metabelian, supersoluble, monomial, A-group
Aliases: C72⋊4C8, C28.6D7, C14.3Dic7, C7⋊(C7⋊C8), (C7×C28).4C2, (C7×C14).3C4, C4.2(C7⋊D7), C2.(C7⋊Dic7), SmallGroup(392,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C72⋊4C8 |
Generators and relations for C72⋊4C8
G = < a,b,c | a7=b7=c8=1, ab=ba, cac-1=a-1, cbc-1=b-1 >
49C8
Smallest permutation representation of C72⋊4C8
►Regular action on 392 points
Generators in S392
(1 144 328 80 264 160 207)(2 208 153 257 73 321 137)(3 138 322 74 258 154 201)(4 202 155 259 75 323 139)(5 140 324 76 260 156 203)(6 204 157 261 77 325 141)(7 142 326 78 262 158 205)(8 206 159 263 79 327 143)(9 98 330 83 268 166 348)(10 349 167 269 84 331 99)(11 100 332 85 270 168 350)(12 351 161 271 86 333 101)(13 102 334 87 272 162 352)(14 345 163 265 88 335 103)(15 104 336 81 266 164 346)(16 347 165 267 82 329 97)(17 63 290 89 279 174 358)(18 359 175 280 90 291 64)(19 57 292 91 273 176 360)(20 353 169 274 92 293 58)(21 59 294 93 275 170 354)(22 355 171 276 94 295 60)(23 61 296 95 277 172 356)(24 357 173 278 96 289 62)(25 215 105 55 287 183 367)(26 368 184 288 56 106 216)(27 209 107 49 281 177 361)(28 362 178 282 50 108 210)(29 211 109 51 283 179 363)(30 364 180 284 52 110 212)(31 213 111 53 285 181 365)(32 366 182 286 54 112 214)(33 372 190 244 302 118 218)(34 219 119 303 245 191 373)(35 374 192 246 304 120 220)(36 221 113 297 247 185 375)(37 376 186 248 298 114 222)(38 223 115 299 241 187 369)(39 370 188 242 300 116 224)(40 217 117 301 243 189 371)(41 382 150 199 308 124 225)(42 226 125 309 200 151 383)(43 384 152 193 310 126 227)(44 228 127 311 194 145 377)(45 378 146 195 312 128 229)(46 230 121 305 196 147 379)(47 380 148 197 306 122 231)(48 232 123 307 198 149 381)(65 314 129 239 392 340 255)(66 256 341 385 240 130 315)(67 316 131 233 386 342 249)(68 250 343 387 234 132 317)(69 318 133 235 388 344 251)(70 252 337 389 236 134 319)(71 320 135 237 390 338 253)(72 254 339 391 238 136 313)
(1 250 305 219 361 275 330)(2 331 276 362 220 306 251)(3 252 307 221 363 277 332)(4 333 278 364 222 308 253)(5 254 309 223 365 279 334)(6 335 280 366 224 310 255)(7 256 311 217 367 273 336)(8 329 274 368 218 312 249)(9 160 317 230 373 281 294)(10 295 282 374 231 318 153)(11 154 319 232 375 283 296)(12 289 284 376 225 320 155)(13 156 313 226 369 285 290)(14 291 286 370 227 314 157)(15 158 315 228 371 287 292)(16 293 288 372 229 316 159)(17 162 76 238 383 241 111)(18 112 242 384 239 77 163)(19 164 78 240 377 243 105)(20 106 244 378 233 79 165)(21 166 80 234 379 245 107)(22 108 246 380 235 73 167)(23 168 74 236 381 247 109)(24 110 248 382 237 75 161)(25 176 81 142 341 194 117)(26 118 195 342 143 82 169)(27 170 83 144 343 196 119)(28 120 197 344 137 84 171)(29 172 85 138 337 198 113)(30 114 199 338 139 86 173)(31 174 87 140 339 200 115)(32 116 193 340 141 88 175)(33 128 67 206 97 92 184)(34 177 93 98 207 68 121)(35 122 69 208 99 94 178)(36 179 95 100 201 70 123)(37 124 71 202 101 96 180)(38 181 89 102 203 72 125)(39 126 65 204 103 90 182)(40 183 91 104 205 66 127)(41 135 259 351 62 52 186)(42 187 53 63 352 260 136)(43 129 261 345 64 54 188)(44 189 55 57 346 262 130)(45 131 263 347 58 56 190)(46 191 49 59 348 264 132)(47 133 257 349 60 50 192)(48 185 51 61 350 258 134)(145 301 215 360 266 326 385)(146 386 327 267 353 216 302)(147 303 209 354 268 328 387)(148 388 321 269 355 210 304)(149 297 211 356 270 322 389)(150 390 323 271 357 212 298)(151 299 213 358 272 324 391)(152 392 325 265 359 214 300)
(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)
G:=sub<Sym(392)| (1,144,328,80,264,160,207)(2,208,153,257,73,321,137)(3,138,322,74,258,154,201)(4,202,155,259,75,323,139)(5,140,324,76,260,156,203)(6,204,157,261,77,325,141)(7,142,326,78,262,158,205)(8,206,159,263,79,327,143)(9,98,330,83,268,166,348)(10,349,167,269,84,331,99)(11,100,332,85,270,168,350)(12,351,161,271,86,333,101)(13,102,334,87,272,162,352)(14,345,163,265,88,335,103)(15,104,336,81,266,164,346)(16,347,165,267,82,329,97)(17,63,290,89,279,174,358)(18,359,175,280,90,291,64)(19,57,292,91,273,176,360)(20,353,169,274,92,293,58)(21,59,294,93,275,170,354)(22,355,171,276,94,295,60)(23,61,296,95,277,172,356)(24,357,173,278,96,289,62)(25,215,105,55,287,183,367)(26,368,184,288,56,106,216)(27,209,107,49,281,177,361)(28,362,178,282,50,108,210)(29,211,109,51,283,179,363)(30,364,180,284,52,110,212)(31,213,111,53,285,181,365)(32,366,182,286,54,112,214)(33,372,190,244,302,118,218)(34,219,119,303,245,191,373)(35,374,192,246,304,120,220)(36,221,113,297,247,185,375)(37,376,186,248,298,114,222)(38,223,115,299,241,187,369)(39,370,188,242,300,116,224)(40,217,117,301,243,189,371)(41,382,150,199,308,124,225)(42,226,125,309,200,151,383)(43,384,152,193,310,126,227)(44,228,127,311,194,145,377)(45,378,146,195,312,128,229)(46,230,121,305,196,147,379)(47,380,148,197,306,122,231)(48,232,123,307,198,149,381)(65,314,129,239,392,340,255)(66,256,341,385,240,130,315)(67,316,131,233,386,342,249)(68,250,343,387,234,132,317)(69,318,133,235,388,344,251)(70,252,337,389,236,134,319)(71,320,135,237,390,338,253)(72,254,339,391,238,136,313), (1,250,305,219,361,275,330)(2,331,276,362,220,306,251)(3,252,307,221,363,277,332)(4,333,278,364,222,308,253)(5,254,309,223,365,279,334)(6,335,280,366,224,310,255)(7,256,311,217,367,273,336)(8,329,274,368,218,312,249)(9,160,317,230,373,281,294)(10,295,282,374,231,318,153)(11,154,319,232,375,283,296)(12,289,284,376,225,320,155)(13,156,313,226,369,285,290)(14,291,286,370,227,314,157)(15,158,315,228,371,287,292)(16,293,288,372,229,316,159)(17,162,76,238,383,241,111)(18,112,242,384,239,77,163)(19,164,78,240,377,243,105)(20,106,244,378,233,79,165)(21,166,80,234,379,245,107)(22,108,246,380,235,73,167)(23,168,74,236,381,247,109)(24,110,248,382,237,75,161)(25,176,81,142,341,194,117)(26,118,195,342,143,82,169)(27,170,83,144,343,196,119)(28,120,197,344,137,84,171)(29,172,85,138,337,198,113)(30,114,199,338,139,86,173)(31,174,87,140,339,200,115)(32,116,193,340,141,88,175)(33,128,67,206,97,92,184)(34,177,93,98,207,68,121)(35,122,69,208,99,94,178)(36,179,95,100,201,70,123)(37,124,71,202,101,96,180)(38,181,89,102,203,72,125)(39,126,65,204,103,90,182)(40,183,91,104,205,66,127)(41,135,259,351,62,52,186)(42,187,53,63,352,260,136)(43,129,261,345,64,54,188)(44,189,55,57,346,262,130)(45,131,263,347,58,56,190)(46,191,49,59,348,264,132)(47,133,257,349,60,50,192)(48,185,51,61,350,258,134)(145,301,215,360,266,326,385)(146,386,327,267,353,216,302)(147,303,209,354,268,328,387)(148,388,321,269,355,210,304)(149,297,211,356,270,322,389)(150,390,323,271,357,212,298)(151,299,213,358,272,324,391)(152,392,325,265,359,214,300), (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)>;
G:=Group( (1,144,328,80,264,160,207)(2,208,153,257,73,321,137)(3,138,322,74,258,154,201)(4,202,155,259,75,323,139)(5,140,324,76,260,156,203)(6,204,157,261,77,325,141)(7,142,326,78,262,158,205)(8,206,159,263,79,327,143)(9,98,330,83,268,166,348)(10,349,167,269,84,331,99)(11,100,332,85,270,168,350)(12,351,161,271,86,333,101)(13,102,334,87,272,162,352)(14,345,163,265,88,335,103)(15,104,336,81,266,164,346)(16,347,165,267,82,329,97)(17,63,290,89,279,174,358)(18,359,175,280,90,291,64)(19,57,292,91,273,176,360)(20,353,169,274,92,293,58)(21,59,294,93,275,170,354)(22,355,171,276,94,295,60)(23,61,296,95,277,172,356)(24,357,173,278,96,289,62)(25,215,105,55,287,183,367)(26,368,184,288,56,106,216)(27,209,107,49,281,177,361)(28,362,178,282,50,108,210)(29,211,109,51,283,179,363)(30,364,180,284,52,110,212)(31,213,111,53,285,181,365)(32,366,182,286,54,112,214)(33,372,190,244,302,118,218)(34,219,119,303,245,191,373)(35,374,192,246,304,120,220)(36,221,113,297,247,185,375)(37,376,186,248,298,114,222)(38,223,115,299,241,187,369)(39,370,188,242,300,116,224)(40,217,117,301,243,189,371)(41,382,150,199,308,124,225)(42,226,125,309,200,151,383)(43,384,152,193,310,126,227)(44,228,127,311,194,145,377)(45,378,146,195,312,128,229)(46,230,121,305,196,147,379)(47,380,148,197,306,122,231)(48,232,123,307,198,149,381)(65,314,129,239,392,340,255)(66,256,341,385,240,130,315)(67,316,131,233,386,342,249)(68,250,343,387,234,132,317)(69,318,133,235,388,344,251)(70,252,337,389,236,134,319)(71,320,135,237,390,338,253)(72,254,339,391,238,136,313), (1,250,305,219,361,275,330)(2,331,276,362,220,306,251)(3,252,307,221,363,277,332)(4,333,278,364,222,308,253)(5,254,309,223,365,279,334)(6,335,280,366,224,310,255)(7,256,311,217,367,273,336)(8,329,274,368,218,312,249)(9,160,317,230,373,281,294)(10,295,282,374,231,318,153)(11,154,319,232,375,283,296)(12,289,284,376,225,320,155)(13,156,313,226,369,285,290)(14,291,286,370,227,314,157)(15,158,315,228,371,287,292)(16,293,288,372,229,316,159)(17,162,76,238,383,241,111)(18,112,242,384,239,77,163)(19,164,78,240,377,243,105)(20,106,244,378,233,79,165)(21,166,80,234,379,245,107)(22,108,246,380,235,73,167)(23,168,74,236,381,247,109)(24,110,248,382,237,75,161)(25,176,81,142,341,194,117)(26,118,195,342,143,82,169)(27,170,83,144,343,196,119)(28,120,197,344,137,84,171)(29,172,85,138,337,198,113)(30,114,199,338,139,86,173)(31,174,87,140,339,200,115)(32,116,193,340,141,88,175)(33,128,67,206,97,92,184)(34,177,93,98,207,68,121)(35,122,69,208,99,94,178)(36,179,95,100,201,70,123)(37,124,71,202,101,96,180)(38,181,89,102,203,72,125)(39,126,65,204,103,90,182)(40,183,91,104,205,66,127)(41,135,259,351,62,52,186)(42,187,53,63,352,260,136)(43,129,261,345,64,54,188)(44,189,55,57,346,262,130)(45,131,263,347,58,56,190)(46,191,49,59,348,264,132)(47,133,257,349,60,50,192)(48,185,51,61,350,258,134)(145,301,215,360,266,326,385)(146,386,327,267,353,216,302)(147,303,209,354,268,328,387)(148,388,321,269,355,210,304)(149,297,211,356,270,322,389)(150,390,323,271,357,212,298)(151,299,213,358,272,324,391)(152,392,325,265,359,214,300), (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) );
G=PermutationGroup([[(1,144,328,80,264,160,207),(2,208,153,257,73,321,137),(3,138,322,74,258,154,201),(4,202,155,259,75,323,139),(5,140,324,76,260,156,203),(6,204,157,261,77,325,141),(7,142,326,78,262,158,205),(8,206,159,263,79,327,143),(9,98,330,83,268,166,348),(10,349,167,269,84,331,99),(11,100,332,85,270,168,350),(12,351,161,271,86,333,101),(13,102,334,87,272,162,352),(14,345,163,265,88,335,103),(15,104,336,81,266,164,346),(16,347,165,267,82,329,97),(17,63,290,89,279,174,358),(18,359,175,280,90,291,64),(19,57,292,91,273,176,360),(20,353,169,274,92,293,58),(21,59,294,93,275,170,354),(22,355,171,276,94,295,60),(23,61,296,95,277,172,356),(24,357,173,278,96,289,62),(25,215,105,55,287,183,367),(26,368,184,288,56,106,216),(27,209,107,49,281,177,361),(28,362,178,282,50,108,210),(29,211,109,51,283,179,363),(30,364,180,284,52,110,212),(31,213,111,53,285,181,365),(32,366,182,286,54,112,214),(33,372,190,244,302,118,218),(34,219,119,303,245,191,373),(35,374,192,246,304,120,220),(36,221,113,297,247,185,375),(37,376,186,248,298,114,222),(38,223,115,299,241,187,369),(39,370,188,242,300,116,224),(40,217,117,301,243,189,371),(41,382,150,199,308,124,225),(42,226,125,309,200,151,383),(43,384,152,193,310,126,227),(44,228,127,311,194,145,377),(45,378,146,195,312,128,229),(46,230,121,305,196,147,379),(47,380,148,197,306,122,231),(48,232,123,307,198,149,381),(65,314,129,239,392,340,255),(66,256,341,385,240,130,315),(67,316,131,233,386,342,249),(68,250,343,387,234,132,317),(69,318,133,235,388,344,251),(70,252,337,389,236,134,319),(71,320,135,237,390,338,253),(72,254,339,391,238,136,313)], [(1,250,305,219,361,275,330),(2,331,276,362,220,306,251),(3,252,307,221,363,277,332),(4,333,278,364,222,308,253),(5,254,309,223,365,279,334),(6,335,280,366,224,310,255),(7,256,311,217,367,273,336),(8,329,274,368,218,312,249),(9,160,317,230,373,281,294),(10,295,282,374,231,318,153),(11,154,319,232,375,283,296),(12,289,284,376,225,320,155),(13,156,313,226,369,285,290),(14,291,286,370,227,314,157),(15,158,315,228,371,287,292),(16,293,288,372,229,316,159),(17,162,76,238,383,241,111),(18,112,242,384,239,77,163),(19,164,78,240,377,243,105),(20,106,244,378,233,79,165),(21,166,80,234,379,245,107),(22,108,246,380,235,73,167),(23,168,74,236,381,247,109),(24,110,248,382,237,75,161),(25,176,81,142,341,194,117),(26,118,195,342,143,82,169),(27,170,83,144,343,196,119),(28,120,197,344,137,84,171),(29,172,85,138,337,198,113),(30,114,199,338,139,86,173),(31,174,87,140,339,200,115),(32,116,193,340,141,88,175),(33,128,67,206,97,92,184),(34,177,93,98,207,68,121),(35,122,69,208,99,94,178),(36,179,95,100,201,70,123),(37,124,71,202,101,96,180),(38,181,89,102,203,72,125),(39,126,65,204,103,90,182),(40,183,91,104,205,66,127),(41,135,259,351,62,52,186),(42,187,53,63,352,260,136),(43,129,261,345,64,54,188),(44,189,55,57,346,262,130),(45,131,263,347,58,56,190),(46,191,49,59,348,264,132),(47,133,257,349,60,50,192),(48,185,51,61,350,258,134),(145,301,215,360,266,326,385),(146,386,327,267,353,216,302),(147,303,209,354,268,328,387),(148,388,321,269,355,210,304),(149,297,211,356,270,322,389),(150,390,323,271,357,212,298),(151,299,213,358,272,324,391),(152,392,325,265,359,214,300)], [(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)]])
104 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7X | 8A | 8B | 8C | 8D | 14A | ··· | 14X | 28A | ··· | 28AV |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 49 | 49 | 49 | 49 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D7 | Dic7 | C7⋊C8 |
| kernel | C72⋊4C8 | C7×C28 | C7×C14 | C72 | C28 | C14 | C7 |
| # reps | 1 | 1 | 2 | 4 | 24 | 24 | 48 |
Matrix representation of C72⋊4C8 ►in GL4(𝔽113) generated by
| 112 | 9 | 0 | 0 |
| 104 | 80 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 9 |
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 9 |
| 23 | 80 | 0 | 0 |
| 61 | 90 | 0 | 0 |
| 0 | 0 | 86 | 11 |
| 0 | 0 | 107 | 27 |
G:=sub<GL(4,GF(113))| [112,104,0,0,9,80,0,0,0,0,0,112,0,0,1,9],[0,112,0,0,1,9,0,0,0,0,0,112,0,0,1,9],[23,61,0,0,80,90,0,0,0,0,86,107,0,0,11,27] >;
C72⋊4C8 in GAP, Magma, Sage, TeX
C_7^2\rtimes_4C_8
% in TeX
G:=Group("C7^2:4C8"); // GroupNames label
G:=SmallGroup(392,15);
// by ID
G=gap.SmallGroup(392,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,10,26,963,8404]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export