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G = Dic22⋊C4order 352 = 25·11

5th semidirect product of Dic22 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic22⋊C4
G = < a,b,c,d | a22=c4=1, b2=a11, d2=c2, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 306 in 70 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, C2×C4, Q8, C11, C42, C4⋊C4, C4⋊C4, C2×Q8, C22, C4×Q8, Dic11, Dic11, C44, C44, C2×C22, Dic22, C2×Dic11, C2×Dic11, C2×C44, C2×C44, C4×Dic11, C4×Dic11, Dic11⋊C4, C11×C4⋊C4, C2×Dic22, Dic22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, D11, C4×Q8, D22, C4×D11, C22×D11, C2×C4×D11, D42D11, Q8×D11, Dic22⋊C4

Smallest permutation representation of Dic22⋊C4
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)
(1 210 12 199)(2 209 13 220)(3 208 14 219)(4 207 15 218)(5 206 16 217)(6 205 17 216)(7 204 18 215)(8 203 19 214)(9 202 20 213)(10 201 21 212)(11 200 22 211)(23 195 34 184)(24 194 35 183)(25 193 36 182)(26 192 37 181)(27 191 38 180)(28 190 39 179)(29 189 40 178)(30 188 41 177)(31 187 42 198)(32 186 43 197)(33 185 44 196)(45 243 56 254)(46 264 57 253)(47 263 58 252)(48 262 59 251)(49 261 60 250)(50 260 61 249)(51 259 62 248)(52 258 63 247)(53 257 64 246)(54 256 65 245)(55 255 66 244)(67 226 78 237)(68 225 79 236)(69 224 80 235)(70 223 81 234)(71 222 82 233)(72 221 83 232)(73 242 84 231)(74 241 85 230)(75 240 86 229)(76 239 87 228)(77 238 88 227)(89 288 100 299)(90 287 101 298)(91 308 102 297)(92 307 103 296)(93 306 104 295)(94 305 105 294)(95 304 106 293)(96 303 107 292)(97 302 108 291)(98 301 109 290)(99 300 110 289)(111 265 122 276)(112 286 123 275)(113 285 124 274)(114 284 125 273)(115 283 126 272)(116 282 127 271)(117 281 128 270)(118 280 129 269)(119 279 130 268)(120 278 131 267)(121 277 132 266)(133 337 144 348)(134 336 145 347)(135 335 146 346)(136 334 147 345)(137 333 148 344)(138 332 149 343)(139 331 150 342)(140 352 151 341)(141 351 152 340)(142 350 153 339)(143 349 154 338)(155 319 166 330)(156 318 167 329)(157 317 168 328)(158 316 169 327)(159 315 170 326)(160 314 171 325)(161 313 172 324)(162 312 173 323)(163 311 174 322)(164 310 175 321)(165 309 176 320)
(1 221 30 254)(2 242 31 253)(3 241 32 252)(4 240 33 251)(5 239 34 250)(6 238 35 249)(7 237 36 248)(8 236 37 247)(9 235 38 246)(10 234 39 245)(11 233 40 244)(12 232 41 243)(13 231 42 264)(14 230 43 263)(15 229 44 262)(16 228 23 261)(17 227 24 260)(18 226 25 259)(19 225 26 258)(20 224 27 257)(21 223 28 256)(22 222 29 255)(45 210 83 188)(46 209 84 187)(47 208 85 186)(48 207 86 185)(49 206 87 184)(50 205 88 183)(51 204 67 182)(52 203 68 181)(53 202 69 180)(54 201 70 179)(55 200 71 178)(56 199 72 177)(57 220 73 198)(58 219 74 197)(59 218 75 196)(60 217 76 195)(61 216 77 194)(62 215 78 193)(63 214 79 192)(64 213 80 191)(65 212 81 190)(66 211 82 189)(89 343 121 321)(90 342 122 320)(91 341 123 319)(92 340 124 318)(93 339 125 317)(94 338 126 316)(95 337 127 315)(96 336 128 314)(97 335 129 313)(98 334 130 312)(99 333 131 311)(100 332 132 310)(101 331 111 309)(102 352 112 330)(103 351 113 329)(104 350 114 328)(105 349 115 327)(106 348 116 326)(107 347 117 325)(108 346 118 324)(109 345 119 323)(110 344 120 322)(133 282 159 293)(134 281 160 292)(135 280 161 291)(136 279 162 290)(137 278 163 289)(138 277 164 288)(139 276 165 287)(140 275 166 308)(141 274 167 307)(142 273 168 306)(143 272 169 305)(144 271 170 304)(145 270 171 303)(146 269 172 302)(147 268 173 301)(148 267 174 300)(149 266 175 299)(150 265 176 298)(151 286 155 297)(152 285 156 296)(153 284 157 295)(154 283 158 294)
(1 111 30 101)(2 112 31 102)(3 113 32 103)(4 114 33 104)(5 115 34 105)(6 116 35 106)(7 117 36 107)(8 118 37 108)(9 119 38 109)(10 120 39 110)(11 121 40 89)(12 122 41 90)(13 123 42 91)(14 124 43 92)(15 125 44 93)(16 126 23 94)(17 127 24 95)(18 128 25 96)(19 129 26 97)(20 130 27 98)(21 131 28 99)(22 132 29 100)(45 176 83 150)(46 155 84 151)(47 156 85 152)(48 157 86 153)(49 158 87 154)(50 159 88 133)(51 160 67 134)(52 161 68 135)(53 162 69 136)(54 163 70 137)(55 164 71 138)(56 165 72 139)(57 166 73 140)(58 167 74 141)(59 168 75 142)(60 169 76 143)(61 170 77 144)(62 171 78 145)(63 172 79 146)(64 173 80 147)(65 174 81 148)(66 175 82 149)(177 287 199 276)(178 288 200 277)(179 289 201 278)(180 290 202 279)(181 291 203 280)(182 292 204 281)(183 293 205 282)(184 294 206 283)(185 295 207 284)(186 296 208 285)(187 297 209 286)(188 298 210 265)(189 299 211 266)(190 300 212 267)(191 301 213 268)(192 302 214 269)(193 303 215 270)(194 304 216 271)(195 305 217 272)(196 306 218 273)(197 307 219 274)(198 308 220 275)(221 331 254 309)(222 332 255 310)(223 333 256 311)(224 334 257 312)(225 335 258 313)(226 336 259 314)(227 337 260 315)(228 338 261 316)(229 339 262 317)(230 340 263 318)(231 341 264 319)(232 342 243 320)(233 343 244 321)(234 344 245 322)(235 345 246 323)(236 346 247 324)(237 347 248 325)(238 348 249 326)(239 349 250 327)(240 350 251 328)(241 351 252 329)(242 352 253 330)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 11A ··· 11E 22A ··· 22O 44A ··· 44AD order 1 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 2 ··· 2 11 11 11 11 22 ··· 22 2 ··· 2 2 ··· 2 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + + - - image C1 C2 C2 C2 C2 C4 Q8 C4○D4 D11 D22 C4×D11 D4⋊2D11 Q8×D11 kernel Dic22⋊C4 C4×Dic11 Dic11⋊C4 C11×C4⋊C4 C2×Dic22 Dic22 Dic11 C22 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 3 2 1 1 8 2 2 5 15 20 5 5

Matrix representation of Dic22⋊C4 in GL4(𝔽89) generated by

 42 88 0 0 1 0 0 0 0 0 88 0 0 0 0 88
,
 46 43 0 0 17 43 0 0 0 0 34 0 0 0 0 34
,
 38 51 0 0 45 51 0 0 0 0 1 68 0 0 34 88
,
 88 0 0 0 0 88 0 0 0 0 73 3 0 0 33 16
G:=sub<GL(4,GF(89))| [42,1,0,0,88,0,0,0,0,0,88,0,0,0,0,88],[46,17,0,0,43,43,0,0,0,0,34,0,0,0,0,34],[38,45,0,0,51,51,0,0,0,0,1,34,0,0,68,88],[88,0,0,0,0,88,0,0,0,0,73,33,0,0,3,16] >;

Dic22⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{22}\rtimes C_4
% in TeX

G:=Group("Dic22:C4");
// GroupNames label

G:=SmallGroup(352,82);
// by ID

G=gap.SmallGroup(352,82);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,55,116,122,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^22=c^4=1,b^2=a^11,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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