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## G = C5×C2.Q32order 320 = 26·5

### Direct product of C5 and C2.Q32

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×C2.Q32
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C2.D8 — C5×C2.Q32
 Lower central C1 — C2 — C4 — C8 — C5×C2.Q32
 Upper central C1 — C2×C10 — C2×C20 — C2×C40 — C5×C2.Q32

Generators and relations for C5×C2.Q32
G = < a,b,c,d | a5=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 114 in 58 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C16, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, C20, C20, C2×C10, C2.D8, C2×C16, C2×Q16, C40, C2×C20, C2×C20, C5×Q8, C2.Q32, C80, C5×C4⋊C4, C2×C40, C5×Q16, C5×Q16, Q8×C10, C5×C2.D8, C2×C80, C10×Q16, C5×C2.Q32
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, D8, SD16, C20, C2×C10, D4⋊C4, SD32, Q32, C2×C20, C5×D4, C2.Q32, C5×C22⋊C4, C5×D8, C5×SD16, C5×D4⋊C4, C5×SD32, C5×Q32, C5×C2.Q32

Smallest permutation representation of C5×C2.Q32
Regular action on 320 points
Generators in S320
(1 79 260 252 129)(2 80 261 253 130)(3 65 262 254 131)(4 66 263 255 132)(5 67 264 256 133)(6 68 265 241 134)(7 69 266 242 135)(8 70 267 243 136)(9 71 268 244 137)(10 72 269 245 138)(11 73 270 246 139)(12 74 271 247 140)(13 75 272 248 141)(14 76 257 249 142)(15 77 258 250 143)(16 78 259 251 144)(17 238 150 274 303)(18 239 151 275 304)(19 240 152 276 289)(20 225 153 277 290)(21 226 154 278 291)(22 227 155 279 292)(23 228 156 280 293)(24 229 157 281 294)(25 230 158 282 295)(26 231 159 283 296)(27 232 160 284 297)(28 233 145 285 298)(29 234 146 286 299)(30 235 147 287 300)(31 236 148 288 301)(32 237 149 273 302)(33 49 193 94 189)(34 50 194 95 190)(35 51 195 96 191)(36 52 196 81 192)(37 53 197 82 177)(38 54 198 83 178)(39 55 199 84 179)(40 56 200 85 180)(41 57 201 86 181)(42 58 202 87 182)(43 59 203 88 183)(44 60 204 89 184)(45 61 205 90 185)(46 62 206 91 186)(47 63 207 92 187)(48 64 208 93 188)(97 162 214 319 128)(98 163 215 320 113)(99 164 216 305 114)(100 165 217 306 115)(101 166 218 307 116)(102 167 219 308 117)(103 168 220 309 118)(104 169 221 310 119)(105 170 222 311 120)(106 171 223 312 121)(107 172 224 313 122)(108 173 209 314 123)(109 174 210 315 124)(110 175 211 316 125)(111 176 212 317 126)(112 161 213 318 127)
(1 181)(2 182)(3 183)(4 184)(5 185)(6 186)(7 187)(8 188)(9 189)(10 190)(11 191)(12 192)(13 177)(14 178)(15 179)(16 180)(17 320)(18 305)(19 306)(20 307)(21 308)(22 309)(23 310)(24 311)(25 312)(26 313)(27 314)(28 315)(29 316)(30 317)(31 318)(32 319)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 268)(50 269)(51 270)(52 271)(53 272)(54 257)(55 258)(56 259)(57 260)(58 261)(59 262)(60 263)(61 264)(62 265)(63 266)(64 267)(81 140)(82 141)(83 142)(84 143)(85 144)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(105 157)(106 158)(107 159)(108 160)(109 145)(110 146)(111 147)(112 148)(113 238)(114 239)(115 240)(116 225)(117 226)(118 227)(119 228)(120 229)(121 230)(122 231)(123 232)(124 233)(125 234)(126 235)(127 236)(128 237)(161 288)(162 273)(163 274)(164 275)(165 276)(166 277)(167 278)(168 279)(169 280)(170 281)(171 282)(172 283)(173 284)(174 285)(175 286)(176 287)(193 244)(194 245)(195 246)(196 247)(197 248)(198 249)(199 250)(200 251)(201 252)(202 253)(203 254)(204 255)(205 256)(206 241)(207 242)(208 243)(209 297)(210 298)(211 299)(212 300)(213 301)(214 302)(215 303)(216 304)(217 289)(218 290)(219 291)(220 292)(221 293)(222 294)(223 295)(224 296)
(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)
(1 159 9 151)(2 106 10 98)(3 157 11 149)(4 104 12 112)(5 155 13 147)(6 102 14 110)(7 153 15 145)(8 100 16 108)(17 202 25 194)(18 252 26 244)(19 200 27 208)(20 250 28 242)(21 198 29 206)(22 248 30 256)(23 196 31 204)(24 246 32 254)(33 164 41 172)(34 274 42 282)(35 162 43 170)(36 288 44 280)(37 176 45 168)(38 286 46 278)(39 174 47 166)(40 284 48 276)(49 216 57 224)(50 303 58 295)(51 214 59 222)(52 301 60 293)(53 212 61 220)(54 299 62 291)(55 210 63 218)(56 297 64 289)(65 281 73 273)(66 169 74 161)(67 279 75 287)(68 167 76 175)(69 277 77 285)(70 165 78 173)(71 275 79 283)(72 163 80 171)(81 236 89 228)(82 126 90 118)(83 234 91 226)(84 124 92 116)(85 232 93 240)(86 122 94 114)(87 230 95 238)(88 120 96 128)(97 183 105 191)(99 181 107 189)(101 179 109 187)(103 177 111 185)(113 130 121 138)(115 144 123 136)(117 142 125 134)(119 140 127 132)(129 231 137 239)(131 229 139 237)(133 227 141 235)(135 225 143 233)(146 186 154 178)(148 184 156 192)(150 182 158 190)(152 180 160 188)(193 305 201 313)(195 319 203 311)(197 317 205 309)(199 315 207 307)(209 267 217 259)(211 265 219 257)(213 263 221 271)(215 261 223 269)(241 308 249 316)(243 306 251 314)(245 320 253 312)(247 318 255 310)(258 298 266 290)(260 296 268 304)(262 294 270 302)(264 292 272 300)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 16A ··· 16H 20A ··· 20H 20I ··· 20X 40A ··· 40P 80A ··· 80AF order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 2 2 8 8 8 8 1 1 1 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D4 D4 SD16 D8 SD32 Q32 C5×D4 C5×D4 C5×SD16 C5×D8 C5×SD32 C5×Q32 kernel C5×C2.Q32 C5×C2.D8 C2×C80 C10×Q16 C5×Q16 C2.Q32 C2.D8 C2×C16 C2×Q16 Q16 C40 C2×C20 C20 C2×C10 C10 C10 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2 2 4 4 4 4 8 8 16 16

Matrix representation of C5×C2.Q32 in GL3(𝔽241) generated by

 1 0 0 0 87 0 0 0 87
,
 240 0 0 0 1 0 0 0 1
,
 177 0 0 0 214 156 0 85 214
,
 1 0 0 0 51 234 0 234 190
G:=sub<GL(3,GF(241))| [1,0,0,0,87,0,0,0,87],[240,0,0,0,1,0,0,0,1],[177,0,0,0,214,85,0,156,214],[1,0,0,0,51,234,0,234,190] >;

C5×C2.Q32 in GAP, Magma, Sage, TeX

C_5\times C_2.Q_{32}
% in TeX

G:=Group("C5xC2.Q32");
// GroupNames label

G:=SmallGroup(320,163);
// by ID

G=gap.SmallGroup(320,163);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,1410,360,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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