7,193,95,144)(8,287,385,336)(9,281,379,330)(10,282,380,331)(11,283,381,332)(12,284,382,333)(13,285,383,334)(14,286,384,335)(15,280,378,329)(16,274,372,323)(17,275,373,324)(18,276,374,325)(19,277,375,326)(20,278,376,327)(21,279,377,328)(22,273,371,322)(23,267,365,316)(24,268,366,317)(25,269,367,318)(26,270,368,319)(27,271,369,320)(28,272,370,321)(29,266,364,315)(30,260,358,309)(31,261,359,310)(32,262,360,311)(33,263,361,312)(34,264,362,313)(35,265,363,314)(36,259,357,308)(37,253,351,302)(38,254,352,303)(39,255,353,304)(40,256,354,305)(41,257,355,306)(42,258,356,307)(43,252,350,301)(44,246,344,295)(45,247,345,296)(46,248,346,297)(47,249,347,298)(48,250,348,299)(49,251,349,300)(50,232,99,148)(51,233,100,149)(52,234,101,150)(53,235,102,151)(54,236,103,152)(55,237,104,153)(56,238,105,154)(57,225,106,155)(58,226,107,156)(59,227,108,157)(60,228,109,158)(61,229,110,159)(62,230,111,160)(63,231,112,161)(64,218,113,162)(65,219,114,163)(66,220,115,164)(67,221,116,165)(68,222,117,166)(69,223,118,167)(70,224,119,168)(71,211,120,169)(72,212,121,170)(73,213,122,171)(74,214,123,172)(75,215,124,173)(76,216,125,174)(77,217,126,175)(78,204,127,176)(79,205,128,177)(80,206,129,178)(81,207,130,179)(82,208,131,180)(83,209,132,181)(84,210,133,182)(85,197,134,183)(86,198,135,184)(87,199,136,185)(88,200,137,186)(89,201,138,187)(90,202,139,188)(91,203,140,189)(239,337,288,386)(240,338,289,387)(241,339,290,388)(242,340,291,389)(243,341,292,390)(244,342,293,391)(245,343,294,392), (1,317,96,268)(2,316,97,267)(3,322,98,273)(4,321,92,272)(5,320,93,271)(6,319,94,270)(7,318,95,269)(8,210,385,182)(9,209,379,181)(10,208,380,180)(11,207,381,179)(12,206,382,178)(13,205,383,177)(14,204,384,176)(15,217,378,175)(16,216,372,174)(17,215,373,173)(18,214,374,172)(19,213,375,171)(20,212,376,170)(21,211,377,169)(22,196,371,147)(23,195,365,146)(24,194,366,145)(25,193,367,144)(26,192,368,143)(27,191,369,142)(28,190,370,141)(29,231,364,161)(30,230,358,160)(31,229,359,159)(32,228,360,158)(33,227,361,157)(34,226,362,156)(35,225,363,155)(36,238,357,154)(37,237,351,153)(38,236,352,152)(39,235,353,151)(40,234,354,150)(41,233,355,149)(42,232,356,148)(43,203,350,189)(44,202,344,188)(45,201,345,187)(46,200,346,186)(47,199,347,185)(48,198,348,184)(49,197,349,183)(50,307,99,258)(51,306,100,257)(52,305,101,256)(53,304,102,255)(54,303,103,254)(55,302,104,253)(56,308,105,259)(57,314,106,265)(58,313,107,264)(59,312,108,263)(60,311,109,262)(61,310,110,261)(62,309,111,260)(63,315,112,266)(64,293,113,244)(65,292,114,243)(66,291,115,242)(67,290,116,241)(68,289,117,240)(69,288,118,239)(70,294,119,245)(71,328,120,279)(72,327,121,278)(73,326,122,277)(74,325,123,276)(75,324,124,275)(76,323,125,274)(77,329,126,280)(78,335,127,286)(79,334,128,285)(80,333,129,284)(81,332,130,283)(82,331,131,282)(83,330,132,281)(84,336,133,287)(85,300,134,251)(86,299,135,250)(87,298,136,249)(88,297,137,248)(89,296,138,247)(90,295,139,246)(91,301,140,252)(162,391,218,342)(163,390,219,341)(164,389,220,340)(165,388,221,339)(166,387,222,338)(167,386,223,337)(168,392,224,343) );
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)], [(1,90,68,75,82,54,61),(2,91,69,76,83,55,62),(3,85,70,77,84,56,63),(4,86,64,71,78,50,57),(5,87,65,72,79,51,58),(6,88,66,73,80,52,59),(7,89,67,74,81,53,60),(8,15,392,49,22,29,36),(9,16,386,43,23,30,37),(10,17,387,44,24,31,38),(11,18,388,45,25,32,39),(12,19,389,46,26,33,40),(13,20,390,47,27,34,41),(14,21,391,48,28,35,42),(92,135,113,120,127,99,106),(93,136,114,121,128,100,107),(94,137,115,122,129,101,108),(95,138,116,123,130,102,109),(96,139,117,124,131,103,110),(97,140,118,125,132,104,111),(98,134,119,126,133,105,112),(141,184,162,169,176,148,155),(142,185,163,170,177,149,156),(143,186,164,171,178,150,157),(144,187,165,172,179,151,158),(145,188,166,173,180,152,159),(146,189,167,174,181,153,160),(147,183,168,175,182,154,161),(190,198,218,211,204,232,225),(191,199,219,212,205,233,226),(192,200,220,213,206,234,227),(193,201,221,214,207,235,228),(194,202,222,215,208,236,229),(195,203,223,216,209,237,230),(196,197,224,217,210,238,231),(239,252,267,260,253,281,274),(240,246,268,261,254,282,275),(241,247,269,262,255,283,276),(242,248,270,263,256,284,277),(243,249,271,264,257,285,278),(244,250,272,265,258,286,279),(245,251,273,266,259,287,280),(288,301,316,309,302,330,323),(289,295,317,310,303,331,324),(290,296,318,311,304,332,325),(291,297,319,312,305,333,326),(292,298,320,313,306,334,327),(293,299,321,314,307,335,328),(294,300,322,315,308,336,329),(337,350,365,358,351,379,372),(338,344,366,359,352,380,373),(339,345,367,360,353,381,374),(340,346,368,361,354,382,375),(341,347,369,362,355,383,376),(342,348,370,363,356,384,377),(343,349,371,364,357,385,378)], [(1,194,96,145),(2,195,97,146),(3,196,98,147),(4,190,92,141),(5,191,93,142),(6,192,94,143),(7,193,95,144),(8,287,385,336),(9,281,379,330),(10,282,380,331),(11,283,381,332),(12,284,382,333),(13,285,383,334),(14,286,384,335),(15,280,378,329),(16,274,372,323),(17,275,373,324),(18,276,374,325),(19,277,375,326),(20,278,376,327),(21,279,377,328),(22,273,371,322),(23,267,365,316),(24,268,366,317),(25,269,367,318),(26,270,368,319),(27,271,369,320),(28,272,370,321),(29,266,364,315),(30,260,358,309),(31,261,359,310),(32,262,360,311),(33,263,361,312),(34,264,362,313),(35,265,363,314),(36,259,357,308),(37,253,351,302),(38,254,352,303),(39,255,353,304),(40,256,354,305),(41,257,355,306),(42,258,356,307),(43,252,350,301),(44,246,344,295),(45,247,345,296),(46,248,346,297),(47,249,347,298),(48,250,348,299),(49,251,349,300),(50,232,99,148),(51,233,100,149),(52,234,101,150),(53,235,102,151),(54,236,103,152),(55,237,104,153),(56,238,105,154),(57,225,106,155),(58,226,107,156),(59,227,108,157),(60,228,109,158),(61,229,110,159),(62,230,111,160),(63,231,112,161),(64,218,113,162),(65,219,114,163),(66,220,115,164),(67,221,116,165),(68,222,117,166),(69,223,118,167),(70,224,119,168),(71,211,120,169),(72,212,121,170),(73,213,122,171),(74,214,123,172),(75,215,124,173),(76,216,125,174),(77,217,126,175),(78,204,127,176),(79,205,128,177),(80,206,129,178),(81,207,130,179),(82,208,131,180),(83,209,132,181),(84,210,133,182),(85,197,134,183),(86,198,135,184),(87,199,136,185),(88,200,137,186),(89,201,138,187),(90,202,139,188),(91,203,140,189),(239,337,288,386),(240,338,289,387),(241,339,290,388),(242,340,291,389),(243,341,292,390),(244,342,293,391),(245,343,294,392)], [(1,317,96,268),(2,316,97,267),(3,322,98,273),(4,321,92,272),(5,320,93,271),(6,319,94,270),(7,318,95,269),(8,210,385,182),(9,209,379,181),(10,208,380,180),(11,207,381,179),(12,206,382,178),(13,205,383,177),(14,204,384,176),(15,217,378,175),(16,216,372,174),(17,215,373,173),(18,214,374,172),(19,213,375,171),(20,212,376,170),(21,211,377,169),(22,196,371,147),(23,195,365,146),(24,194,366,145),(25,193,367,144),(26,192,368,143),(27,191,369,142),(28,190,370,141),(29,231,364,161),(30,230,358,160),(31,229,359,159),(32,228,360,158),(33,227,361,157),(34,226,362,156),(35,225,363,155),(36,238,357,154),(37,237,351,153),(38,236,352,152),(39,235,353,151),(40,234,354,150),(41,233,355,149),(42,232,356,148),(43,203,350,189),(44,202,344,188),(45,201,345,187),(46,200,346,186),(47,199,347,185),(48,198,348,184),(49,197,349,183),(50,307,99,258),(51,306,100,257),(52,305,101,256),(53,304,102,255),(54,303,103,254),(55,302,104,253),(56,308,105,259),(57,314,106,265),(58,313,107,264),(59,312,108,263),(60,311,109,262),(61,310,110,261),(62,309,111,260),(63,315,112,266),(64,293,113,244),(65,292,114,243),(66,291,115,242),(67,290,116,241),(68,289,117,240),(69,288,118,239),(70,294,119,245),(71,328,120,279),(72,327,121,278),(73,326,122,277),(74,325,123,276),(75,324,124,275),(76,323,125,274),(77,329,126,280),(78,335,127,286),(79,334,128,285),(80,333,129,284),(81,332,130,283),(82,331,131,282),(83,330,132,281),(84,336,133,287),(85,300,134,251),(86,299,135,250),(87,298,136,249),(88,297,137,248),(89,296,138,247),(90,295,139,246),(91,301,140,252),(162,391,218,342),(163,390,219,341),(164,389,220,340),(165,388,221,339),(166,387,222,338),(167,386,223,337),(168,392,224,343)]])
101 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | ··· | 7X | 14A | ··· | 14X | 28A | ··· | 28AV |
| order | 1 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 98 | 98 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
101 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D7 | D14 | Dic14 |
| kernel | C72⋊4Q8 | C7⋊Dic7 | C7×C28 | C72 | C28 | C14 | C7 |
| # reps | 1 | 2 | 1 | 1 | 24 | 24 | 48 |
Matrix representation of C72⋊4Q8 ►in GL4(𝔽29) generated by
| 16 | 0 | 0 | 0 |
| 27 | 20 | 0 | 0 |
| 0 | 0 | 19 | 22 |
| 0 | 0 | 7 | 28 |
| 16 | 0 | 0 | 0 |
| 27 | 20 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 7 |
| 17 | 0 | 0 | 0 |
| 17 | 12 | 0 | 0 |
| 0 | 0 | 8 | 6 |
| 0 | 0 | 23 | 21 |
| 3 | 23 | 0 | 0 |
| 21 | 26 | 0 | 0 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 3 | 25 |
G:=sub<GL(4,GF(29))| [16,27,0,0,0,20,0,0,0,0,19,7,0,0,22,28],[16,27,0,0,0,20,0,0,0,0,0,28,0,0,1,7],[17,17,0,0,0,12,0,0,0,0,8,23,0,0,6,21],[3,21,0,0,23,26,0,0,0,0,4,3,0,0,4,25] >;
C72⋊4Q8 in GAP, Magma, Sage, TeX
C_7^2\rtimes_4Q_8
% in TeX
G:=Group("C7^2:4Q8"); // GroupNames label
G:=SmallGroup(392,28);
// by ID
G=gap.SmallGroup(392,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,20,61,26,963,8404]);
// 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,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations