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## G = (Q8×C10)⋊17C4order 320 = 26·5

### 3rd semidirect product of Q8×C10 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (Q8×C10)⋊17C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — (Q8×C10)⋊17C4
 Lower central C5 — C2×C10 — (Q8×C10)⋊17C4
 Upper central C1 — C23 — C22×Q8

Generators and relations for (Q8×C10)⋊17C4
G = < a,b,c,d | a10=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, dbd-1=a5b-1, cd=dc >

Subgroups: 510 in 186 conjugacy classes, 91 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C5, C2×C4 [×10], C2×C4 [×18], Q8 [×8], C23, C10 [×3], C10 [×4], C42 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], Dic5 [×6], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×10], C2×C20 [×4], C5×Q8 [×8], C22×C10, C23.67C23, C4×Dic5 [×2], C4⋊Dic5 [×2], C22×Dic5 [×4], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C10.10C42 [×4], C2×C4×Dic5, C2×C4⋊Dic5, Q8×C2×C10, (Q8×C10)⋊17C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.67C23, C23.D5 [×4], Q8×D5 [×2], Q82D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], Dic5⋊Q8, Q8×Dic5 [×2], D103Q8 [×2], C20.23D4, C2×C23.D5, (Q8×C10)⋊17C4

Smallest permutation representation of (Q8×C10)⋊17C4
Regular action on 320 points
Generators in S320
(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 65 22 57)(2 66 23 58)(3 67 24 59)(4 68 25 60)(5 69 26 51)(6 70 27 52)(7 61 28 53)(8 62 29 54)(9 63 30 55)(10 64 21 56)(11 277 317 284)(12 278 318 285)(13 279 319 286)(14 280 320 287)(15 271 311 288)(16 272 312 289)(17 273 313 290)(18 274 314 281)(19 275 315 282)(20 276 316 283)(31 76 49 84)(32 77 50 85)(33 78 41 86)(34 79 42 87)(35 80 43 88)(36 71 44 89)(37 72 45 90)(38 73 46 81)(39 74 47 82)(40 75 48 83)(91 144 109 136)(92 145 110 137)(93 146 101 138)(94 147 102 139)(95 148 103 140)(96 149 104 131)(97 150 105 132)(98 141 106 133)(99 142 107 134)(100 143 108 135)(111 163 129 156)(112 164 130 157)(113 165 121 158)(114 166 122 159)(115 167 123 160)(116 168 124 151)(117 169 125 152)(118 170 126 153)(119 161 127 154)(120 162 128 155)(171 228 188 211)(172 229 189 212)(173 230 190 213)(174 221 181 214)(175 222 182 215)(176 223 183 216)(177 224 184 217)(178 225 185 218)(179 226 186 219)(180 227 187 220)(191 248 208 231)(192 249 209 232)(193 250 210 233)(194 241 201 234)(195 242 202 235)(196 243 203 236)(197 244 204 237)(198 245 205 238)(199 246 206 239)(200 247 207 240)(251 291 268 308)(252 292 269 309)(253 293 270 310)(254 294 261 301)(255 295 262 302)(256 296 263 303)(257 297 264 304)(258 298 265 305)(259 299 266 306)(260 300 267 307)
(1 125 22 117)(2 126 23 118)(3 127 24 119)(4 128 25 120)(5 129 26 111)(6 130 27 112)(7 121 28 113)(8 122 29 114)(9 123 30 115)(10 124 21 116)(11 212 317 229)(12 213 318 230)(13 214 319 221)(14 215 320 222)(15 216 311 223)(16 217 312 224)(17 218 313 225)(18 219 314 226)(19 220 315 227)(20 211 316 228)(31 96 49 104)(32 97 50 105)(33 98 41 106)(34 99 42 107)(35 100 43 108)(36 91 44 109)(37 92 45 110)(38 93 46 101)(39 94 47 102)(40 95 48 103)(51 156 69 163)(52 157 70 164)(53 158 61 165)(54 159 62 166)(55 160 63 167)(56 151 64 168)(57 152 65 169)(58 153 66 170)(59 154 67 161)(60 155 68 162)(71 136 89 144)(72 137 90 145)(73 138 81 146)(74 139 82 147)(75 140 83 148)(76 131 84 149)(77 132 85 150)(78 133 86 141)(79 134 87 142)(80 135 88 143)(171 276 188 283)(172 277 189 284)(173 278 190 285)(174 279 181 286)(175 280 182 287)(176 271 183 288)(177 272 184 289)(178 273 185 290)(179 274 186 281)(180 275 187 282)(191 256 208 263)(192 257 209 264)(193 258 210 265)(194 259 201 266)(195 260 202 267)(196 251 203 268)(197 252 204 269)(198 253 205 270)(199 254 206 261)(200 255 207 262)(231 296 248 303)(232 297 249 304)(233 298 250 305)(234 299 241 306)(235 300 242 307)(236 291 243 308)(237 292 244 309)(238 293 245 310)(239 294 246 301)(240 295 247 302)
(1 205 50 185)(2 204 41 184)(3 203 42 183)(4 202 43 182)(5 201 44 181)(6 210 45 190)(7 209 46 189)(8 208 47 188)(9 207 48 187)(10 206 49 186)(11 170 304 141)(12 169 305 150)(13 168 306 149)(14 167 307 148)(15 166 308 147)(16 165 309 146)(17 164 310 145)(18 163 301 144)(19 162 302 143)(20 161 303 142)(21 199 31 179)(22 198 32 178)(23 197 33 177)(24 196 34 176)(25 195 35 175)(26 194 36 174)(27 193 37 173)(28 192 38 172)(29 191 39 171)(30 200 40 180)(51 239 71 219)(52 238 72 218)(53 237 73 217)(54 236 74 216)(55 235 75 215)(56 234 76 214)(57 233 77 213)(58 232 78 212)(59 231 79 211)(60 240 80 220)(61 244 81 224)(62 243 82 223)(63 242 83 222)(64 241 84 221)(65 250 85 230)(66 249 86 229)(67 248 87 228)(68 247 88 227)(69 246 89 226)(70 245 90 225)(91 279 111 259)(92 278 112 258)(93 277 113 257)(94 276 114 256)(95 275 115 255)(96 274 116 254)(97 273 117 253)(98 272 118 252)(99 271 119 251)(100 280 120 260)(101 284 121 264)(102 283 122 263)(103 282 123 262)(104 281 124 261)(105 290 125 270)(106 289 126 269)(107 288 127 268)(108 287 128 267)(109 286 129 266)(110 285 130 265)(131 319 151 299)(132 318 152 298)(133 317 153 297)(134 316 154 296)(135 315 155 295)(136 314 156 294)(137 313 157 293)(138 312 158 292)(139 311 159 291)(140 320 160 300)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + - + + + - - + image C1 C2 C2 C2 C2 C4 Q8 D4 D5 C4○D4 D10 Dic5 C5⋊D4 Q8×D5 Q8⋊2D5 kernel (Q8×C10)⋊17C4 C10.10C42 C2×C4×Dic5 C2×C4⋊Dic5 Q8×C2×C10 Q8×C10 C2×Dic5 C2×C20 C22×Q8 C2×C10 C22×C4 C2×Q8 C2×C4 C22 C22 # reps 1 4 1 1 1 8 4 4 2 4 6 8 16 4 4

Matrix representation of (Q8×C10)⋊17C4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 0 0 0 0 0 0 10 0 0 0 0 0 0 7 35 0 0 0 0 7 0
,
 1 2 0 0 0 0 40 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 17 6 0 0 0 0 34 24
,
 23 40 0 0 0 0 38 18 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 23 40 0 0 0 0 38 18 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 22 32 0 0 0 0 22 19

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,10,0,0,0,0,0,0,7,7,0,0,0,0,35,0],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,34,0,0,0,0,6,24],[23,38,0,0,0,0,40,18,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[23,38,0,0,0,0,40,18,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,22,22,0,0,0,0,32,19] >;

(Q8×C10)⋊17C4 in GAP, Magma, Sage, TeX

(Q_8\times C_{10})\rtimes_{17}C_4
% in TeX

G:=Group("(Q8xC10):17C4");
// GroupNames label

G:=SmallGroup(320,857);
// by ID

G=gap.SmallGroup(320,857);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^5*b^-1,c*d=d*c>;
// generators/relations

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