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## G = C2×C5⋊Q32order 320 = 26·5

### Direct product of C2 and C5⋊Q32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C2×C5⋊Q32
 Chief series C1 — C5 — C10 — C20 — C40 — Dic20 — C2×Dic20 — C2×C5⋊Q32
 Lower central C5 — C10 — C20 — C40 — C2×C5⋊Q32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for C2×C5⋊Q32
G = < a,b,c,d | a2=b5=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 286 in 82 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C10, C16, C2×C8, Q16, Q16, C2×Q8, Dic5, C20, C20, C2×C10, C2×C16, Q32, C2×Q16, C2×Q16, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×Q32, C52C16, Dic20, Dic20, C2×C40, C5×Q16, C5×Q16, C2×Dic10, Q8×C10, C2×C52C16, C5⋊Q32, C2×Dic20, C10×Q16, C2×C5⋊Q32
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, Q32, C2×D8, C5⋊D4, C22×D5, C2×Q32, D4⋊D5, C2×C5⋊D4, C5⋊Q32, C2×D4⋊D5, C2×C5⋊Q32

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 Q32 C5⋊D4 C5⋊D4 D4⋊D5 D4⋊D5 C5⋊Q32 kernel C2×C5⋊Q32 C2×C5⋊2C16 C5⋊Q32 C2×Dic20 C10×Q16 C40 C2×C20 C2×Q16 C20 C2×C10 C2×C8 Q16 C10 C8 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 1 1 2 2 2 2 4 8 4 4 2 2 8

Matrix representation of C2×C5⋊Q32 in GL5(𝔽241)

 240 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 190 240 0 0 0 191 240
,
 240 0 0 0 0 0 85 214 0 0 0 27 85 0 0 0 0 0 45 230 0 0 0 228 196
,
 1 0 0 0 0 0 234 190 0 0 0 190 7 0 0 0 0 0 165 103 0 0 0 89 76

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,190,191,0,0,0,240,240],[240,0,0,0,0,0,85,27,0,0,0,214,85,0,0,0,0,0,45,228,0,0,0,230,196],[1,0,0,0,0,0,234,190,0,0,0,190,7,0,0,0,0,0,165,89,0,0,0,103,76] >;

C2×C5⋊Q32 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC5:Q32");
// GroupNames label

G:=SmallGroup(320,807);
// by ID

G=gap.SmallGroup(320,807);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,184,675,185,192,1684,438,102,12550]);
// Polycyclic

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

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