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G = C2×Dic5.Q8order 320 = 26·5

Direct product of C2 and Dic5.Q8

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

Aliases: C2×Dic5.Q8, C4⋊C4.259D10, Dic5.1(C2×Q8), C22.31(Q8×D5), (C2×C10).42C24, (C2×Dic5).26Q8, C102(C42.C2), C10.22(C22×Q8), (C2×C20).133C23, (C22×C4).174D10, C22.80(C23×D5), C22.75(C4○D20), C4⋊Dic5.291C22, C23.322(C22×D5), C22.71(D42D5), (C22×C20).356C22, (C22×C10).391C23, (C4×Dic5).332C22, (C2×Dic5).192C23, C10.D4.103C22, (C22×Dic5).230C22, C2.6(C2×Q8×D5), C52(C2×C42.C2), (C2×C4⋊C4).25D5, (C10×C4⋊C4).18C2, C2.20(C2×C4○D20), C10.18(C2×C4○D4), (C2×C10).91(C2×Q8), (C2×C4×Dic5).45C2, C2.13(C2×D42D5), (C2×C4⋊Dic5).27C2, (C5×C4⋊C4).291C22, (C2×C4).138(C22×D5), (C2×C10).105(C4○D4), (C2×C10.D4).31C2, SmallGroup(320,1170)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×Dic5.Q8
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C4×Dic5 — C2×Dic5.Q8
C5C2×C10 — C2×Dic5.Q8
C1C23C2×C4⋊C4

Generators and relations for C2×Dic5.Q8
 G = < a,b,c,d,e | a2=b10=d4=1, c2=b5, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b5c, ce=ec, ede-1=b5d-1 >

Subgroups: 606 in 226 conjugacy classes, 119 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×16], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×24], C23, C10 [×3], C10 [×4], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×4], Dic5 [×4], Dic5 [×6], C20 [×6], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×5], C42.C2 [×8], C2×Dic5 [×12], C2×Dic5 [×6], C2×C20 [×6], C2×C20 [×6], C22×C10, C2×C42.C2, C4×Dic5 [×4], C10.D4 [×16], C4⋊Dic5 [×4], C5×C4⋊C4 [×4], C22×Dic5 [×4], C22×C20 [×3], Dic5.Q8 [×8], C2×C4×Dic5, C2×C10.D4 [×4], C2×C4⋊Dic5, C10×C4⋊C4, C2×Dic5.Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C22×D5 [×7], C2×C42.C2, C4○D20 [×2], D42D5 [×2], Q8×D5 [×2], C23×D5, Dic5.Q8 [×4], C2×C4○D20, C2×D42D5, C2×Q8×D5, C2×Dic5.Q8

Smallest permutation representation of C2×Dic5.Q8
Regular action on 320 points
Generators in S320
(1 96)(2 97)(3 98)(4 99)(5 100)(6 91)(7 92)(8 93)(9 94)(10 95)(11 239)(12 240)(13 231)(14 232)(15 233)(16 234)(17 235)(18 236)(19 237)(20 238)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 101)(28 102)(29 103)(30 104)(31 127)(32 128)(33 129)(34 130)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 111)(48 112)(49 113)(50 114)(51 147)(52 148)(53 149)(54 150)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 131)(68 132)(69 133)(70 134)(71 170)(72 161)(73 162)(74 163)(75 164)(76 165)(77 166)(78 167)(79 168)(80 169)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 151)(88 152)(89 153)(90 154)(171 270)(172 261)(173 262)(174 263)(175 264)(176 265)(177 266)(178 267)(179 268)(180 269)(181 252)(182 253)(183 254)(184 255)(185 256)(186 257)(187 258)(188 259)(189 260)(190 251)(191 290)(192 281)(193 282)(194 283)(195 284)(196 285)(197 286)(198 287)(199 288)(200 289)(201 272)(202 273)(203 274)(204 275)(205 276)(206 277)(207 278)(208 279)(209 280)(210 271)(211 310)(212 301)(213 302)(214 303)(215 304)(216 305)(217 306)(218 307)(219 308)(220 309)(221 292)(222 293)(223 294)(224 295)(225 296)(226 297)(227 298)(228 299)(229 300)(230 291)(241 312)(242 313)(243 314)(244 315)(245 316)(246 317)(247 318)(248 319)(249 320)(250 311)
(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)
(1 219 6 214)(2 218 7 213)(3 217 8 212)(4 216 9 211)(5 215 10 220)(11 111 16 116)(12 120 17 115)(13 119 18 114)(14 118 19 113)(15 117 20 112)(21 225 26 230)(22 224 27 229)(23 223 28 228)(24 222 29 227)(25 221 30 226)(31 243 36 248)(32 242 37 247)(33 241 38 246)(34 250 39 245)(35 249 40 244)(41 240 46 235)(42 239 47 234)(43 238 48 233)(44 237 49 232)(45 236 50 231)(51 179 56 174)(52 178 57 173)(53 177 58 172)(54 176 59 171)(55 175 60 180)(61 184 66 189)(62 183 67 188)(63 182 68 187)(64 181 69 186)(65 190 70 185)(71 199 76 194)(72 198 77 193)(73 197 78 192)(74 196 79 191)(75 195 80 200)(81 204 86 209)(82 203 87 208)(83 202 88 207)(84 201 89 206)(85 210 90 205)(91 303 96 308)(92 302 97 307)(93 301 98 306)(94 310 99 305)(95 309 100 304)(101 300 106 295)(102 299 107 294)(103 298 108 293)(104 297 109 292)(105 296 110 291)(121 320 126 315)(122 319 127 314)(123 318 128 313)(124 317 129 312)(125 316 130 311)(131 259 136 254)(132 258 137 253)(133 257 138 252)(134 256 139 251)(135 255 140 260)(141 264 146 269)(142 263 147 268)(143 262 148 267)(144 261 149 266)(145 270 150 265)(151 279 156 274)(152 278 157 273)(153 277 158 272)(154 276 159 271)(155 275 160 280)(161 287 166 282)(162 286 167 281)(163 285 168 290)(164 284 169 289)(165 283 170 288)
(1 127 28 111)(2 128 29 112)(3 129 30 113)(4 130 21 114)(5 121 22 115)(6 122 23 116)(7 123 24 117)(8 124 25 118)(9 125 26 119)(10 126 27 120)(11 219 319 228)(12 220 320 229)(13 211 311 230)(14 212 312 221)(15 213 313 222)(16 214 314 223)(17 215 315 224)(18 216 316 225)(19 217 317 226)(20 218 318 227)(31 102 47 96)(32 103 48 97)(33 104 49 98)(34 105 50 99)(35 106 41 100)(36 107 42 91)(37 108 43 92)(38 109 44 93)(39 110 45 94)(40 101 46 95)(51 165 67 156)(52 166 68 157)(53 167 69 158)(54 168 70 159)(55 169 61 160)(56 170 62 151)(57 161 63 152)(58 162 64 153)(59 163 65 154)(60 164 66 155)(71 136 87 142)(72 137 88 143)(73 138 89 144)(74 139 90 145)(75 140 81 146)(76 131 82 147)(77 132 83 148)(78 133 84 149)(79 134 85 150)(80 135 86 141)(171 290 190 271)(172 281 181 272)(173 282 182 273)(174 283 183 274)(175 284 184 275)(176 285 185 276)(177 286 186 277)(178 287 187 278)(179 288 188 279)(180 289 189 280)(191 251 210 270)(192 252 201 261)(193 253 202 262)(194 254 203 263)(195 255 204 264)(196 256 205 265)(197 257 206 266)(198 258 207 267)(199 259 208 268)(200 260 209 269)(231 310 250 291)(232 301 241 292)(233 302 242 293)(234 303 243 294)(235 304 244 295)(236 305 245 296)(237 306 246 297)(238 307 247 298)(239 308 248 299)(240 309 249 300)
(1 174 28 183)(2 173 29 182)(3 172 30 181)(4 171 21 190)(5 180 22 189)(6 179 23 188)(7 178 24 187)(8 177 25 186)(9 176 26 185)(10 175 27 184)(11 170 319 151)(12 169 320 160)(13 168 311 159)(14 167 312 158)(15 166 313 157)(16 165 314 156)(17 164 315 155)(18 163 316 154)(19 162 317 153)(20 161 318 152)(31 208 47 199)(32 207 48 198)(33 206 49 197)(34 205 50 196)(35 204 41 195)(36 203 42 194)(37 202 43 193)(38 201 44 192)(39 210 45 191)(40 209 46 200)(51 228 67 219)(52 227 68 218)(53 226 69 217)(54 225 70 216)(55 224 61 215)(56 223 62 214)(57 222 63 213)(58 221 64 212)(59 230 65 211)(60 229 66 220)(71 248 87 239)(72 247 88 238)(73 246 89 237)(74 245 90 236)(75 244 81 235)(76 243 82 234)(77 242 83 233)(78 241 84 232)(79 250 85 231)(80 249 86 240)(91 268 107 259)(92 267 108 258)(93 266 109 257)(94 265 110 256)(95 264 101 255)(96 263 102 254)(97 262 103 253)(98 261 104 252)(99 270 105 251)(100 269 106 260)(111 288 127 279)(112 287 128 278)(113 286 129 277)(114 285 130 276)(115 284 121 275)(116 283 122 274)(117 282 123 273)(118 281 124 272)(119 290 125 271)(120 289 126 280)(131 308 147 299)(132 307 148 298)(133 306 149 297)(134 305 150 296)(135 304 141 295)(136 303 142 294)(137 302 143 293)(138 301 144 292)(139 310 145 291)(140 309 146 300)

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

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

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

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T5A5B10A···10N20A···20X
order12···2444444444···444445510···1020···20
size11···12222444410···1020202020222···24···4

68 irreducible representations

dim11111122222244
type++++++-+++--
imageC1C2C2C2C2C2Q8D5C4○D4D10D10C4○D20D42D5Q8×D5
kernelC2×Dic5.Q8Dic5.Q8C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C10×C4⋊C4C2×Dic5C2×C4⋊C4C2×C10C4⋊C4C22×C4C22C22C22
# reps181411428861644

Matrix representation of C2×Dic5.Q8 in GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
0074000
0084000
000010
000001
,
2920000
30120000
00383600
0018300
000010
000001
,
4020000
4010000
0040000
0004000
0000040
000010
,
3200000
0320000
003500
00233800
0000147
0000727

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,8,0,0,0,0,40,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[29,30,0,0,0,0,2,12,0,0,0,0,0,0,38,18,0,0,0,0,36,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,3,23,0,0,0,0,5,38,0,0,0,0,0,0,14,7,0,0,0,0,7,27] >;

C2×Dic5.Q8 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5.Q_8
% in TeX

G:=Group("C2xDic5.Q8");
// GroupNames label

G:=SmallGroup(320,1170);
// by ID

G=gap.SmallGroup(320,1170);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,675,297,136,12550]);
// Polycyclic

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

׿
×
𝔽