direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C4⋊Q8, C44⋊4Q8, C44.40D4, C42.5C22, C4⋊(Q8×C11), C4⋊C4.5C22, C4.5(D4×C11), C2.5(Q8×C22), (C4×C44).11C2, C22.73(C2×D4), C2.10(D4×C22), (Q8×C22).8C2, (C2×Q8).3C22, C22.22(C2×Q8), (C2×C22).83C23, (C2×C44).126C22, C22.18(C22×C22), (C2×C4).9(C2×C22), (C11×C4⋊C4).12C2, SmallGroup(352,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C4⋊Q8
G = < a,b,c,d | a11=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 84 in 68 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C11, C42, C4⋊C4, C2×Q8, C22, C22, C4⋊Q8, C44, C44, C2×C22, C2×C44, C2×C44, Q8×C11, C4×C44, C11×C4⋊C4, Q8×C22, C11×C4⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C11, C2×D4, C2×Q8, C22, C4⋊Q8, C2×C22, D4×C11, Q8×C11, C22×C22, D4×C22, Q8×C22, C11×C4⋊Q8
►Regular action on 352 points
Generators in S352
(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)
(1 167 63 178)(2 168 64 179)(3 169 65 180)(4 170 66 181)(5 171 56 182)(6 172 57 183)(7 173 58 184)(8 174 59 185)(9 175 60 186)(10 176 61 187)(11 166 62 177)(12 216 34 227)(13 217 35 228)(14 218 36 229)(15 219 37 230)(16 220 38 231)(17 210 39 221)(18 211 40 222)(19 212 41 223)(20 213 42 224)(21 214 43 225)(22 215 44 226)(23 235 350 245)(24 236 351 246)(25 237 352 247)(26 238 342 248)(27 239 343 249)(28 240 344 250)(29 241 345 251)(30 242 346 252)(31 232 347 253)(32 233 348 243)(33 234 349 244)(45 205 72 195)(46 206 73 196)(47 207 74 197)(48 208 75 198)(49 209 76 188)(50 199 77 189)(51 200 67 190)(52 201 68 191)(53 202 69 192)(54 203 70 193)(55 204 71 194)(78 133 89 122)(79 134 90 123)(80 135 91 124)(81 136 92 125)(82 137 93 126)(83 138 94 127)(84 139 95 128)(85 140 96 129)(86 141 97 130)(87 142 98 131)(88 143 99 132)(100 165 121 144)(101 155 111 145)(102 156 112 146)(103 157 113 147)(104 158 114 148)(105 159 115 149)(106 160 116 150)(107 161 117 151)(108 162 118 152)(109 163 119 153)(110 164 120 154)(254 298 265 309)(255 299 266 310)(256 300 267 311)(257 301 268 312)(258 302 269 313)(259 303 270 314)(260 304 271 315)(261 305 272 316)(262 306 273 317)(263 307 274 318)(264 308 275 319)(276 320 297 341)(277 321 287 331)(278 322 288 332)(279 323 289 333)(280 324 290 334)(281 325 291 335)(282 326 292 336)(283 327 293 337)(284 328 294 338)(285 329 295 339)(286 330 296 340)
(1 123 45 161)(2 124 46 162)(3 125 47 163)(4 126 48 164)(5 127 49 165)(6 128 50 155)(7 129 51 156)(8 130 52 157)(9 131 53 158)(10 132 54 159)(11 122 55 160)(12 260 348 287)(13 261 349 288)(14 262 350 289)(15 263 351 290)(16 264 352 291)(17 254 342 292)(18 255 343 293)(19 256 344 294)(20 257 345 295)(21 258 346 296)(22 259 347 297)(23 279 36 273)(24 280 37 274)(25 281 38 275)(26 282 39 265)(27 283 40 266)(28 284 41 267)(29 285 42 268)(30 286 43 269)(31 276 44 270)(32 277 34 271)(33 278 35 272)(56 138 76 144)(57 139 77 145)(58 140 67 146)(59 141 68 147)(60 142 69 148)(61 143 70 149)(62 133 71 150)(63 134 72 151)(64 135 73 152)(65 136 74 153)(66 137 75 154)(78 204 116 166)(79 205 117 167)(80 206 118 168)(81 207 119 169)(82 208 120 170)(83 209 121 171)(84 199 111 172)(85 200 112 173)(86 201 113 174)(87 202 114 175)(88 203 115 176)(89 194 106 177)(90 195 107 178)(91 196 108 179)(92 197 109 180)(93 198 110 181)(94 188 100 182)(95 189 101 183)(96 190 102 184)(97 191 103 185)(98 192 104 186)(99 193 105 187)(210 298 248 336)(211 299 249 337)(212 300 250 338)(213 301 251 339)(214 302 252 340)(215 303 253 341)(216 304 243 331)(217 305 244 332)(218 306 245 333)(219 307 246 334)(220 308 247 335)(221 309 238 326)(222 310 239 327)(223 311 240 328)(224 312 241 329)(225 313 242 330)(226 314 232 320)(227 315 233 321)(228 316 234 322)(229 317 235 323)(230 318 236 324)(231 319 237 325)
(1 211 45 249)(2 212 46 250)(3 213 47 251)(4 214 48 252)(5 215 49 253)(6 216 50 243)(7 217 51 244)(8 218 52 245)(9 219 53 246)(10 220 54 247)(11 210 55 248)(12 199 348 172)(13 200 349 173)(14 201 350 174)(15 202 351 175)(16 203 352 176)(17 204 342 166)(18 205 343 167)(19 206 344 168)(20 207 345 169)(21 208 346 170)(22 209 347 171)(23 185 36 191)(24 186 37 192)(25 187 38 193)(26 177 39 194)(27 178 40 195)(28 179 41 196)(29 180 42 197)(30 181 43 198)(31 182 44 188)(32 183 34 189)(33 184 35 190)(56 226 76 232)(57 227 77 233)(58 228 67 234)(59 229 68 235)(60 230 69 236)(61 231 70 237)(62 221 71 238)(63 222 72 239)(64 223 73 240)(65 224 74 241)(66 225 75 242)(78 292 116 254)(79 293 117 255)(80 294 118 256)(81 295 119 257)(82 296 120 258)(83 297 121 259)(84 287 111 260)(85 288 112 261)(86 289 113 262)(87 290 114 263)(88 291 115 264)(89 282 106 265)(90 283 107 266)(91 284 108 267)(92 285 109 268)(93 286 110 269)(94 276 100 270)(95 277 101 271)(96 278 102 272)(97 279 103 273)(98 280 104 274)(99 281 105 275)(122 336 160 298)(123 337 161 299)(124 338 162 300)(125 339 163 301)(126 340 164 302)(127 341 165 303)(128 331 155 304)(129 332 156 305)(130 333 157 306)(131 334 158 307)(132 335 159 308)(133 326 150 309)(134 327 151 310)(135 328 152 311)(136 329 153 312)(137 330 154 313)(138 320 144 314)(139 321 145 315)(140 322 146 316)(141 323 147 317)(142 324 148 318)(143 325 149 319)
G:=sub<Sym(352)| (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), (1,167,63,178)(2,168,64,179)(3,169,65,180)(4,170,66,181)(5,171,56,182)(6,172,57,183)(7,173,58,184)(8,174,59,185)(9,175,60,186)(10,176,61,187)(11,166,62,177)(12,216,34,227)(13,217,35,228)(14,218,36,229)(15,219,37,230)(16,220,38,231)(17,210,39,221)(18,211,40,222)(19,212,41,223)(20,213,42,224)(21,214,43,225)(22,215,44,226)(23,235,350,245)(24,236,351,246)(25,237,352,247)(26,238,342,248)(27,239,343,249)(28,240,344,250)(29,241,345,251)(30,242,346,252)(31,232,347,253)(32,233,348,243)(33,234,349,244)(45,205,72,195)(46,206,73,196)(47,207,74,197)(48,208,75,198)(49,209,76,188)(50,199,77,189)(51,200,67,190)(52,201,68,191)(53,202,69,192)(54,203,70,193)(55,204,71,194)(78,133,89,122)(79,134,90,123)(80,135,91,124)(81,136,92,125)(82,137,93,126)(83,138,94,127)(84,139,95,128)(85,140,96,129)(86,141,97,130)(87,142,98,131)(88,143,99,132)(100,165,121,144)(101,155,111,145)(102,156,112,146)(103,157,113,147)(104,158,114,148)(105,159,115,149)(106,160,116,150)(107,161,117,151)(108,162,118,152)(109,163,119,153)(110,164,120,154)(254,298,265,309)(255,299,266,310)(256,300,267,311)(257,301,268,312)(258,302,269,313)(259,303,270,314)(260,304,271,315)(261,305,272,316)(262,306,273,317)(263,307,274,318)(264,308,275,319)(276,320,297,341)(277,321,287,331)(278,322,288,332)(279,323,289,333)(280,324,290,334)(281,325,291,335)(282,326,292,336)(283,327,293,337)(284,328,294,338)(285,329,295,339)(286,330,296,340), (1,123,45,161)(2,124,46,162)(3,125,47,163)(4,126,48,164)(5,127,49,165)(6,128,50,155)(7,129,51,156)(8,130,52,157)(9,131,53,158)(10,132,54,159)(11,122,55,160)(12,260,348,287)(13,261,349,288)(14,262,350,289)(15,263,351,290)(16,264,352,291)(17,254,342,292)(18,255,343,293)(19,256,344,294)(20,257,345,295)(21,258,346,296)(22,259,347,297)(23,279,36,273)(24,280,37,274)(25,281,38,275)(26,282,39,265)(27,283,40,266)(28,284,41,267)(29,285,42,268)(30,286,43,269)(31,276,44,270)(32,277,34,271)(33,278,35,272)(56,138,76,144)(57,139,77,145)(58,140,67,146)(59,141,68,147)(60,142,69,148)(61,143,70,149)(62,133,71,150)(63,134,72,151)(64,135,73,152)(65,136,74,153)(66,137,75,154)(78,204,116,166)(79,205,117,167)(80,206,118,168)(81,207,119,169)(82,208,120,170)(83,209,121,171)(84,199,111,172)(85,200,112,173)(86,201,113,174)(87,202,114,175)(88,203,115,176)(89,194,106,177)(90,195,107,178)(91,196,108,179)(92,197,109,180)(93,198,110,181)(94,188,100,182)(95,189,101,183)(96,190,102,184)(97,191,103,185)(98,192,104,186)(99,193,105,187)(210,298,248,336)(211,299,249,337)(212,300,250,338)(213,301,251,339)(214,302,252,340)(215,303,253,341)(216,304,243,331)(217,305,244,332)(218,306,245,333)(219,307,246,334)(220,308,247,335)(221,309,238,326)(222,310,239,327)(223,311,240,328)(224,312,241,329)(225,313,242,330)(226,314,232,320)(227,315,233,321)(228,316,234,322)(229,317,235,323)(230,318,236,324)(231,319,237,325), (1,211,45,249)(2,212,46,250)(3,213,47,251)(4,214,48,252)(5,215,49,253)(6,216,50,243)(7,217,51,244)(8,218,52,245)(9,219,53,246)(10,220,54,247)(11,210,55,248)(12,199,348,172)(13,200,349,173)(14,201,350,174)(15,202,351,175)(16,203,352,176)(17,204,342,166)(18,205,343,167)(19,206,344,168)(20,207,345,169)(21,208,346,170)(22,209,347,171)(23,185,36,191)(24,186,37,192)(25,187,38,193)(26,177,39,194)(27,178,40,195)(28,179,41,196)(29,180,42,197)(30,181,43,198)(31,182,44,188)(32,183,34,189)(33,184,35,190)(56,226,76,232)(57,227,77,233)(58,228,67,234)(59,229,68,235)(60,230,69,236)(61,231,70,237)(62,221,71,238)(63,222,72,239)(64,223,73,240)(65,224,74,241)(66,225,75,242)(78,292,116,254)(79,293,117,255)(80,294,118,256)(81,295,119,257)(82,296,120,258)(83,297,121,259)(84,287,111,260)(85,288,112,261)(86,289,113,262)(87,290,114,263)(88,291,115,264)(89,282,106,265)(90,283,107,266)(91,284,108,267)(92,285,109,268)(93,286,110,269)(94,276,100,270)(95,277,101,271)(96,278,102,272)(97,279,103,273)(98,280,104,274)(99,281,105,275)(122,336,160,298)(123,337,161,299)(124,338,162,300)(125,339,163,301)(126,340,164,302)(127,341,165,303)(128,331,155,304)(129,332,156,305)(130,333,157,306)(131,334,158,307)(132,335,159,308)(133,326,150,309)(134,327,151,310)(135,328,152,311)(136,329,153,312)(137,330,154,313)(138,320,144,314)(139,321,145,315)(140,322,146,316)(141,323,147,317)(142,324,148,318)(143,325,149,319)>;
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), (1,167,63,178)(2,168,64,179)(3,169,65,180)(4,170,66,181)(5,171,56,182)(6,172,57,183)(7,173,58,184)(8,174,59,185)(9,175,60,186)(10,176,61,187)(11,166,62,177)(12,216,34,227)(13,217,35,228)(14,218,36,229)(15,219,37,230)(16,220,38,231)(17,210,39,221)(18,211,40,222)(19,212,41,223)(20,213,42,224)(21,214,43,225)(22,215,44,226)(23,235,350,245)(24,236,351,246)(25,237,352,247)(26,238,342,248)(27,239,343,249)(28,240,344,250)(29,241,345,251)(30,242,346,252)(31,232,347,253)(32,233,348,243)(33,234,349,244)(45,205,72,195)(46,206,73,196)(47,207,74,197)(48,208,75,198)(49,209,76,188)(50,199,77,189)(51,200,67,190)(52,201,68,191)(53,202,69,192)(54,203,70,193)(55,204,71,194)(78,133,89,122)(79,134,90,123)(80,135,91,124)(81,136,92,125)(82,137,93,126)(83,138,94,127)(84,139,95,128)(85,140,96,129)(86,141,97,130)(87,142,98,131)(88,143,99,132)(100,165,121,144)(101,155,111,145)(102,156,112,146)(103,157,113,147)(104,158,114,148)(105,159,115,149)(106,160,116,150)(107,161,117,151)(108,162,118,152)(109,163,119,153)(110,164,120,154)(254,298,265,309)(255,299,266,310)(256,300,267,311)(257,301,268,312)(258,302,269,313)(259,303,270,314)(260,304,271,315)(261,305,272,316)(262,306,273,317)(263,307,274,318)(264,308,275,319)(276,320,297,341)(277,321,287,331)(278,322,288,332)(279,323,289,333)(280,324,290,334)(281,325,291,335)(282,326,292,336)(283,327,293,337)(284,328,294,338)(285,329,295,339)(286,330,296,340), (1,123,45,161)(2,124,46,162)(3,125,47,163)(4,126,48,164)(5,127,49,165)(6,128,50,155)(7,129,51,156)(8,130,52,157)(9,131,53,158)(10,132,54,159)(11,122,55,160)(12,260,348,287)(13,261,349,288)(14,262,350,289)(15,263,351,290)(16,264,352,291)(17,254,342,292)(18,255,343,293)(19,256,344,294)(20,257,345,295)(21,258,346,296)(22,259,347,297)(23,279,36,273)(24,280,37,274)(25,281,38,275)(26,282,39,265)(27,283,40,266)(28,284,41,267)(29,285,42,268)(30,286,43,269)(31,276,44,270)(32,277,34,271)(33,278,35,272)(56,138,76,144)(57,139,77,145)(58,140,67,146)(59,141,68,147)(60,142,69,148)(61,143,70,149)(62,133,71,150)(63,134,72,151)(64,135,73,152)(65,136,74,153)(66,137,75,154)(78,204,116,166)(79,205,117,167)(80,206,118,168)(81,207,119,169)(82,208,120,170)(83,209,121,171)(84,199,111,172)(85,200,112,173)(86,201,113,174)(87,202,114,175)(88,203,115,176)(89,194,106,177)(90,195,107,178)(91,196,108,179)(92,197,109,180)(93,198,110,181)(94,188,100,182)(95,189,101,183)(96,190,102,184)(97,191,103,185)(98,192,104,186)(99,193,105,187)(210,298,248,336)(211,299,249,337)(212,300,250,338)(213,301,251,339)(214,302,252,340)(215,303,253,341)(216,304,243,331)(217,305,244,332)(218,306,245,333)(219,307,246,334)(220,308,247,335)(221,309,238,326)(222,310,239,327)(223,311,240,328)(224,312,241,329)(225,313,242,330)(226,314,232,320)(227,315,233,321)(228,316,234,322)(229,317,235,323)(230,318,236,324)(231,319,237,325), (1,211,45,249)(2,212,46,250)(3,213,47,251)(4,214,48,252)(5,215,49,253)(6,216,50,243)(7,217,51,244)(8,218,52,245)(9,219,53,246)(10,220,54,247)(11,210,55,248)(12,199,348,172)(13,200,349,173)(14,201,350,174)(15,202,351,175)(16,203,352,176)(17,204,342,166)(18,205,343,167)(19,206,344,168)(20,207,345,169)(21,208,346,170)(22,209,347,171)(23,185,36,191)(24,186,37,192)(25,187,38,193)(26,177,39,194)(27,178,40,195)(28,179,41,196)(29,180,42,197)(30,181,43,198)(31,182,44,188)(32,183,34,189)(33,184,35,190)(56,226,76,232)(57,227,77,233)(58,228,67,234)(59,229,68,235)(60,230,69,236)(61,231,70,237)(62,221,71,238)(63,222,72,239)(64,223,73,240)(65,224,74,241)(66,225,75,242)(78,292,116,254)(79,293,117,255)(80,294,118,256)(81,295,119,257)(82,296,120,258)(83,297,121,259)(84,287,111,260)(85,288,112,261)(86,289,113,262)(87,290,114,263)(88,291,115,264)(89,282,106,265)(90,283,107,266)(91,284,108,267)(92,285,109,268)(93,286,110,269)(94,276,100,270)(95,277,101,271)(96,278,102,272)(97,279,103,273)(98,280,104,274)(99,281,105,275)(122,336,160,298)(123,337,161,299)(124,338,162,300)(125,339,163,301)(126,340,164,302)(127,341,165,303)(128,331,155,304)(129,332,156,305)(130,333,157,306)(131,334,158,307)(132,335,159,308)(133,326,150,309)(134,327,151,310)(135,328,152,311)(136,329,153,312)(137,330,154,313)(138,320,144,314)(139,321,145,315)(140,322,146,316)(141,323,147,317)(142,324,148,318)(143,325,149,319) );
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)], [(1,167,63,178),(2,168,64,179),(3,169,65,180),(4,170,66,181),(5,171,56,182),(6,172,57,183),(7,173,58,184),(8,174,59,185),(9,175,60,186),(10,176,61,187),(11,166,62,177),(12,216,34,227),(13,217,35,228),(14,218,36,229),(15,219,37,230),(16,220,38,231),(17,210,39,221),(18,211,40,222),(19,212,41,223),(20,213,42,224),(21,214,43,225),(22,215,44,226),(23,235,350,245),(24,236,351,246),(25,237,352,247),(26,238,342,248),(27,239,343,249),(28,240,344,250),(29,241,345,251),(30,242,346,252),(31,232,347,253),(32,233,348,243),(33,234,349,244),(45,205,72,195),(46,206,73,196),(47,207,74,197),(48,208,75,198),(49,209,76,188),(50,199,77,189),(51,200,67,190),(52,201,68,191),(53,202,69,192),(54,203,70,193),(55,204,71,194),(78,133,89,122),(79,134,90,123),(80,135,91,124),(81,136,92,125),(82,137,93,126),(83,138,94,127),(84,139,95,128),(85,140,96,129),(86,141,97,130),(87,142,98,131),(88,143,99,132),(100,165,121,144),(101,155,111,145),(102,156,112,146),(103,157,113,147),(104,158,114,148),(105,159,115,149),(106,160,116,150),(107,161,117,151),(108,162,118,152),(109,163,119,153),(110,164,120,154),(254,298,265,309),(255,299,266,310),(256,300,267,311),(257,301,268,312),(258,302,269,313),(259,303,270,314),(260,304,271,315),(261,305,272,316),(262,306,273,317),(263,307,274,318),(264,308,275,319),(276,320,297,341),(277,321,287,331),(278,322,288,332),(279,323,289,333),(280,324,290,334),(281,325,291,335),(282,326,292,336),(283,327,293,337),(284,328,294,338),(285,329,295,339),(286,330,296,340)], [(1,123,45,161),(2,124,46,162),(3,125,47,163),(4,126,48,164),(5,127,49,165),(6,128,50,155),(7,129,51,156),(8,130,52,157),(9,131,53,158),(10,132,54,159),(11,122,55,160),(12,260,348,287),(13,261,349,288),(14,262,350,289),(15,263,351,290),(16,264,352,291),(17,254,342,292),(18,255,343,293),(19,256,344,294),(20,257,345,295),(21,258,346,296),(22,259,347,297),(23,279,36,273),(24,280,37,274),(25,281,38,275),(26,282,39,265),(27,283,40,266),(28,284,41,267),(29,285,42,268),(30,286,43,269),(31,276,44,270),(32,277,34,271),(33,278,35,272),(56,138,76,144),(57,139,77,145),(58,140,67,146),(59,141,68,147),(60,142,69,148),(61,143,70,149),(62,133,71,150),(63,134,72,151),(64,135,73,152),(65,136,74,153),(66,137,75,154),(78,204,116,166),(79,205,117,167),(80,206,118,168),(81,207,119,169),(82,208,120,170),(83,209,121,171),(84,199,111,172),(85,200,112,173),(86,201,113,174),(87,202,114,175),(88,203,115,176),(89,194,106,177),(90,195,107,178),(91,196,108,179),(92,197,109,180),(93,198,110,181),(94,188,100,182),(95,189,101,183),(96,190,102,184),(97,191,103,185),(98,192,104,186),(99,193,105,187),(210,298,248,336),(211,299,249,337),(212,300,250,338),(213,301,251,339),(214,302,252,340),(215,303,253,341),(216,304,243,331),(217,305,244,332),(218,306,245,333),(219,307,246,334),(220,308,247,335),(221,309,238,326),(222,310,239,327),(223,311,240,328),(224,312,241,329),(225,313,242,330),(226,314,232,320),(227,315,233,321),(228,316,234,322),(229,317,235,323),(230,318,236,324),(231,319,237,325)], [(1,211,45,249),(2,212,46,250),(3,213,47,251),(4,214,48,252),(5,215,49,253),(6,216,50,243),(7,217,51,244),(8,218,52,245),(9,219,53,246),(10,220,54,247),(11,210,55,248),(12,199,348,172),(13,200,349,173),(14,201,350,174),(15,202,351,175),(16,203,352,176),(17,204,342,166),(18,205,343,167),(19,206,344,168),(20,207,345,169),(21,208,346,170),(22,209,347,171),(23,185,36,191),(24,186,37,192),(25,187,38,193),(26,177,39,194),(27,178,40,195),(28,179,41,196),(29,180,42,197),(30,181,43,198),(31,182,44,188),(32,183,34,189),(33,184,35,190),(56,226,76,232),(57,227,77,233),(58,228,67,234),(59,229,68,235),(60,230,69,236),(61,231,70,237),(62,221,71,238),(63,222,72,239),(64,223,73,240),(65,224,74,241),(66,225,75,242),(78,292,116,254),(79,293,117,255),(80,294,118,256),(81,295,119,257),(82,296,120,258),(83,297,121,259),(84,287,111,260),(85,288,112,261),(86,289,113,262),(87,290,114,263),(88,291,115,264),(89,282,106,265),(90,283,107,266),(91,284,108,267),(92,285,109,268),(93,286,110,269),(94,276,100,270),(95,277,101,271),(96,278,102,272),(97,279,103,273),(98,280,104,274),(99,281,105,275),(122,336,160,298),(123,337,161,299),(124,338,162,300),(125,339,163,301),(126,340,164,302),(127,341,165,303),(128,331,155,304),(129,332,156,305),(130,333,157,306),(131,334,158,307),(132,335,159,308),(133,326,150,309),(134,327,151,310),(135,328,152,311),(136,329,153,312),(137,330,154,313),(138,320,144,314),(139,321,145,315),(140,322,146,316),(141,323,147,317),(142,324,148,318),(143,325,149,319)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 11A | ··· | 11J | 22A | ··· | 22AD | 44A | ··· | 44BH | 44BI | ··· | 44CV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | D4 | Q8 | D4×C11 | Q8×C11 |
| kernel | C11×C4⋊Q8 | C4×C44 | C11×C4⋊C4 | Q8×C22 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C44 | C44 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 10 | 10 | 40 | 20 | 2 | 4 | 20 | 40 |
Matrix representation of C11×C4⋊Q8 ►in GL4(𝔽89) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 45 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 1 | 0 |
| 0 | 88 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 54 | 51 | 0 | 0 |
| 51 | 35 | 0 | 0 |
| 0 | 0 | 72 | 54 |
| 0 | 0 | 54 | 17 |
G:=sub<GL(4,GF(89))| [4,0,0,0,0,4,0,0,0,0,45,0,0,0,0,45],[88,0,0,0,0,88,0,0,0,0,0,1,0,0,88,0],[0,1,0,0,88,0,0,0,0,0,1,0,0,0,0,1],[54,51,0,0,51,35,0,0,0,0,72,54,0,0,54,17] >;
C11×C4⋊Q8 in GAP, Magma, Sage, TeX
C_{11}\times C_4\rtimes Q_8 % in TeX
G:=Group("C11xC4:Q8"); // GroupNames label
G:=SmallGroup(352,163);
// by ID
G=gap.SmallGroup(352,163);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,528,1081,535,3242,806]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations