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## G = C22×C5⋊C16order 320 = 26·5

### Direct product of C22 and C5⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×C5⋊C16
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16 — C22×C5⋊C16
 Lower central C5 — C22×C5⋊C16
 Upper central C1 — C22×C4

Generators and relations for C22×C5⋊C16
G = < a,b,c,d | a2=b2=c5=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 186 in 98 conjugacy classes, 76 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C23, C10, C10, C16, C2×C8, C22×C4, C20, C20, C2×C10, C2×C16, C22×C8, C52C8, C52C8, C2×C20, C22×C10, C22×C16, C5⋊C16, C2×C52C8, C22×C20, C2×C5⋊C16, C22×C52C8, C22×C5⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C16, C2×C8, C22×C4, F5, C2×C16, C22×C8, C5⋊C8, C2×F5, C22×C16, C5⋊C16, C2×C5⋊C8, C22×F5, C2×C5⋊C16, C22×C5⋊C8, C22×C5⋊C16

Smallest permutation representation of C22×C5⋊C16
Regular action on 320 points
Generators in S320
(1 174)(2 175)(3 176)(4 161)(5 162)(6 163)(7 164)(8 165)(9 166)(10 167)(11 168)(12 169)(13 170)(14 171)(15 172)(16 173)(17 211)(18 212)(19 213)(20 214)(21 215)(22 216)(23 217)(24 218)(25 219)(26 220)(27 221)(28 222)(29 223)(30 224)(31 209)(32 210)(33 182)(34 183)(35 184)(36 185)(37 186)(38 187)(39 188)(40 189)(41 190)(42 191)(43 192)(44 177)(45 178)(46 179)(47 180)(48 181)(49 301)(50 302)(51 303)(52 304)(53 289)(54 290)(55 291)(56 292)(57 293)(58 294)(59 295)(60 296)(61 297)(62 298)(63 299)(64 300)(65 317)(66 318)(67 319)(68 320)(69 305)(70 306)(71 307)(72 308)(73 309)(74 310)(75 311)(76 312)(77 313)(78 314)(79 315)(80 316)(81 143)(82 144)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 246)(98 247)(99 248)(100 249)(101 250)(102 251)(103 252)(104 253)(105 254)(106 255)(107 256)(108 241)(109 242)(110 243)(111 244)(112 245)(113 270)(114 271)(115 272)(116 257)(117 258)(118 259)(119 260)(120 261)(121 262)(122 263)(123 264)(124 265)(125 266)(126 267)(127 268)(128 269)(145 287)(146 288)(147 273)(148 274)(149 275)(150 276)(151 277)(152 278)(153 279)(154 280)(155 281)(156 282)(157 283)(158 284)(159 285)(160 286)(193 232)(194 233)(195 234)(196 235)(197 236)(198 237)(199 238)(200 239)(201 240)(202 225)(203 226)(204 227)(205 228)(206 229)(207 230)(208 231)
(1 297)(2 298)(3 299)(4 300)(5 301)(6 302)(7 303)(8 304)(9 289)(10 290)(11 291)(12 292)(13 293)(14 294)(15 295)(16 296)(17 206)(18 207)(19 208)(20 193)(21 194)(22 195)(23 196)(24 197)(25 198)(26 199)(27 200)(28 201)(29 202)(30 203)(31 204)(32 205)(33 156)(34 157)(35 158)(36 159)(37 160)(38 145)(39 146)(40 147)(41 148)(42 149)(43 150)(44 151)(45 152)(46 153)(47 154)(48 155)(49 162)(50 163)(51 164)(52 165)(53 166)(54 167)(55 168)(56 169)(57 170)(58 171)(59 172)(60 173)(61 174)(62 175)(63 176)(64 161)(65 247)(66 248)(67 249)(68 250)(69 251)(70 252)(71 253)(72 254)(73 255)(74 256)(75 241)(76 242)(77 243)(78 244)(79 245)(80 246)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 121)(97 316)(98 317)(99 318)(100 319)(101 320)(102 305)(103 306)(104 307)(105 308)(106 309)(107 310)(108 311)(109 312)(110 313)(111 314)(112 315)(129 265)(130 266)(131 267)(132 268)(133 269)(134 270)(135 271)(136 272)(137 257)(138 258)(139 259)(140 260)(141 261)(142 262)(143 263)(144 264)(177 277)(178 278)(179 279)(180 280)(181 281)(182 282)(183 283)(184 284)(185 285)(186 286)(187 287)(188 288)(189 273)(190 274)(191 275)(192 276)(209 227)(210 228)(211 229)(212 230)(213 231)(214 232)(215 233)(216 234)(217 235)(218 236)(219 237)(220 238)(221 239)(222 240)(223 225)(224 226)
(1 29 46 122 101)(2 123 30 102 47)(3 103 124 48 31)(4 33 104 32 125)(5 17 34 126 105)(6 127 18 106 35)(7 107 128 36 19)(8 37 108 20 113)(9 21 38 114 109)(10 115 22 110 39)(11 111 116 40 23)(12 41 112 24 117)(13 25 42 118 97)(14 119 26 98 43)(15 99 120 44 27)(16 45 100 28 121)(49 229 283 131 72)(50 132 230 73 284)(51 74 133 285 231)(52 286 75 232 134)(53 233 287 135 76)(54 136 234 77 288)(55 78 137 273 235)(56 274 79 236 138)(57 237 275 139 80)(58 140 238 65 276)(59 66 141 277 239)(60 278 67 240 142)(61 225 279 143 68)(62 144 226 69 280)(63 70 129 281 227)(64 282 71 228 130)(81 320 297 202 153)(82 203 305 154 298)(83 155 204 299 306)(84 300 156 307 205)(85 308 301 206 157)(86 207 309 158 302)(87 159 208 303 310)(88 304 160 311 193)(89 312 289 194 145)(90 195 313 146 290)(91 147 196 291 314)(92 292 148 315 197)(93 316 293 198 149)(94 199 317 150 294)(95 151 200 295 318)(96 296 152 319 201)(161 182 253 210 266)(162 211 183 267 254)(163 268 212 255 184)(164 256 269 185 213)(165 186 241 214 270)(166 215 187 271 242)(167 272 216 243 188)(168 244 257 189 217)(169 190 245 218 258)(170 219 191 259 246)(171 260 220 247 192)(172 248 261 177 221)(173 178 249 222 262)(174 223 179 263 250)(175 264 224 251 180)(176 252 265 181 209)
(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)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5 8A ··· 8P 10A ··· 10G 16A ··· 16AF 20A ··· 20H order 1 2 ··· 2 4 ··· 4 5 8 ··· 8 10 ··· 10 16 ··· 16 20 ··· 20 size 1 1 ··· 1 1 ··· 1 4 5 ··· 5 4 ··· 4 5 ··· 5 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 F5 C5⋊C8 C2×F5 C5⋊C8 C5⋊C16 kernel C22×C5⋊C16 C2×C5⋊C16 C22×C5⋊2C8 C2×C5⋊2C8 C22×C20 C2×C20 C22×C10 C2×C10 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 12 4 32 1 3 3 1 8

Matrix representation of C22×C5⋊C16 in GL6(𝔽241)

 1 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 240 0 0 0 1 0 240 0 0 0 0 1 240
,
 44 0 0 0 0 0 0 177 0 0 0 0 0 0 29 104 160 85 0 0 189 189 23 114 0 0 52 218 127 33 0 0 156 137 212 137

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[44,0,0,0,0,0,0,177,0,0,0,0,0,0,29,189,52,156,0,0,104,189,218,137,0,0,160,23,127,212,0,0,85,114,33,137] >;

C22×C5⋊C16 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes C_{16}
% in TeX

G:=Group("C2^2xC5:C16");
// GroupNames label

G:=SmallGroup(320,1080);
// by ID

G=gap.SmallGroup(320,1080);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^5=d^16=1,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^3>;
// generators/relations

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