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G = C22×Dic22order 352 = 25·11

Direct product of C22 and Dic22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C22×Dic22
 Chief series C1 — C11 — C22 — Dic11 — C2×Dic11 — C22×Dic11 — C22×Dic22
 Lower central C11 — C22 — C22×Dic22
 Upper central C1 — C23 — C22×C4

Generators and relations for C22×Dic22
G = < a,b,c,d | a2=b2=c44=1, d2=c22, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 666 in 156 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C11, C22×C4, C22×C4, C2×Q8, C22, C22, C22×Q8, Dic11, C44, C2×C22, Dic22, C2×Dic11, C2×C44, C22×C22, C2×Dic22, C22×Dic11, C22×C44, C22×Dic22
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, D11, C22×Q8, D22, Dic22, C22×D11, C2×Dic22, C23×D11, C22×Dic22

Smallest permutation representation of C22×Dic22
Regular action on 352 points
Generators in S352
(1 85)(2 86)(3 87)(4 88)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(89 247)(90 248)(91 249)(92 250)(93 251)(94 252)(95 253)(96 254)(97 255)(98 256)(99 257)(100 258)(101 259)(102 260)(103 261)(104 262)(105 263)(106 264)(107 221)(108 222)(109 223)(110 224)(111 225)(112 226)(113 227)(114 228)(115 229)(116 230)(117 231)(118 232)(119 233)(120 234)(121 235)(122 236)(123 237)(124 238)(125 239)(126 240)(127 241)(128 242)(129 243)(130 244)(131 245)(132 246)(133 307)(134 308)(135 265)(136 266)(137 267)(138 268)(139 269)(140 270)(141 271)(142 272)(143 273)(144 274)(145 275)(146 276)(147 277)(148 278)(149 279)(150 280)(151 281)(152 282)(153 283)(154 284)(155 285)(156 286)(157 287)(158 288)(159 289)(160 290)(161 291)(162 292)(163 293)(164 294)(165 295)(166 296)(167 297)(168 298)(169 299)(170 300)(171 301)(172 302)(173 303)(174 304)(175 305)(176 306)(177 332)(178 333)(179 334)(180 335)(181 336)(182 337)(183 338)(184 339)(185 340)(186 341)(187 342)(188 343)(189 344)(190 345)(191 346)(192 347)(193 348)(194 349)(195 350)(196 351)(197 352)(198 309)(199 310)(200 311)(201 312)(202 313)(203 314)(204 315)(205 316)(206 317)(207 318)(208 319)(209 320)(210 321)(211 322)(212 323)(213 324)(214 325)(215 326)(216 327)(217 328)(218 329)(219 330)(220 331)
(1 269)(2 270)(3 271)(4 272)(5 273)(6 274)(7 275)(8 276)(9 277)(10 278)(11 279)(12 280)(13 281)(14 282)(15 283)(16 284)(17 285)(18 286)(19 287)(20 288)(21 289)(22 290)(23 291)(24 292)(25 293)(26 294)(27 295)(28 296)(29 297)(30 298)(31 299)(32 300)(33 301)(34 302)(35 303)(36 304)(37 305)(38 306)(39 307)(40 308)(41 265)(42 266)(43 267)(44 268)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 159)(62 160)(63 161)(64 162)(65 163)(66 164)(67 165)(68 166)(69 167)(70 168)(71 169)(72 170)(73 171)(74 172)(75 173)(76 174)(77 175)(78 176)(79 133)(80 134)(81 135)(82 136)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(89 207)(90 208)(91 209)(92 210)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(99 217)(100 218)(101 219)(102 220)(103 177)(104 178)(105 179)(106 180)(107 181)(108 182)(109 183)(110 184)(111 185)(112 186)(113 187)(114 188)(115 189)(116 190)(117 191)(118 192)(119 193)(120 194)(121 195)(122 196)(123 197)(124 198)(125 199)(126 200)(127 201)(128 202)(129 203)(130 204)(131 205)(132 206)(221 336)(222 337)(223 338)(224 339)(225 340)(226 341)(227 342)(228 343)(229 344)(230 345)(231 346)(232 347)(233 348)(234 349)(235 350)(236 351)(237 352)(238 309)(239 310)(240 311)(241 312)(242 313)(243 314)(244 315)(245 316)(246 317)(247 318)(248 319)(249 320)(250 321)(251 322)(252 323)(253 324)(254 325)(255 326)(256 327)(257 328)(258 329)(259 330)(260 331)(261 332)(262 333)(263 334)(264 335)
(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)
(1 102 23 124)(2 101 24 123)(3 100 25 122)(4 99 26 121)(5 98 27 120)(6 97 28 119)(7 96 29 118)(8 95 30 117)(9 94 31 116)(10 93 32 115)(11 92 33 114)(12 91 34 113)(13 90 35 112)(14 89 36 111)(15 132 37 110)(16 131 38 109)(17 130 39 108)(18 129 40 107)(19 128 41 106)(20 127 42 105)(21 126 43 104)(22 125 44 103)(45 256 67 234)(46 255 68 233)(47 254 69 232)(48 253 70 231)(49 252 71 230)(50 251 72 229)(51 250 73 228)(52 249 74 227)(53 248 75 226)(54 247 76 225)(55 246 77 224)(56 245 78 223)(57 244 79 222)(58 243 80 221)(59 242 81 264)(60 241 82 263)(61 240 83 262)(62 239 84 261)(63 238 85 260)(64 237 86 259)(65 236 87 258)(66 235 88 257)(133 337 155 315)(134 336 156 314)(135 335 157 313)(136 334 158 312)(137 333 159 311)(138 332 160 310)(139 331 161 309)(140 330 162 352)(141 329 163 351)(142 328 164 350)(143 327 165 349)(144 326 166 348)(145 325 167 347)(146 324 168 346)(147 323 169 345)(148 322 170 344)(149 321 171 343)(150 320 172 342)(151 319 173 341)(152 318 174 340)(153 317 175 339)(154 316 176 338)(177 290 199 268)(178 289 200 267)(179 288 201 266)(180 287 202 265)(181 286 203 308)(182 285 204 307)(183 284 205 306)(184 283 206 305)(185 282 207 304)(186 281 208 303)(187 280 209 302)(188 279 210 301)(189 278 211 300)(190 277 212 299)(191 276 213 298)(192 275 214 297)(193 274 215 296)(194 273 216 295)(195 272 217 294)(196 271 218 293)(197 270 219 292)(198 269 220 291)

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

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

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

100 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 11A ··· 11E 22A ··· 22AI 44A ··· 44AN order 1 2 ··· 2 4 4 4 4 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 ··· 1 2 2 2 2 22 ··· 22 2 ··· 2 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + - + + + - image C1 C2 C2 C2 Q8 D11 D22 D22 Dic22 kernel C22×Dic22 C2×Dic22 C22×Dic11 C22×C44 C2×C22 C22×C4 C2×C4 C23 C22 # reps 1 12 2 1 4 5 30 5 40

Matrix representation of C22×Dic22 in GL6(𝔽89)

 88 0 0 0 0 0 0 88 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 88 0 0 0 0 0 0 88
,
 88 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 0 0 88
,
 0 88 0 0 0 0 1 18 0 0 0 0 0 0 18 1 0 0 0 0 66 68 0 0 0 0 0 0 59 86 0 0 0 0 3 24
,
 25 75 0 0 0 0 70 64 0 0 0 0 0 0 58 8 0 0 0 0 58 31 0 0 0 0 0 0 50 9 0 0 0 0 88 39

G:=sub<GL(6,GF(89))| [88,0,0,0,0,0,0,88,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[88,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[0,1,0,0,0,0,88,18,0,0,0,0,0,0,18,66,0,0,0,0,1,68,0,0,0,0,0,0,59,3,0,0,0,0,86,24],[25,70,0,0,0,0,75,64,0,0,0,0,0,0,58,58,0,0,0,0,8,31,0,0,0,0,0,0,50,88,0,0,0,0,9,39] >;

C22×Dic22 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{22}
% in TeX

G:=Group("C2^2xDic22");
// GroupNames label

G:=SmallGroup(352,173);
// by ID

G=gap.SmallGroup(352,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,579,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^44=1,d^2=c^22,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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