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G = C10×C4⋊Q8order 320 = 26·5

Direct product of C10 and C4⋊Q8

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

Aliases: C10×C4⋊Q8, C207(C2×Q8), C41(Q8×C10), (C2×C20)⋊15Q8, C4.14(D4×C10), (C2×C20).431D4, C20.321(C2×D4), (C2×C42).19C10, C42.89(C2×C10), C22.64(D4×C10), (C22×Q8).8C10, C10.59(C22×Q8), C22.19(Q8×C10), (C2×C10).351C24, (C2×C20).962C23, (C4×C20).374C22, C10.187(C22×D4), C23.73(C22×C10), C22.25(C23×C10), (Q8×C10).267C22, (C22×C10).470C23, (C22×C20).445C22, (C2×C4)⋊4(C5×Q8), C2.5(Q8×C2×C10), (C2×C4×C20).42C2, C2.11(D4×C2×C10), (C2×C4).87(C5×D4), (C2×C4⋊C4).19C10, (C10×C4⋊C4).48C2, (Q8×C2×C10).18C2, C4⋊C4.65(C2×C10), (C2×C10).685(C2×D4), (C2×Q8).54(C2×C10), (C2×C10).117(C2×Q8), (C5×C4⋊C4).388C22, (C22×C4).55(C2×C10), (C2×C4).18(C22×C10), SmallGroup(320,1533)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C4⋊Q8
C1C2C22C2×C10C2×C20Q8×C10C5×C4⋊Q8 — C10×C4⋊Q8
C1C22 — C10×C4⋊Q8
C1C22×C10 — C10×C4⋊Q8

Generators and relations for C10×C4⋊Q8
 G = < a,b,c,d | a10=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 370 in 290 conjugacy classes, 210 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C5, C2×C4 [×26], C2×C4 [×8], Q8 [×16], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×6], C2×Q8 [×8], C2×Q8 [×8], C20 [×12], C20 [×8], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C22×Q8 [×2], C2×C20 [×26], C2×C20 [×8], C5×Q8 [×16], C22×C10, C2×C4⋊Q8, C4×C20 [×4], C5×C4⋊C4 [×16], C22×C20, C22×C20 [×6], Q8×C10 [×8], Q8×C10 [×8], C2×C4×C20, C10×C4⋊C4 [×4], C5×C4⋊Q8 [×8], Q8×C2×C10 [×2], C10×C4⋊Q8
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], Q8 [×8], C23 [×15], C10 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C2×C10 [×35], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C5×D4 [×4], C5×Q8 [×8], C22×C10 [×15], C2×C4⋊Q8, D4×C10 [×6], Q8×C10 [×12], C23×C10, C5×C4⋊Q8 [×4], D4×C2×C10, Q8×C2×C10 [×2], C10×C4⋊Q8

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

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

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

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

140 conjugacy classes

class 1 2A···2G4A···4L4M···4T5A5B5C5D10A···10AB20A···20AV20AW···20CB
order12···24···44···4555510···1020···2020···20
size11···12···24···411111···12···24···4

140 irreducible representations

dim11111111112222
type++++++-
imageC1C2C2C2C2C5C10C10C10C10D4Q8C5×D4C5×Q8
kernelC10×C4⋊Q8C2×C4×C20C10×C4⋊C4C5×C4⋊Q8Q8×C2×C10C2×C4⋊Q8C2×C42C2×C4⋊C4C4⋊Q8C22×Q8C2×C20C2×C20C2×C4C2×C4
# reps114824416328481632

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

400000
040000
0010000
0001000
0000230
0000023
,
010000
4000000
0032000
0025900
0000040
000010
,
4000000
0400000
001000
000100
000001
0000400
,
29120000
12120000
00403700
000100
000062
0000235

G:=sub<GL(6,GF(41))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,23,0,0,0,0,0,0,23],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,25,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[29,12,0,0,0,0,12,12,0,0,0,0,0,0,40,0,0,0,0,0,37,1,0,0,0,0,0,0,6,2,0,0,0,0,2,35] >;

C10×C4⋊Q8 in GAP, Magma, Sage, TeX

C_{10}\times C_4\rtimes Q_8
% in TeX

G:=Group("C10xC4:Q8");
// GroupNames label

G:=SmallGroup(320,1533);
// by ID

G=gap.SmallGroup(320,1533);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446,856]);
// Polycyclic

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

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