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## G = Q8×C5⋊C8order 320 = 26·5

### Direct product of Q8 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Q8×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — Q8×C5⋊C8
 Lower central C5 — C10 — Q8×C5⋊C8
 Upper central C1 — C22 — C2×Q8

Generators and relations for Q8×C5⋊C8
G = < a,b,c,d | a4=c5=d8=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 266 in 102 conjugacy classes, 64 normal (18 characteristic)
C1, C2 [×3], C4 [×6], C4 [×5], C22, C5, C8 [×5], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×C8 [×4], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×6], C2×C10, C4×C8 [×3], C4⋊C8 [×3], C4×Q8, C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5, C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C8×Q8, C4×Dic5 [×3], C4⋊Dic5 [×3], C2×C5⋊C8, C2×C5⋊C8 [×3], Q8×C10, C4×C5⋊C8 [×3], C20⋊C8 [×3], Q8×Dic5, Q8×C5⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, C2×C8 [×6], C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C22×C8, C8○D4, C5⋊C8 [×4], C2×F5 [×3], C8×Q8, C2×C5⋊C8 [×6], C22×F5, Q8.F5, Q8×F5, C22×C5⋊C8, Q8×C5⋊C8

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

50 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 5 8A ··· 8H 8I ··· 8T 10A 10B 10C 20A ··· 20F order 1 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 8 ··· 8 8 ··· 8 10 10 10 20 ··· 20 size 1 1 1 1 2 ··· 2 5 5 5 5 10 ··· 10 4 5 ··· 5 10 ··· 10 4 4 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 4 8 8 type + + + + - + + - + - image C1 C2 C2 C2 C4 C4 C8 Q8 C4○D4 C8○D4 F5 C2×F5 C5⋊C8 Q8.F5 Q8×F5 kernel Q8×C5⋊C8 C4×C5⋊C8 C20⋊C8 Q8×Dic5 C4⋊Dic5 Q8×C10 C5×Q8 C5⋊C8 Dic5 C10 C2×Q8 C2×C4 Q8 C2 C2 # reps 1 3 3 1 6 2 16 2 2 4 1 3 4 1 1

Matrix representation of Q8×C5⋊C8 in GL8(𝔽41)

 1 2 0 0 0 0 0 0 40 40 0 0 0 0 0 0 0 0 26 5 0 0 0 0 0 0 4 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 32 23 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 38 0 0 0 0 0 0 0 0 38 0 0 0 0 0 0 0 0 19 9 16 31 0 0 0 0 7 22 32 10 0 0 0 0 10 29 19 26 0 0 0 0 31 38 12 22

G:=sub<GL(8,GF(41))| [1,40,0,0,0,0,0,0,2,40,0,0,0,0,0,0,0,0,26,4,0,0,0,0,0,0,5,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,0,23,9,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,19,7,10,31,0,0,0,0,9,22,29,38,0,0,0,0,16,32,19,12,0,0,0,0,31,10,26,22] >;

Q8×C5⋊C8 in GAP, Magma, Sage, TeX

Q_8\times C_5\rtimes C_8
% in TeX

G:=Group("Q8xC5:C8");
// GroupNames label

G:=SmallGroup(320,1124);
// by ID

G=gap.SmallGroup(320,1124);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,219,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^5=d^8=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

׿
×
𝔽