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## G = C5×Q8⋊C8order 320 = 26·5

### Direct product of C5 and Q8⋊C8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×Q8⋊C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C20 — C5×C4⋊C8 — C5×Q8⋊C8
 Lower central C1 — C2 — C4 — C5×Q8⋊C8
 Upper central C1 — C2×C20 — C4×C20 — C5×Q8⋊C8

Generators and relations for C5×Q8⋊C8
G = < a,b,c,d | a5=b4=d8=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 106 in 70 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C4 [×4], C4 [×4], C22, C5, C8 [×3], C2×C4 [×3], C2×C4 [×2], Q8 [×2], Q8, C10 [×3], C42, C42, C4⋊C4, C4⋊C4, C2×C8 [×2], C2×Q8, C20 [×4], C20 [×4], C2×C10, C4×C8, C4⋊C8, C4×Q8, C40 [×3], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C5×Q8, Q8⋊C8, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40 [×2], Q8×C10, C4×C40, C5×C4⋊C8, Q8×C20, C5×Q8⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C8 [×2], C2×C4, D4 [×2], C10 [×3], C22⋊C4, C2×C8, M4(2), SD16, Q16, C20 [×2], C2×C10, C22⋊C8, Q8⋊C4, C4≀C2, C40 [×2], C2×C20, C5×D4 [×2], Q8⋊C8, C5×C22⋊C4, C2×C40, C5×M4(2), C5×SD16, C5×Q16, C5×C22⋊C8, C5×Q8⋊C4, C5×C4≀C2, C5×Q8⋊C8

Smallest permutation representation of C5×Q8⋊C8
Regular action on 320 points
Generators in S320
(1 55 47 39 31)(2 56 48 40 32)(3 49 41 33 25)(4 50 42 34 26)(5 51 43 35 27)(6 52 44 36 28)(7 53 45 37 29)(8 54 46 38 30)(9 85 77 69 61)(10 86 78 70 62)(11 87 79 71 63)(12 88 80 72 64)(13 81 73 65 57)(14 82 74 66 58)(15 83 75 67 59)(16 84 76 68 60)(17 233 225 217 176)(18 234 226 218 169)(19 235 227 219 170)(20 236 228 220 171)(21 237 229 221 172)(22 238 230 222 173)(23 239 231 223 174)(24 240 232 224 175)(89 121 113 105 97)(90 122 114 106 98)(91 123 115 107 99)(92 124 116 108 100)(93 125 117 109 101)(94 126 118 110 102)(95 127 119 111 103)(96 128 120 112 104)(129 167 153 145 137)(130 168 154 146 138)(131 161 155 147 139)(132 162 156 148 140)(133 163 157 149 141)(134 164 158 150 142)(135 165 159 151 143)(136 166 160 152 144)(177 209 201 193 185)(178 210 202 194 186)(179 211 203 195 187)(180 212 204 196 188)(181 213 205 197 189)(182 214 206 198 190)(183 215 207 199 191)(184 216 208 200 192)(241 273 265 257 249)(242 274 266 258 250)(243 275 267 259 251)(244 276 268 260 252)(245 277 269 261 253)(246 278 270 262 254)(247 279 271 263 255)(248 280 272 264 256)(281 313 305 297 289)(282 314 306 298 290)(283 315 307 299 291)(284 316 308 300 292)(285 317 309 301 293)(286 318 310 302 294)(287 319 311 303 295)(288 320 312 304 296)
(1 95 247 183)(2 184 248 96)(3 89 241 177)(4 178 242 90)(5 91 243 179)(6 180 244 92)(7 93 245 181)(8 182 246 94)(9 172 317 163)(10 164 318 173)(11 174 319 165)(12 166 320 175)(13 176 313 167)(14 168 314 169)(15 170 315 161)(16 162 316 171)(17 305 153 81)(18 82 154 306)(19 307 155 83)(20 84 156 308)(21 309 157 85)(22 86 158 310)(23 311 159 87)(24 88 160 312)(25 97 249 185)(26 186 250 98)(27 99 251 187)(28 188 252 100)(29 101 253 189)(30 190 254 102)(31 103 255 191)(32 192 256 104)(33 105 257 193)(34 194 258 106)(35 107 259 195)(36 196 260 108)(37 109 261 197)(38 198 262 110)(39 111 263 199)(40 200 264 112)(41 113 265 201)(42 202 266 114)(43 115 267 203)(44 204 268 116)(45 117 269 205)(46 206 270 118)(47 119 271 207)(48 208 272 120)(49 121 273 209)(50 210 274 122)(51 123 275 211)(52 212 276 124)(53 125 277 213)(54 214 278 126)(55 127 279 215)(56 216 280 128)(57 217 281 129)(58 130 282 218)(59 219 283 131)(60 132 284 220)(61 221 285 133)(62 134 286 222)(63 223 287 135)(64 136 288 224)(65 225 289 137)(66 138 290 226)(67 227 291 139)(68 140 292 228)(69 229 293 141)(70 142 294 230)(71 231 295 143)(72 144 296 232)(73 233 297 145)(74 146 298 234)(75 235 299 147)(76 148 300 236)(77 237 301 149)(78 150 302 238)(79 239 303 151)(80 152 304 240)
(1 283 247 59)(2 132 248 220)(3 285 241 61)(4 134 242 222)(5 287 243 63)(6 136 244 224)(7 281 245 57)(8 130 246 218)(9 49 317 273)(10 122 318 210)(11 51 319 275)(12 124 320 212)(13 53 313 277)(14 126 314 214)(15 55 315 279)(16 128 316 216)(17 205 153 117)(18 46 154 270)(19 207 155 119)(20 48 156 272)(21 201 157 113)(22 42 158 266)(23 203 159 115)(24 44 160 268)(25 293 249 69)(26 142 250 230)(27 295 251 71)(28 144 252 232)(29 289 253 65)(30 138 254 226)(31 291 255 67)(32 140 256 228)(33 301 257 77)(34 150 258 238)(35 303 259 79)(36 152 260 240)(37 297 261 73)(38 146 262 234)(39 299 263 75)(40 148 264 236)(41 309 265 85)(43 311 267 87)(45 305 269 81)(47 307 271 83)(50 164 274 173)(52 166 276 175)(54 168 278 169)(56 162 280 171)(58 94 282 182)(60 96 284 184)(62 90 286 178)(64 92 288 180)(66 102 290 190)(68 104 292 192)(70 98 294 186)(72 100 296 188)(74 110 298 198)(76 112 300 200)(78 106 302 194)(80 108 304 196)(82 118 306 206)(84 120 308 208)(86 114 310 202)(88 116 312 204)(89 221 177 133)(91 223 179 135)(93 217 181 129)(95 219 183 131)(97 229 185 141)(99 231 187 143)(101 225 189 137)(103 227 191 139)(105 237 193 149)(107 239 195 151)(109 233 197 145)(111 235 199 147)(121 172 209 163)(123 174 211 165)(125 176 213 167)(127 170 215 161)
(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)| (1,55,47,39,31)(2,56,48,40,32)(3,49,41,33,25)(4,50,42,34,26)(5,51,43,35,27)(6,52,44,36,28)(7,53,45,37,29)(8,54,46,38,30)(9,85,77,69,61)(10,86,78,70,62)(11,87,79,71,63)(12,88,80,72,64)(13,81,73,65,57)(14,82,74,66,58)(15,83,75,67,59)(16,84,76,68,60)(17,233,225,217,176)(18,234,226,218,169)(19,235,227,219,170)(20,236,228,220,171)(21,237,229,221,172)(22,238,230,222,173)(23,239,231,223,174)(24,240,232,224,175)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,167,153,145,137)(130,168,154,146,138)(131,161,155,147,139)(132,162,156,148,140)(133,163,157,149,141)(134,164,158,150,142)(135,165,159,151,143)(136,166,160,152,144)(177,209,201,193,185)(178,210,202,194,186)(179,211,203,195,187)(180,212,204,196,188)(181,213,205,197,189)(182,214,206,198,190)(183,215,207,199,191)(184,216,208,200,192)(241,273,265,257,249)(242,274,266,258,250)(243,275,267,259,251)(244,276,268,260,252)(245,277,269,261,253)(246,278,270,262,254)(247,279,271,263,255)(248,280,272,264,256)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,95,247,183)(2,184,248,96)(3,89,241,177)(4,178,242,90)(5,91,243,179)(6,180,244,92)(7,93,245,181)(8,182,246,94)(9,172,317,163)(10,164,318,173)(11,174,319,165)(12,166,320,175)(13,176,313,167)(14,168,314,169)(15,170,315,161)(16,162,316,171)(17,305,153,81)(18,82,154,306)(19,307,155,83)(20,84,156,308)(21,309,157,85)(22,86,158,310)(23,311,159,87)(24,88,160,312)(25,97,249,185)(26,186,250,98)(27,99,251,187)(28,188,252,100)(29,101,253,189)(30,190,254,102)(31,103,255,191)(32,192,256,104)(33,105,257,193)(34,194,258,106)(35,107,259,195)(36,196,260,108)(37,109,261,197)(38,198,262,110)(39,111,263,199)(40,200,264,112)(41,113,265,201)(42,202,266,114)(43,115,267,203)(44,204,268,116)(45,117,269,205)(46,206,270,118)(47,119,271,207)(48,208,272,120)(49,121,273,209)(50,210,274,122)(51,123,275,211)(52,212,276,124)(53,125,277,213)(54,214,278,126)(55,127,279,215)(56,216,280,128)(57,217,281,129)(58,130,282,218)(59,219,283,131)(60,132,284,220)(61,221,285,133)(62,134,286,222)(63,223,287,135)(64,136,288,224)(65,225,289,137)(66,138,290,226)(67,227,291,139)(68,140,292,228)(69,229,293,141)(70,142,294,230)(71,231,295,143)(72,144,296,232)(73,233,297,145)(74,146,298,234)(75,235,299,147)(76,148,300,236)(77,237,301,149)(78,150,302,238)(79,239,303,151)(80,152,304,240), (1,283,247,59)(2,132,248,220)(3,285,241,61)(4,134,242,222)(5,287,243,63)(6,136,244,224)(7,281,245,57)(8,130,246,218)(9,49,317,273)(10,122,318,210)(11,51,319,275)(12,124,320,212)(13,53,313,277)(14,126,314,214)(15,55,315,279)(16,128,316,216)(17,205,153,117)(18,46,154,270)(19,207,155,119)(20,48,156,272)(21,201,157,113)(22,42,158,266)(23,203,159,115)(24,44,160,268)(25,293,249,69)(26,142,250,230)(27,295,251,71)(28,144,252,232)(29,289,253,65)(30,138,254,226)(31,291,255,67)(32,140,256,228)(33,301,257,77)(34,150,258,238)(35,303,259,79)(36,152,260,240)(37,297,261,73)(38,146,262,234)(39,299,263,75)(40,148,264,236)(41,309,265,85)(43,311,267,87)(45,305,269,81)(47,307,271,83)(50,164,274,173)(52,166,276,175)(54,168,278,169)(56,162,280,171)(58,94,282,182)(60,96,284,184)(62,90,286,178)(64,92,288,180)(66,102,290,190)(68,104,292,192)(70,98,294,186)(72,100,296,188)(74,110,298,198)(76,112,300,200)(78,106,302,194)(80,108,304,196)(82,118,306,206)(84,120,308,208)(86,114,310,202)(88,116,312,204)(89,221,177,133)(91,223,179,135)(93,217,181,129)(95,219,183,131)(97,229,185,141)(99,231,187,143)(101,225,189,137)(103,227,191,139)(105,237,193,149)(107,239,195,151)(109,233,197,145)(111,235,199,147)(121,172,209,163)(123,174,211,165)(125,176,213,167)(127,170,215,161), (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( (1,55,47,39,31)(2,56,48,40,32)(3,49,41,33,25)(4,50,42,34,26)(5,51,43,35,27)(6,52,44,36,28)(7,53,45,37,29)(8,54,46,38,30)(9,85,77,69,61)(10,86,78,70,62)(11,87,79,71,63)(12,88,80,72,64)(13,81,73,65,57)(14,82,74,66,58)(15,83,75,67,59)(16,84,76,68,60)(17,233,225,217,176)(18,234,226,218,169)(19,235,227,219,170)(20,236,228,220,171)(21,237,229,221,172)(22,238,230,222,173)(23,239,231,223,174)(24,240,232,224,175)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,167,153,145,137)(130,168,154,146,138)(131,161,155,147,139)(132,162,156,148,140)(133,163,157,149,141)(134,164,158,150,142)(135,165,159,151,143)(136,166,160,152,144)(177,209,201,193,185)(178,210,202,194,186)(179,211,203,195,187)(180,212,204,196,188)(181,213,205,197,189)(182,214,206,198,190)(183,215,207,199,191)(184,216,208,200,192)(241,273,265,257,249)(242,274,266,258,250)(243,275,267,259,251)(244,276,268,260,252)(245,277,269,261,253)(246,278,270,262,254)(247,279,271,263,255)(248,280,272,264,256)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,95,247,183)(2,184,248,96)(3,89,241,177)(4,178,242,90)(5,91,243,179)(6,180,244,92)(7,93,245,181)(8,182,246,94)(9,172,317,163)(10,164,318,173)(11,174,319,165)(12,166,320,175)(13,176,313,167)(14,168,314,169)(15,170,315,161)(16,162,316,171)(17,305,153,81)(18,82,154,306)(19,307,155,83)(20,84,156,308)(21,309,157,85)(22,86,158,310)(23,311,159,87)(24,88,160,312)(25,97,249,185)(26,186,250,98)(27,99,251,187)(28,188,252,100)(29,101,253,189)(30,190,254,102)(31,103,255,191)(32,192,256,104)(33,105,257,193)(34,194,258,106)(35,107,259,195)(36,196,260,108)(37,109,261,197)(38,198,262,110)(39,111,263,199)(40,200,264,112)(41,113,265,201)(42,202,266,114)(43,115,267,203)(44,204,268,116)(45,117,269,205)(46,206,270,118)(47,119,271,207)(48,208,272,120)(49,121,273,209)(50,210,274,122)(51,123,275,211)(52,212,276,124)(53,125,277,213)(54,214,278,126)(55,127,279,215)(56,216,280,128)(57,217,281,129)(58,130,282,218)(59,219,283,131)(60,132,284,220)(61,221,285,133)(62,134,286,222)(63,223,287,135)(64,136,288,224)(65,225,289,137)(66,138,290,226)(67,227,291,139)(68,140,292,228)(69,229,293,141)(70,142,294,230)(71,231,295,143)(72,144,296,232)(73,233,297,145)(74,146,298,234)(75,235,299,147)(76,148,300,236)(77,237,301,149)(78,150,302,238)(79,239,303,151)(80,152,304,240), (1,283,247,59)(2,132,248,220)(3,285,241,61)(4,134,242,222)(5,287,243,63)(6,136,244,224)(7,281,245,57)(8,130,246,218)(9,49,317,273)(10,122,318,210)(11,51,319,275)(12,124,320,212)(13,53,313,277)(14,126,314,214)(15,55,315,279)(16,128,316,216)(17,205,153,117)(18,46,154,270)(19,207,155,119)(20,48,156,272)(21,201,157,113)(22,42,158,266)(23,203,159,115)(24,44,160,268)(25,293,249,69)(26,142,250,230)(27,295,251,71)(28,144,252,232)(29,289,253,65)(30,138,254,226)(31,291,255,67)(32,140,256,228)(33,301,257,77)(34,150,258,238)(35,303,259,79)(36,152,260,240)(37,297,261,73)(38,146,262,234)(39,299,263,75)(40,148,264,236)(41,309,265,85)(43,311,267,87)(45,305,269,81)(47,307,271,83)(50,164,274,173)(52,166,276,175)(54,168,278,169)(56,162,280,171)(58,94,282,182)(60,96,284,184)(62,90,286,178)(64,92,288,180)(66,102,290,190)(68,104,292,192)(70,98,294,186)(72,100,296,188)(74,110,298,198)(76,112,300,200)(78,106,302,194)(80,108,304,196)(82,118,306,206)(84,120,308,208)(86,114,310,202)(88,116,312,204)(89,221,177,133)(91,223,179,135)(93,217,181,129)(95,219,183,131)(97,229,185,141)(99,231,187,143)(101,225,189,137)(103,227,191,139)(105,237,193,149)(107,239,195,151)(109,233,197,145)(111,235,199,147)(121,172,209,163)(123,174,211,165)(125,176,213,167)(127,170,215,161), (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([(1,55,47,39,31),(2,56,48,40,32),(3,49,41,33,25),(4,50,42,34,26),(5,51,43,35,27),(6,52,44,36,28),(7,53,45,37,29),(8,54,46,38,30),(9,85,77,69,61),(10,86,78,70,62),(11,87,79,71,63),(12,88,80,72,64),(13,81,73,65,57),(14,82,74,66,58),(15,83,75,67,59),(16,84,76,68,60),(17,233,225,217,176),(18,234,226,218,169),(19,235,227,219,170),(20,236,228,220,171),(21,237,229,221,172),(22,238,230,222,173),(23,239,231,223,174),(24,240,232,224,175),(89,121,113,105,97),(90,122,114,106,98),(91,123,115,107,99),(92,124,116,108,100),(93,125,117,109,101),(94,126,118,110,102),(95,127,119,111,103),(96,128,120,112,104),(129,167,153,145,137),(130,168,154,146,138),(131,161,155,147,139),(132,162,156,148,140),(133,163,157,149,141),(134,164,158,150,142),(135,165,159,151,143),(136,166,160,152,144),(177,209,201,193,185),(178,210,202,194,186),(179,211,203,195,187),(180,212,204,196,188),(181,213,205,197,189),(182,214,206,198,190),(183,215,207,199,191),(184,216,208,200,192),(241,273,265,257,249),(242,274,266,258,250),(243,275,267,259,251),(244,276,268,260,252),(245,277,269,261,253),(246,278,270,262,254),(247,279,271,263,255),(248,280,272,264,256),(281,313,305,297,289),(282,314,306,298,290),(283,315,307,299,291),(284,316,308,300,292),(285,317,309,301,293),(286,318,310,302,294),(287,319,311,303,295),(288,320,312,304,296)], [(1,95,247,183),(2,184,248,96),(3,89,241,177),(4,178,242,90),(5,91,243,179),(6,180,244,92),(7,93,245,181),(8,182,246,94),(9,172,317,163),(10,164,318,173),(11,174,319,165),(12,166,320,175),(13,176,313,167),(14,168,314,169),(15,170,315,161),(16,162,316,171),(17,305,153,81),(18,82,154,306),(19,307,155,83),(20,84,156,308),(21,309,157,85),(22,86,158,310),(23,311,159,87),(24,88,160,312),(25,97,249,185),(26,186,250,98),(27,99,251,187),(28,188,252,100),(29,101,253,189),(30,190,254,102),(31,103,255,191),(32,192,256,104),(33,105,257,193),(34,194,258,106),(35,107,259,195),(36,196,260,108),(37,109,261,197),(38,198,262,110),(39,111,263,199),(40,200,264,112),(41,113,265,201),(42,202,266,114),(43,115,267,203),(44,204,268,116),(45,117,269,205),(46,206,270,118),(47,119,271,207),(48,208,272,120),(49,121,273,209),(50,210,274,122),(51,123,275,211),(52,212,276,124),(53,125,277,213),(54,214,278,126),(55,127,279,215),(56,216,280,128),(57,217,281,129),(58,130,282,218),(59,219,283,131),(60,132,284,220),(61,221,285,133),(62,134,286,222),(63,223,287,135),(64,136,288,224),(65,225,289,137),(66,138,290,226),(67,227,291,139),(68,140,292,228),(69,229,293,141),(70,142,294,230),(71,231,295,143),(72,144,296,232),(73,233,297,145),(74,146,298,234),(75,235,299,147),(76,148,300,236),(77,237,301,149),(78,150,302,238),(79,239,303,151),(80,152,304,240)], [(1,283,247,59),(2,132,248,220),(3,285,241,61),(4,134,242,222),(5,287,243,63),(6,136,244,224),(7,281,245,57),(8,130,246,218),(9,49,317,273),(10,122,318,210),(11,51,319,275),(12,124,320,212),(13,53,313,277),(14,126,314,214),(15,55,315,279),(16,128,316,216),(17,205,153,117),(18,46,154,270),(19,207,155,119),(20,48,156,272),(21,201,157,113),(22,42,158,266),(23,203,159,115),(24,44,160,268),(25,293,249,69),(26,142,250,230),(27,295,251,71),(28,144,252,232),(29,289,253,65),(30,138,254,226),(31,291,255,67),(32,140,256,228),(33,301,257,77),(34,150,258,238),(35,303,259,79),(36,152,260,240),(37,297,261,73),(38,146,262,234),(39,299,263,75),(40,148,264,236),(41,309,265,85),(43,311,267,87),(45,305,269,81),(47,307,271,83),(50,164,274,173),(52,166,276,175),(54,168,278,169),(56,162,280,171),(58,94,282,182),(60,96,284,184),(62,90,286,178),(64,92,288,180),(66,102,290,190),(68,104,292,192),(70,98,294,186),(72,100,296,188),(74,110,298,198),(76,112,300,200),(78,106,302,194),(80,108,304,196),(82,118,306,206),(84,120,308,208),(86,114,310,202),(88,116,312,204),(89,221,177,133),(91,223,179,135),(93,217,181,129),(95,219,183,131),(97,229,185,141),(99,231,187,143),(101,225,189,137),(103,227,191,139),(105,237,193,149),(107,239,195,151),(109,233,197,145),(111,235,199,147),(121,172,209,163),(123,174,211,165),(125,176,213,167),(127,170,215,161)], [(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)])

140 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 5C 5D 8A ··· 8H 8I 8J 8K 8L 10A ··· 10L 20A ··· 20P 20Q ··· 20AF 20AG ··· 20AV 40A ··· 40AF 40AG ··· 40AV order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C5 C8 C10 C10 C10 C20 C20 C40 D4 M4(2) SD16 Q16 C4≀C2 C5×D4 C5×M4(2) C5×SD16 C5×Q16 C5×C4≀C2 kernel C5×Q8⋊C8 C4×C40 C5×C4⋊C8 Q8×C20 C5×C4⋊C4 Q8×C10 Q8⋊C8 C5×Q8 C4×C8 C4⋊C8 C4×Q8 C4⋊C4 C2×Q8 Q8 C2×C20 C20 C20 C20 C10 C2×C4 C4 C4 C4 C2 # reps 1 1 1 1 2 2 4 8 4 4 4 8 8 32 2 2 2 2 4 8 8 8 8 16

Matrix representation of C5×Q8⋊C8 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 37 0 0 0 0 37
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 0 40 0 0 40 0 0 0 0 0 27 7 0 0 7 14
,
 21 9 0 0 32 20 0 0 0 0 13 2 0 0 2 28
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,27,7,0,0,7,14],[21,32,0,0,9,20,0,0,0,0,13,2,0,0,2,28] >;

C5×Q8⋊C8 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes C_8
% in TeX

G:=Group("C5xQ8:C8");
// GroupNames label

G:=SmallGroup(320,131);
// by ID

G=gap.SmallGroup(320,131);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,1410,136,172]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=d^8=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

׿
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