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## G = Q8×C2×C22order 352 = 25·11

### Direct product of C2×C22 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C2×C22
 Chief series C1 — C2 — C22 — C44 — Q8×C11 — Q8×C22 — Q8×C2×C22
 Lower central C1 — C2 — Q8×C2×C22
 Upper central C1 — C22×C22 — Q8×C2×C22

Generators and relations for Q8×C2×C22
G = < a,b,c,d | a2=b22=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 156, all normal (8 characteristic)
C1, C2, C2, C4, C22, C2×C4, Q8, C23, C11, C22×C4, C2×Q8, C22, C22, C22×Q8, C44, C2×C22, C2×C44, Q8×C11, C22×C22, C22×C44, Q8×C22, Q8×C2×C22
Quotients: C1, C2, C22, Q8, C23, C11, C2×Q8, C24, C22, C22×Q8, C2×C22, Q8×C11, C22×C22, Q8×C22, C23×C22, Q8×C2×C22

Smallest permutation representation of Q8×C2×C22
Regular action on 352 points
Generators in S352
(1 279)(2 280)(3 281)(4 282)(5 283)(6 284)(7 285)(8 286)(9 265)(10 266)(11 267)(12 268)(13 269)(14 270)(15 271)(16 272)(17 273)(18 274)(19 275)(20 276)(21 277)(22 278)(23 249)(24 250)(25 251)(26 252)(27 253)(28 254)(29 255)(30 256)(31 257)(32 258)(33 259)(34 260)(35 261)(36 262)(37 263)(38 264)(39 243)(40 244)(41 245)(42 246)(43 247)(44 248)(45 225)(46 226)(47 227)(48 228)(49 229)(50 230)(51 231)(52 232)(53 233)(54 234)(55 235)(56 236)(57 237)(58 238)(59 239)(60 240)(61 241)(62 242)(63 221)(64 222)(65 223)(66 224)(67 189)(68 190)(69 191)(70 192)(71 193)(72 194)(73 195)(74 196)(75 197)(76 198)(77 177)(78 178)(79 179)(80 180)(81 181)(82 182)(83 183)(84 184)(85 185)(86 186)(87 187)(88 188)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)(109 145)(110 146)(111 175)(112 176)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 161)(120 162)(121 163)(122 164)(123 165)(124 166)(125 167)(126 168)(127 169)(128 170)(129 171)(130 172)(131 173)(132 174)(199 331)(200 332)(201 333)(202 334)(203 335)(204 336)(205 337)(206 338)(207 339)(208 340)(209 341)(210 342)(211 343)(212 344)(213 345)(214 346)(215 347)(216 348)(217 349)(218 350)(219 351)(220 352)(287 325)(288 326)(289 327)(290 328)(291 329)(292 330)(293 309)(294 310)(295 311)(296 312)(297 313)(298 314)(299 315)(300 316)(301 317)(302 318)(303 319)(304 320)(305 321)(306 322)(307 323)(308 324)
(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)
(1 148 60 248)(2 149 61 249)(3 150 62 250)(4 151 63 251)(5 152 64 252)(6 153 65 253)(7 154 66 254)(8 133 45 255)(9 134 46 256)(10 135 47 257)(11 136 48 258)(12 137 49 259)(13 138 50 260)(14 139 51 261)(15 140 52 262)(16 141 53 263)(17 142 54 264)(18 143 55 243)(19 144 56 244)(20 145 57 245)(21 146 58 246)(22 147 59 247)(23 280 91 241)(24 281 92 242)(25 282 93 221)(26 283 94 222)(27 284 95 223)(28 285 96 224)(29 286 97 225)(30 265 98 226)(31 266 99 227)(32 267 100 228)(33 268 101 229)(34 269 102 230)(35 270 103 231)(36 271 104 232)(37 272 105 233)(38 273 106 234)(39 274 107 235)(40 275 108 236)(41 276 109 237)(42 277 110 238)(43 278 89 239)(44 279 90 240)(67 201 125 326)(68 202 126 327)(69 203 127 328)(70 204 128 329)(71 205 129 330)(72 206 130 309)(73 207 131 310)(74 208 132 311)(75 209 111 312)(76 210 112 313)(77 211 113 314)(78 212 114 315)(79 213 115 316)(80 214 116 317)(81 215 117 318)(82 216 118 319)(83 217 119 320)(84 218 120 321)(85 219 121 322)(86 220 122 323)(87 199 123 324)(88 200 124 325)(155 298 177 343)(156 299 178 344)(157 300 179 345)(158 301 180 346)(159 302 181 347)(160 303 182 348)(161 304 183 349)(162 305 184 350)(163 306 185 351)(164 307 186 352)(165 308 187 331)(166 287 188 332)(167 288 189 333)(168 289 190 334)(169 290 191 335)(170 291 192 336)(171 292 193 337)(172 293 194 338)(173 294 195 339)(174 295 196 340)(175 296 197 341)(176 297 198 342)
(1 350 60 305)(2 351 61 306)(3 352 62 307)(4 331 63 308)(5 332 64 287)(6 333 65 288)(7 334 66 289)(8 335 45 290)(9 336 46 291)(10 337 47 292)(11 338 48 293)(12 339 49 294)(13 340 50 295)(14 341 51 296)(15 342 52 297)(16 343 53 298)(17 344 54 299)(18 345 55 300)(19 346 56 301)(20 347 57 302)(21 348 58 303)(22 349 59 304)(23 121 91 85)(24 122 92 86)(25 123 93 87)(26 124 94 88)(27 125 95 67)(28 126 96 68)(29 127 97 69)(30 128 98 70)(31 129 99 71)(32 130 100 72)(33 131 101 73)(34 132 102 74)(35 111 103 75)(36 112 104 76)(37 113 105 77)(38 114 106 78)(39 115 107 79)(40 116 108 80)(41 117 109 81)(42 118 110 82)(43 119 89 83)(44 120 90 84)(133 191 255 169)(134 192 256 170)(135 193 257 171)(136 194 258 172)(137 195 259 173)(138 196 260 174)(139 197 261 175)(140 198 262 176)(141 177 263 155)(142 178 264 156)(143 179 243 157)(144 180 244 158)(145 181 245 159)(146 182 246 160)(147 183 247 161)(148 184 248 162)(149 185 249 163)(150 186 250 164)(151 187 251 165)(152 188 252 166)(153 189 253 167)(154 190 254 168)(199 221 324 282)(200 222 325 283)(201 223 326 284)(202 224 327 285)(203 225 328 286)(204 226 329 265)(205 227 330 266)(206 228 309 267)(207 229 310 268)(208 230 311 269)(209 231 312 270)(210 232 313 271)(211 233 314 272)(212 234 315 273)(213 235 316 274)(214 236 317 275)(215 237 318 276)(216 238 319 277)(217 239 320 278)(218 240 321 279)(219 241 322 280)(220 242 323 281)

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

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

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

220 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 11A ··· 11J 22A ··· 22BR 44A ··· 44DP order 1 2 ··· 2 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

220 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C11 C22 C22 Q8 Q8×C11 kernel Q8×C2×C22 C22×C44 Q8×C22 C22×Q8 C22×C4 C2×Q8 C2×C22 C22 # reps 1 3 12 10 30 120 4 40

Matrix representation of Q8×C2×C22 in GL4(𝔽89) generated by

 88 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
,
 88 0 0 0 0 88 0 0 0 0 39 0 0 0 0 39
,
 88 0 0 0 0 1 0 0 0 0 0 1 0 0 88 0
,
 1 0 0 0 0 1 0 0 0 0 69 20 0 0 20 20
G:=sub<GL(4,GF(89))| [88,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[88,0,0,0,0,88,0,0,0,0,39,0,0,0,0,39],[88,0,0,0,0,1,0,0,0,0,0,88,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,69,20,0,0,20,20] >;

Q8×C2×C22 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_{22}
% in TeX

G:=Group("Q8xC2xC22");
// GroupNames label

G:=SmallGroup(352,190);
// by ID

G=gap.SmallGroup(352,190);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-11,-2,1056,2137,1063]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^22=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

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