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G = C2×C10.Q16order 320 = 26·5

Direct product of C2 and C10.Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C10.Q16, C4.61(C2×D20), C4⋊C4.229D10, (C2×C20).135D4, (C2×C4).140D20, C20.141(C2×D4), (C2×C10).16Q16, C10.32(C2×Q16), C102(Q8⋊C4), Dic1028(C2×C4), (C2×Dic10)⋊20C4, C10.49(C2×SD16), (C2×C10).32SD16, C4.8(D10⋊C4), C20.68(C22⋊C4), C20.120(C22×C4), (C2×C20).322C23, (C22×C4).330D10, (C22×C10).187D4, C23.98(C5⋊D4), C22.8(C5⋊Q16), C22.11(D4.D5), (C22×C20).137C22, (C22×Dic10).11C2, C22.46(D10⋊C4), (C2×Dic10).269C22, C4.50(C2×C4×D5), (C2×C4⋊C4).7D5, C53(C2×Q8⋊C4), (C10×C4⋊C4).6C2, (C2×C4).76(C4×D5), C2.2(C2×D4.D5), C2.2(C2×C5⋊Q16), (C2×C20).252(C2×C4), (C2×C10).442(C2×D4), C10.81(C2×C22⋊C4), (C22×C52C8).5C2, C22.59(C2×C5⋊D4), C2.13(C2×D10⋊C4), (C2×C4).125(C5⋊D4), (C5×C4⋊C4).260C22, (C2×C4).422(C22×D5), (C2×C52C8).249C22, (C2×C10).123(C22⋊C4), SmallGroup(320,596)

Series: Derived Chief Lower central Upper central

C1C20 — C2×C10.Q16
C1C5C10C2×C10C2×C20C2×Dic10C22×Dic10 — C2×C10.Q16
C5C10C20 — C2×C10.Q16
C1C23C22×C4C2×C4⋊C4

Generators and relations for C2×C10.Q16
 G = < a,b,c,d | a2=b10=c8=1, d2=b5c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b5c-1 >

Subgroups: 510 in 162 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×10], C23, C10 [×3], C10 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×9], Dic5 [×4], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], Q8⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×Q8, C52C8 [×2], Dic10 [×4], Dic10 [×6], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×C10, C2×Q8⋊C4, C2×C52C8 [×2], C2×C52C8 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×6], C2×Dic10 [×3], C22×Dic5, C22×C20, C22×C20, C10.Q16 [×4], C22×C52C8, C10×C4⋊C4, C22×Dic10, C2×C10.Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], D10 [×3], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×Q8⋊C4, D10⋊C4 [×4], D4.D5 [×2], C5⋊Q16 [×2], C2×C4×D5, C2×D20, C2×C5⋊D4, C10.Q16 [×4], C2×D10⋊C4, C2×D4.D5, C2×C5⋊Q16, C2×C10.Q16

Smallest permutation representation of C2×C10.Q16
Regular action on 320 points
Generators in S320
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 101)(11 126)(12 127)(13 128)(14 129)(15 130)(16 121)(17 122)(18 123)(19 124)(20 125)(21 219)(22 220)(23 211)(24 212)(25 213)(26 214)(27 215)(28 216)(29 217)(30 218)(31 119)(32 120)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 96)(42 97)(43 98)(44 99)(45 100)(46 91)(47 92)(48 93)(49 94)(50 95)(51 165)(52 166)(53 167)(54 168)(55 169)(56 170)(57 161)(58 162)(59 163)(60 164)(61 156)(62 157)(63 158)(64 159)(65 160)(66 151)(67 152)(68 153)(69 154)(70 155)(71 146)(72 147)(73 148)(74 149)(75 150)(76 141)(77 142)(78 143)(79 144)(80 145)(81 136)(82 137)(83 138)(84 139)(85 140)(86 131)(87 132)(88 133)(89 134)(90 135)(171 286)(172 287)(173 288)(174 289)(175 290)(176 281)(177 282)(178 283)(179 284)(180 285)(181 276)(182 277)(183 278)(184 279)(185 280)(186 271)(187 272)(188 273)(189 274)(190 275)(191 266)(192 267)(193 268)(194 269)(195 270)(196 261)(197 262)(198 263)(199 264)(200 265)(201 256)(202 257)(203 258)(204 259)(205 260)(206 251)(207 252)(208 253)(209 254)(210 255)(221 316)(222 317)(223 318)(224 319)(225 320)(226 311)(227 312)(228 313)(229 314)(230 315)(231 306)(232 307)(233 308)(234 309)(235 310)(236 301)(237 302)(238 303)(239 304)(240 305)(241 296)(242 297)(243 298)(244 299)(245 300)(246 291)(247 292)(248 293)(249 294)(250 295)
(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 27 45 308 39 295 20 313)(2 26 46 307 40 294 11 312)(3 25 47 306 31 293 12 311)(4 24 48 305 32 292 13 320)(5 23 49 304 33 291 14 319)(6 22 50 303 34 300 15 318)(7 21 41 302 35 299 16 317)(8 30 42 301 36 298 17 316)(9 29 43 310 37 297 18 315)(10 28 44 309 38 296 19 314)(51 277 63 259 86 262 78 284)(52 276 64 258 87 261 79 283)(53 275 65 257 88 270 80 282)(54 274 66 256 89 269 71 281)(55 273 67 255 90 268 72 290)(56 272 68 254 81 267 73 289)(57 271 69 253 82 266 74 288)(58 280 70 252 83 265 75 287)(59 279 61 251 84 264 76 286)(60 278 62 260 85 263 77 285)(91 232 118 249 126 227 103 214)(92 231 119 248 127 226 104 213)(93 240 120 247 128 225 105 212)(94 239 111 246 129 224 106 211)(95 238 112 245 130 223 107 220)(96 237 113 244 121 222 108 219)(97 236 114 243 122 221 109 218)(98 235 115 242 123 230 110 217)(99 234 116 241 124 229 101 216)(100 233 117 250 125 228 102 215)(131 197 143 179 165 182 158 204)(132 196 144 178 166 181 159 203)(133 195 145 177 167 190 160 202)(134 194 146 176 168 189 151 201)(135 193 147 175 169 188 152 210)(136 192 148 174 170 187 153 209)(137 191 149 173 161 186 154 208)(138 200 150 172 162 185 155 207)(139 199 141 171 163 184 156 206)(140 198 142 180 164 183 157 205)
(1 147 34 157)(2 148 35 158)(3 149 36 159)(4 150 37 160)(5 141 38 151)(6 142 39 152)(7 143 40 153)(8 144 31 154)(9 145 32 155)(10 146 33 156)(11 170 41 131)(12 161 42 132)(13 162 43 133)(14 163 44 134)(15 164 45 135)(16 165 46 136)(17 166 47 137)(18 167 48 138)(19 168 49 139)(20 169 50 140)(21 192 294 182)(22 193 295 183)(23 194 296 184)(24 195 297 185)(25 196 298 186)(26 197 299 187)(27 198 300 188)(28 199 291 189)(29 200 292 190)(30 191 293 181)(51 91 81 121)(52 92 82 122)(53 93 83 123)(54 94 84 124)(55 95 85 125)(56 96 86 126)(57 97 87 127)(58 98 88 128)(59 99 89 129)(60 100 90 130)(61 101 71 111)(62 102 72 112)(63 103 73 113)(64 104 74 114)(65 105 75 115)(66 106 76 116)(67 107 77 117)(68 108 78 118)(69 109 79 119)(70 110 80 120)(171 304 201 314)(172 305 202 315)(173 306 203 316)(174 307 204 317)(175 308 205 318)(176 309 206 319)(177 310 207 320)(178 301 208 311)(179 302 209 312)(180 303 210 313)(211 269 241 279)(212 270 242 280)(213 261 243 271)(214 262 244 272)(215 263 245 273)(216 264 246 274)(217 265 247 275)(218 266 248 276)(219 267 249 277)(220 268 250 278)(221 288 231 258)(222 289 232 259)(223 290 233 260)(224 281 234 251)(225 282 235 252)(226 283 236 253)(227 284 237 254)(228 285 238 255)(229 286 239 256)(230 287 240 257)

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

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

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

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H10A···10N20A···20X
order12···2444444444444558···810···1020···20
size11···122224444202020202210···102···24···4

68 irreducible representations

dim1111112222222222244
type++++++++-+++--
imageC1C2C2C2C2C4D4D4D5SD16Q16D10D10C4×D5D20C5⋊D4C5⋊D4D4.D5C5⋊Q16
kernelC2×C10.Q16C10.Q16C22×C52C8C10×C4⋊C4C22×Dic10C2×Dic10C2×C20C22×C10C2×C4⋊C4C2×C10C2×C10C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C22C22
# reps1411183124442884444

Matrix representation of C2×C10.Q16 in GL7(𝔽41)

1000000
0100000
0010000
00040000
00004000
0000010
0000001
,
40000000
04000000
00400000
0001000
0000100
00000356
000003540
,
9000000
015260000
015150000
00011700
000244000
000002121
000001820
,
32000000
029290000
029120000
00004000
00040000
000002335
00000618

G:=sub<GL(7,GF(41))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,35,35,0,0,0,0,0,6,40],[9,0,0,0,0,0,0,0,15,15,0,0,0,0,0,26,15,0,0,0,0,0,0,0,1,24,0,0,0,0,0,17,40,0,0,0,0,0,0,0,21,18,0,0,0,0,0,21,20],[32,0,0,0,0,0,0,0,29,29,0,0,0,0,0,29,12,0,0,0,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,0,0,23,6,0,0,0,0,0,35,18] >;

C2×C10.Q16 in GAP, Magma, Sage, TeX

C_2\times C_{10}.Q_{16}
% in TeX

G:=Group("C2xC10.Q16");
// GroupNames label

G:=SmallGroup(320,596);
// by ID

G=gap.SmallGroup(320,596);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^8=1,d^2=b^5*c^4,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=b^5*c^-1>;
// generators/relations

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