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## G = C22×C5⋊2C16order 320 = 26·5

### Direct product of C22 and C5⋊2C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×C5⋊2C16
 Chief series C1 — C5 — C10 — C20 — C40 — C5⋊2C16 — C2×C5⋊2C16 — C22×C5⋊2C16
 Lower central C5 — C22×C5⋊2C16
 Upper central C1 — C22×C8

Generators and relations for C22×C52C16
G = < a,b,c,d | a2=b2=c5=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 142 in 98 conjugacy classes, 87 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, C23, C10, C10, C16, C2×C8, C22×C4, C20, C20, C2×C10, C2×C16, C22×C8, C40, C40, C2×C20, C22×C10, C22×C16, C52C16, C2×C40, C22×C20, C2×C52C16, C22×C40, C22×C52C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C16, C2×C8, C22×C4, Dic5, D10, C2×C16, C22×C8, C52C8, C2×Dic5, C22×D5, C22×C16, C52C16, C2×C52C8, C22×Dic5, C2×C52C16, C22×C52C8, C22×C52C16

Smallest permutation representation of C22×C52C16
Regular action on 320 points
Generators in S320
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 215)(18 216)(19 217)(20 218)(21 219)(22 220)(23 221)(24 222)(25 223)(26 224)(27 209)(28 210)(29 211)(30 212)(31 213)(32 214)(33 250)(34 251)(35 252)(36 253)(37 254)(38 255)(39 256)(40 241)(41 242)(42 243)(43 244)(44 245)(45 246)(46 247)(47 248)(48 249)(49 288)(50 273)(51 274)(52 275)(53 276)(54 277)(55 278)(56 279)(57 280)(58 281)(59 282)(60 283)(61 284)(62 285)(63 286)(64 287)(65 258)(66 259)(67 260)(68 261)(69 262)(70 263)(71 264)(72 265)(73 266)(74 267)(75 268)(76 269)(77 270)(78 271)(79 272)(80 257)(81 138)(82 139)(83 140)(84 141)(85 142)(86 143)(87 144)(88 129)(89 130)(90 131)(91 132)(92 133)(93 134)(94 135)(95 136)(96 137)(113 293)(114 294)(115 295)(116 296)(117 297)(118 298)(119 299)(120 300)(121 301)(122 302)(123 303)(124 304)(125 289)(126 290)(127 291)(128 292)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 177)(152 178)(153 179)(154 180)(155 181)(156 182)(157 183)(158 184)(159 185)(160 186)(161 313)(162 314)(163 315)(164 316)(165 317)(166 318)(167 319)(168 320)(169 305)(170 306)(171 307)(172 308)(173 309)(174 310)(175 311)(176 312)(193 227)(194 228)(195 229)(196 230)(197 231)(198 232)(199 233)(200 234)(201 235)(202 236)(203 237)(204 238)(205 239)(206 240)(207 225)(208 226)
(1 233)(2 234)(3 235)(4 236)(5 237)(6 238)(7 239)(8 240)(9 225)(10 226)(11 227)(12 228)(13 229)(14 230)(15 231)(16 232)(17 261)(18 262)(19 263)(20 264)(21 265)(22 266)(23 267)(24 268)(25 269)(26 270)(27 271)(28 272)(29 257)(30 258)(31 259)(32 260)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(65 212)(66 213)(67 214)(68 215)(69 216)(70 217)(71 218)(72 219)(73 220)(74 221)(75 222)(76 223)(77 224)(78 209)(79 210)(80 211)(81 314)(82 315)(83 316)(84 317)(85 318)(86 319)(87 320)(88 305)(89 306)(90 307)(91 308)(92 309)(93 310)(94 311)(95 312)(96 313)(97 193)(98 194)(99 195)(100 196)(101 197)(102 198)(103 199)(104 200)(105 201)(106 202)(107 203)(108 204)(109 205)(110 206)(111 207)(112 208)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)(121 158)(122 159)(123 160)(124 145)(125 146)(126 147)(127 148)(128 149)(129 169)(130 170)(131 171)(132 172)(133 173)(134 174)(135 175)(136 176)(137 161)(138 162)(139 163)(140 164)(141 165)(142 166)(143 167)(144 168)(177 294)(178 295)(179 296)(180 297)(181 298)(182 299)(183 300)(184 301)(185 302)(186 303)(187 304)(188 289)(189 290)(190 291)(191 292)(192 293)(241 286)(242 287)(243 288)(244 273)(245 274)(246 275)(247 276)(248 277)(249 278)(250 279)(251 280)(252 281)(253 282)(254 283)(255 284)(256 285)
(1 23 241 188 310)(2 311 189 242 24)(3 25 243 190 312)(4 313 191 244 26)(5 27 245 192 314)(6 315 177 246 28)(7 29 247 178 316)(8 317 179 248 30)(9 31 249 180 318)(10 319 181 250 32)(11 17 251 182 320)(12 305 183 252 18)(13 19 253 184 306)(14 307 185 254 20)(15 21 255 186 308)(16 309 187 256 22)(33 214 112 167 155)(34 156 168 97 215)(35 216 98 169 157)(36 158 170 99 217)(37 218 100 171 159)(38 160 172 101 219)(39 220 102 173 145)(40 146 174 103 221)(41 222 104 175 147)(42 148 176 105 223)(43 224 106 161 149)(44 150 162 107 209)(45 210 108 163 151)(46 152 164 109 211)(47 212 110 165 153)(48 154 166 111 213)(49 127 136 201 76)(50 77 202 137 128)(51 113 138 203 78)(52 79 204 139 114)(53 115 140 205 80)(54 65 206 141 116)(55 117 142 207 66)(56 67 208 143 118)(57 119 144 193 68)(58 69 194 129 120)(59 121 130 195 70)(60 71 196 131 122)(61 123 132 197 72)(62 73 198 133 124)(63 125 134 199 74)(64 75 200 135 126)(81 237 271 274 293)(82 294 275 272 238)(83 239 257 276 295)(84 296 277 258 240)(85 225 259 278 297)(86 298 279 260 226)(87 227 261 280 299)(88 300 281 262 228)(89 229 263 282 301)(90 302 283 264 230)(91 231 265 284 303)(92 304 285 266 232)(93 233 267 286 289)(94 290 287 268 234)(95 235 269 288 291)(96 292 273 270 236)
(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)| (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,215)(18,216)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,209)(28,210)(29,211)(30,212)(31,213)(32,214)(33,250)(34,251)(35,252)(36,253)(37,254)(38,255)(39,256)(40,241)(41,242)(42,243)(43,244)(44,245)(45,246)(46,247)(47,248)(48,249)(49,288)(50,273)(51,274)(52,275)(53,276)(54,277)(55,278)(56,279)(57,280)(58,281)(59,282)(60,283)(61,284)(62,285)(63,286)(64,287)(65,258)(66,259)(67,260)(68,261)(69,262)(70,263)(71,264)(72,265)(73,266)(74,267)(75,268)(76,269)(77,270)(78,271)(79,272)(80,257)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(113,293)(114,294)(115,295)(116,296)(117,297)(118,298)(119,299)(120,300)(121,301)(122,302)(123,303)(124,304)(125,289)(126,290)(127,291)(128,292)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,177)(152,178)(153,179)(154,180)(155,181)(156,182)(157,183)(158,184)(159,185)(160,186)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(169,305)(170,306)(171,307)(172,308)(173,309)(174,310)(175,311)(176,312)(193,227)(194,228)(195,229)(196,230)(197,231)(198,232)(199,233)(200,234)(201,235)(202,236)(203,237)(204,238)(205,239)(206,240)(207,225)(208,226), (1,233)(2,234)(3,235)(4,236)(5,237)(6,238)(7,239)(8,240)(9,225)(10,226)(11,227)(12,228)(13,229)(14,230)(15,231)(16,232)(17,261)(18,262)(19,263)(20,264)(21,265)(22,266)(23,267)(24,268)(25,269)(26,270)(27,271)(28,272)(29,257)(30,258)(31,259)(32,260)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(65,212)(66,213)(67,214)(68,215)(69,216)(70,217)(71,218)(72,219)(73,220)(74,221)(75,222)(76,223)(77,224)(78,209)(79,210)(80,211)(81,314)(82,315)(83,316)(84,317)(85,318)(86,319)(87,320)(88,305)(89,306)(90,307)(91,308)(92,309)(93,310)(94,311)(95,312)(96,313)(97,193)(98,194)(99,195)(100,196)(101,197)(102,198)(103,199)(104,200)(105,201)(106,202)(107,203)(108,204)(109,205)(110,206)(111,207)(112,208)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,145)(125,146)(126,147)(127,148)(128,149)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,161)(138,162)(139,163)(140,164)(141,165)(142,166)(143,167)(144,168)(177,294)(178,295)(179,296)(180,297)(181,298)(182,299)(183,300)(184,301)(185,302)(186,303)(187,304)(188,289)(189,290)(190,291)(191,292)(192,293)(241,286)(242,287)(243,288)(244,273)(245,274)(246,275)(247,276)(248,277)(249,278)(250,279)(251,280)(252,281)(253,282)(254,283)(255,284)(256,285), (1,23,241,188,310)(2,311,189,242,24)(3,25,243,190,312)(4,313,191,244,26)(5,27,245,192,314)(6,315,177,246,28)(7,29,247,178,316)(8,317,179,248,30)(9,31,249,180,318)(10,319,181,250,32)(11,17,251,182,320)(12,305,183,252,18)(13,19,253,184,306)(14,307,185,254,20)(15,21,255,186,308)(16,309,187,256,22)(33,214,112,167,155)(34,156,168,97,215)(35,216,98,169,157)(36,158,170,99,217)(37,218,100,171,159)(38,160,172,101,219)(39,220,102,173,145)(40,146,174,103,221)(41,222,104,175,147)(42,148,176,105,223)(43,224,106,161,149)(44,150,162,107,209)(45,210,108,163,151)(46,152,164,109,211)(47,212,110,165,153)(48,154,166,111,213)(49,127,136,201,76)(50,77,202,137,128)(51,113,138,203,78)(52,79,204,139,114)(53,115,140,205,80)(54,65,206,141,116)(55,117,142,207,66)(56,67,208,143,118)(57,119,144,193,68)(58,69,194,129,120)(59,121,130,195,70)(60,71,196,131,122)(61,123,132,197,72)(62,73,198,133,124)(63,125,134,199,74)(64,75,200,135,126)(81,237,271,274,293)(82,294,275,272,238)(83,239,257,276,295)(84,296,277,258,240)(85,225,259,278,297)(86,298,279,260,226)(87,227,261,280,299)(88,300,281,262,228)(89,229,263,282,301)(90,302,283,264,230)(91,231,265,284,303)(92,304,285,266,232)(93,233,267,286,289)(94,290,287,268,234)(95,235,269,288,291)(96,292,273,270,236), (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( (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,111)(10,112)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,215)(18,216)(19,217)(20,218)(21,219)(22,220)(23,221)(24,222)(25,223)(26,224)(27,209)(28,210)(29,211)(30,212)(31,213)(32,214)(33,250)(34,251)(35,252)(36,253)(37,254)(38,255)(39,256)(40,241)(41,242)(42,243)(43,244)(44,245)(45,246)(46,247)(47,248)(48,249)(49,288)(50,273)(51,274)(52,275)(53,276)(54,277)(55,278)(56,279)(57,280)(58,281)(59,282)(60,283)(61,284)(62,285)(63,286)(64,287)(65,258)(66,259)(67,260)(68,261)(69,262)(70,263)(71,264)(72,265)(73,266)(74,267)(75,268)(76,269)(77,270)(78,271)(79,272)(80,257)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(113,293)(114,294)(115,295)(116,296)(117,297)(118,298)(119,299)(120,300)(121,301)(122,302)(123,303)(124,304)(125,289)(126,290)(127,291)(128,292)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,177)(152,178)(153,179)(154,180)(155,181)(156,182)(157,183)(158,184)(159,185)(160,186)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(169,305)(170,306)(171,307)(172,308)(173,309)(174,310)(175,311)(176,312)(193,227)(194,228)(195,229)(196,230)(197,231)(198,232)(199,233)(200,234)(201,235)(202,236)(203,237)(204,238)(205,239)(206,240)(207,225)(208,226), (1,233)(2,234)(3,235)(4,236)(5,237)(6,238)(7,239)(8,240)(9,225)(10,226)(11,227)(12,228)(13,229)(14,230)(15,231)(16,232)(17,261)(18,262)(19,263)(20,264)(21,265)(22,266)(23,267)(24,268)(25,269)(26,270)(27,271)(28,272)(29,257)(30,258)(31,259)(32,260)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(65,212)(66,213)(67,214)(68,215)(69,216)(70,217)(71,218)(72,219)(73,220)(74,221)(75,222)(76,223)(77,224)(78,209)(79,210)(80,211)(81,314)(82,315)(83,316)(84,317)(85,318)(86,319)(87,320)(88,305)(89,306)(90,307)(91,308)(92,309)(93,310)(94,311)(95,312)(96,313)(97,193)(98,194)(99,195)(100,196)(101,197)(102,198)(103,199)(104,200)(105,201)(106,202)(107,203)(108,204)(109,205)(110,206)(111,207)(112,208)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,145)(125,146)(126,147)(127,148)(128,149)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,161)(138,162)(139,163)(140,164)(141,165)(142,166)(143,167)(144,168)(177,294)(178,295)(179,296)(180,297)(181,298)(182,299)(183,300)(184,301)(185,302)(186,303)(187,304)(188,289)(189,290)(190,291)(191,292)(192,293)(241,286)(242,287)(243,288)(244,273)(245,274)(246,275)(247,276)(248,277)(249,278)(250,279)(251,280)(252,281)(253,282)(254,283)(255,284)(256,285), (1,23,241,188,310)(2,311,189,242,24)(3,25,243,190,312)(4,313,191,244,26)(5,27,245,192,314)(6,315,177,246,28)(7,29,247,178,316)(8,317,179,248,30)(9,31,249,180,318)(10,319,181,250,32)(11,17,251,182,320)(12,305,183,252,18)(13,19,253,184,306)(14,307,185,254,20)(15,21,255,186,308)(16,309,187,256,22)(33,214,112,167,155)(34,156,168,97,215)(35,216,98,169,157)(36,158,170,99,217)(37,218,100,171,159)(38,160,172,101,219)(39,220,102,173,145)(40,146,174,103,221)(41,222,104,175,147)(42,148,176,105,223)(43,224,106,161,149)(44,150,162,107,209)(45,210,108,163,151)(46,152,164,109,211)(47,212,110,165,153)(48,154,166,111,213)(49,127,136,201,76)(50,77,202,137,128)(51,113,138,203,78)(52,79,204,139,114)(53,115,140,205,80)(54,65,206,141,116)(55,117,142,207,66)(56,67,208,143,118)(57,119,144,193,68)(58,69,194,129,120)(59,121,130,195,70)(60,71,196,131,122)(61,123,132,197,72)(62,73,198,133,124)(63,125,134,199,74)(64,75,200,135,126)(81,237,271,274,293)(82,294,275,272,238)(83,239,257,276,295)(84,296,277,258,240)(85,225,259,278,297)(86,298,279,260,226)(87,227,261,280,299)(88,300,281,262,228)(89,229,263,282,301)(90,302,283,264,230)(91,231,265,284,303)(92,304,285,266,232)(93,233,267,286,289)(94,290,287,268,234)(95,235,269,288,291)(96,292,273,270,236), (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([[(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,111),(10,112),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,215),(18,216),(19,217),(20,218),(21,219),(22,220),(23,221),(24,222),(25,223),(26,224),(27,209),(28,210),(29,211),(30,212),(31,213),(32,214),(33,250),(34,251),(35,252),(36,253),(37,254),(38,255),(39,256),(40,241),(41,242),(42,243),(43,244),(44,245),(45,246),(46,247),(47,248),(48,249),(49,288),(50,273),(51,274),(52,275),(53,276),(54,277),(55,278),(56,279),(57,280),(58,281),(59,282),(60,283),(61,284),(62,285),(63,286),(64,287),(65,258),(66,259),(67,260),(68,261),(69,262),(70,263),(71,264),(72,265),(73,266),(74,267),(75,268),(76,269),(77,270),(78,271),(79,272),(80,257),(81,138),(82,139),(83,140),(84,141),(85,142),(86,143),(87,144),(88,129),(89,130),(90,131),(91,132),(92,133),(93,134),(94,135),(95,136),(96,137),(113,293),(114,294),(115,295),(116,296),(117,297),(118,298),(119,299),(120,300),(121,301),(122,302),(123,303),(124,304),(125,289),(126,290),(127,291),(128,292),(145,187),(146,188),(147,189),(148,190),(149,191),(150,192),(151,177),(152,178),(153,179),(154,180),(155,181),(156,182),(157,183),(158,184),(159,185),(160,186),(161,313),(162,314),(163,315),(164,316),(165,317),(166,318),(167,319),(168,320),(169,305),(170,306),(171,307),(172,308),(173,309),(174,310),(175,311),(176,312),(193,227),(194,228),(195,229),(196,230),(197,231),(198,232),(199,233),(200,234),(201,235),(202,236),(203,237),(204,238),(205,239),(206,240),(207,225),(208,226)], [(1,233),(2,234),(3,235),(4,236),(5,237),(6,238),(7,239),(8,240),(9,225),(10,226),(11,227),(12,228),(13,229),(14,230),(15,231),(16,232),(17,261),(18,262),(19,263),(20,264),(21,265),(22,266),(23,267),(24,268),(25,269),(26,270),(27,271),(28,272),(29,257),(30,258),(31,259),(32,260),(33,56),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(65,212),(66,213),(67,214),(68,215),(69,216),(70,217),(71,218),(72,219),(73,220),(74,221),(75,222),(76,223),(77,224),(78,209),(79,210),(80,211),(81,314),(82,315),(83,316),(84,317),(85,318),(86,319),(87,320),(88,305),(89,306),(90,307),(91,308),(92,309),(93,310),(94,311),(95,312),(96,313),(97,193),(98,194),(99,195),(100,196),(101,197),(102,198),(103,199),(104,200),(105,201),(106,202),(107,203),(108,204),(109,205),(110,206),(111,207),(112,208),(113,150),(114,151),(115,152),(116,153),(117,154),(118,155),(119,156),(120,157),(121,158),(122,159),(123,160),(124,145),(125,146),(126,147),(127,148),(128,149),(129,169),(130,170),(131,171),(132,172),(133,173),(134,174),(135,175),(136,176),(137,161),(138,162),(139,163),(140,164),(141,165),(142,166),(143,167),(144,168),(177,294),(178,295),(179,296),(180,297),(181,298),(182,299),(183,300),(184,301),(185,302),(186,303),(187,304),(188,289),(189,290),(190,291),(191,292),(192,293),(241,286),(242,287),(243,288),(244,273),(245,274),(246,275),(247,276),(248,277),(249,278),(250,279),(251,280),(252,281),(253,282),(254,283),(255,284),(256,285)], [(1,23,241,188,310),(2,311,189,242,24),(3,25,243,190,312),(4,313,191,244,26),(5,27,245,192,314),(6,315,177,246,28),(7,29,247,178,316),(8,317,179,248,30),(9,31,249,180,318),(10,319,181,250,32),(11,17,251,182,320),(12,305,183,252,18),(13,19,253,184,306),(14,307,185,254,20),(15,21,255,186,308),(16,309,187,256,22),(33,214,112,167,155),(34,156,168,97,215),(35,216,98,169,157),(36,158,170,99,217),(37,218,100,171,159),(38,160,172,101,219),(39,220,102,173,145),(40,146,174,103,221),(41,222,104,175,147),(42,148,176,105,223),(43,224,106,161,149),(44,150,162,107,209),(45,210,108,163,151),(46,152,164,109,211),(47,212,110,165,153),(48,154,166,111,213),(49,127,136,201,76),(50,77,202,137,128),(51,113,138,203,78),(52,79,204,139,114),(53,115,140,205,80),(54,65,206,141,116),(55,117,142,207,66),(56,67,208,143,118),(57,119,144,193,68),(58,69,194,129,120),(59,121,130,195,70),(60,71,196,131,122),(61,123,132,197,72),(62,73,198,133,124),(63,125,134,199,74),(64,75,200,135,126),(81,237,271,274,293),(82,294,275,272,238),(83,239,257,276,295),(84,296,277,258,240),(85,225,259,278,297),(86,298,279,260,226),(87,227,261,280,299),(88,300,281,262,228),(89,229,263,282,301),(90,302,283,264,230),(91,231,265,284,303),(92,304,285,266,232),(93,233,267,286,289),(94,290,287,268,234),(95,235,269,288,291),(96,292,273,270,236)], [(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)]])

128 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5A 5B 8A ··· 8P 10A ··· 10N 16A ··· 16AF 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 4 ··· 4 5 5 8 ··· 8 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 size 1 1 ··· 1 1 ··· 1 2 2 1 ··· 1 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 D5 Dic5 D10 Dic5 C5⋊2C8 C5⋊2C8 C5⋊2C16 kernel C22×C5⋊2C16 C2×C5⋊2C16 C22×C40 C2×C40 C22×C20 C2×C20 C22×C10 C2×C10 C22×C8 C2×C8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 12 4 32 2 6 6 2 12 4 32

Matrix representation of C22×C52C16 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 190 240 0 0 191 240
,
 197 0 0 0 0 240 0 0 0 0 115 101 0 0 45 126
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,190,191,0,0,240,240],[197,0,0,0,0,240,0,0,0,0,115,45,0,0,101,126] >;

C22×C52C16 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes_2C_{16}
% in TeX

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

G:=SmallGroup(320,723);
// by ID

G=gap.SmallGroup(320,723);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,80,102,12550]);
// 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^-1>;
// generators/relations

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