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G = C11×Q32order 352 = 25·11

Direct product of C11 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C11×Q32, C16.C22, Q16.C22, C176.3C2, C22.17D8, C44.38D4, C88.26C22, C8.4(C2×C22), C4.3(D4×C11), C2.5(C11×D8), (C11×Q16).2C2, SmallGroup(352,62)

Series: Derived Chief Lower central Upper central

C1C8 — C11×Q32
C1C2C4C8C88C11×Q16 — C11×Q32
C1C2C4C8 — C11×Q32
C1C22C44C88 — C11×Q32

Generators and relations for C11×Q32
 G = < a,b,c | a11=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

4C4
4C4
2Q8
2Q8
4C44
4C44
2Q8×C11
2Q8×C11

Smallest permutation representation of C11×Q32
Regular action on 352 points
Generators in S352
(1 139 105 199 235 295 317 270 325 282 217)(2 140 106 200 236 296 318 271 326 283 218)(3 141 107 201 237 297 319 272 327 284 219)(4 142 108 202 238 298 320 257 328 285 220)(5 143 109 203 239 299 305 258 329 286 221)(6 144 110 204 240 300 306 259 330 287 222)(7 129 111 205 225 301 307 260 331 288 223)(8 130 112 206 226 302 308 261 332 273 224)(9 131 97 207 227 303 309 262 333 274 209)(10 132 98 208 228 304 310 263 334 275 210)(11 133 99 193 229 289 311 264 335 276 211)(12 134 100 194 230 290 312 265 336 277 212)(13 135 101 195 231 291 313 266 321 278 213)(14 136 102 196 232 292 314 267 322 279 214)(15 137 103 197 233 293 315 268 323 280 215)(16 138 104 198 234 294 316 269 324 281 216)(17 161 147 89 343 127 36 78 184 246 54)(18 162 148 90 344 128 37 79 185 247 55)(19 163 149 91 345 113 38 80 186 248 56)(20 164 150 92 346 114 39 65 187 249 57)(21 165 151 93 347 115 40 66 188 250 58)(22 166 152 94 348 116 41 67 189 251 59)(23 167 153 95 349 117 42 68 190 252 60)(24 168 154 96 350 118 43 69 191 253 61)(25 169 155 81 351 119 44 70 192 254 62)(26 170 156 82 352 120 45 71 177 255 63)(27 171 157 83 337 121 46 72 178 256 64)(28 172 158 84 338 122 47 73 179 241 49)(29 173 159 85 339 123 48 74 180 242 50)(30 174 160 86 340 124 33 75 181 243 51)(31 175 145 87 341 125 34 76 182 244 52)(32 176 146 88 342 126 35 77 183 245 53)
(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)
(1 348 9 340)(2 347 10 339)(3 346 11 338)(4 345 12 337)(5 344 13 352)(6 343 14 351)(7 342 15 350)(8 341 16 349)(17 267 25 259)(18 266 26 258)(19 265 27 257)(20 264 28 272)(21 263 29 271)(22 262 30 270)(23 261 31 269)(24 260 32 268)(33 105 41 97)(34 104 42 112)(35 103 43 111)(36 102 44 110)(37 101 45 109)(38 100 46 108)(39 99 47 107)(40 98 48 106)(49 319 57 311)(50 318 58 310)(51 317 59 309)(52 316 60 308)(53 315 61 307)(54 314 62 306)(55 313 63 305)(56 312 64 320)(65 193 73 201)(66 208 74 200)(67 207 75 199)(68 206 76 198)(69 205 77 197)(70 204 78 196)(71 203 79 195)(72 202 80 194)(81 222 89 214)(82 221 90 213)(83 220 91 212)(84 219 92 211)(85 218 93 210)(86 217 94 209)(87 216 95 224)(88 215 96 223)(113 134 121 142)(114 133 122 141)(115 132 123 140)(116 131 124 139)(117 130 125 138)(118 129 126 137)(119 144 127 136)(120 143 128 135)(145 281 153 273)(146 280 154 288)(147 279 155 287)(148 278 156 286)(149 277 157 285)(150 276 158 284)(151 275 159 283)(152 274 160 282)(161 322 169 330)(162 321 170 329)(163 336 171 328)(164 335 172 327)(165 334 173 326)(166 333 174 325)(167 332 175 324)(168 331 176 323)(177 239 185 231)(178 238 186 230)(179 237 187 229)(180 236 188 228)(181 235 189 227)(182 234 190 226)(183 233 191 225)(184 232 192 240)(241 297 249 289)(242 296 250 304)(243 295 251 303)(244 294 252 302)(245 293 253 301)(246 292 254 300)(247 291 255 299)(248 290 256 298)

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

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

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

121 conjugacy classes

class 1  2 4A4B4C8A8B11A···11J16A16B16C16D22A···22J44A···44J44K···44AD88A···88T176A···176AN
order124448811···111616161622···2244···4444···4488···88176···176
size11288221···122221···12···28···82···22···2

121 irreducible representations

dim111111222222
type+++++-
imageC1C2C2C11C22C22D4D8Q32D4×C11C11×D8C11×Q32
kernelC11×Q32C176C11×Q16Q32C16Q16C44C22C11C4C2C1
# reps112101020124102040

Matrix representation of C11×Q32 in GL2(𝔽353) generated by

220
022
,
18320
333183
,
302218
21851
G:=sub<GL(2,GF(353))| [22,0,0,22],[183,333,20,183],[302,218,218,51] >;

C11×Q32 in GAP, Magma, Sage, TeX

C_{11}\times Q_{32}
% in TeX

G:=Group("C11xQ32");
// GroupNames label

G:=SmallGroup(352,62);
// by ID

G=gap.SmallGroup(352,62);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,1056,553,1063,3171,1593,165,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c|a^11=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×Q32 in TeX

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