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G = C10×Q8⋊C4order 320 = 26·5

Direct product of C10 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C10×Q8⋊C4, Q83(C2×C20), (C2×Q8)⋊5C20, (Q8×C10)⋊25C4, C4.52(D4×C10), C2.1(C10×Q16), C20.459(C2×D4), (C2×C20).415D4, C4.2(C22×C20), (C22×C8).5C10, C10.48(C2×Q16), (C2×C10).20Q16, C2.2(C10×SD16), C23.55(C5×D4), C22.5(C5×Q16), (C22×C40).11C2, (C2×C10).45SD16, C10.82(C2×SD16), C22.42(D4×C10), (C22×Q8).4C10, (C2×C40).357C22, C20.206(C22×C4), (C2×C20).891C23, (C22×C10).216D4, C22.11(C5×SD16), C20.128(C22⋊C4), (Q8×C10).250C22, (C22×C20).582C22, (C5×Q8)⋊30(C2×C4), (C2×C4).69(C5×D4), (C10×C4⋊C4).41C2, (C2×C4⋊C4).12C10, (Q8×C2×C10).14C2, C4⋊C4.37(C2×C10), (C2×C8).60(C2×C10), (C2×C4).48(C2×C20), C4.13(C5×C22⋊C4), (C2×C20).442(C2×C4), (C2×C10).618(C2×D4), C2.18(C10×C22⋊C4), (C2×Q8).35(C2×C10), (C5×C4⋊C4).358C22, C10.147(C2×C22⋊C4), (C2×C4).66(C22×C10), C22.34(C5×C22⋊C4), (C22×C4).111(C2×C10), (C2×C10).203(C22⋊C4), SmallGroup(320,916)

Series: Derived Chief Lower central Upper central

C1C4 — C10×Q8⋊C4
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×Q8⋊C4 — C10×Q8⋊C4
C1C2C4 — C10×Q8⋊C4
C1C22×C10C22×C20 — C10×Q8⋊C4

Generators and relations for C10×Q8⋊C4
 G = < a,b,c,d | a10=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 242 in 162 conjugacy classes, 98 normal (30 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 [×4], Q8 [×6], C23, C10 [×3], C10 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×3], C20 [×2], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], Q8⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×Q8, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×10], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C2×Q8⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C2×C40 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×6], Q8×C10 [×3], C5×Q8⋊C4 [×4], C10×C4⋊C4, C22×C40, Q8×C2×C10, C10×Q8⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C2×C20 [×6], C5×D4 [×4], C22×C10, C2×Q8⋊C4, C5×C22⋊C4 [×4], C5×SD16 [×2], C5×Q16 [×2], C22×C20, D4×C10 [×2], C5×Q8⋊C4 [×4], C10×C22⋊C4, C10×SD16, C10×Q16, C10×Q8⋊C4

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

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

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

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

140 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L5A5B5C5D8A···8H10A···10AB20A···20P20Q···20AV40A···40AF
order12···244444···455558···810···1020···2020···2040···40
size11···122224···411112···21···12···24···42···2

140 irreducible representations

dim11111111111122222222
type+++++++-
imageC1C2C2C2C2C4C5C10C10C10C10C20D4D4SD16Q16C5×D4C5×D4C5×SD16C5×Q16
kernelC10×Q8⋊C4C5×Q8⋊C4C10×C4⋊C4C22×C40Q8×C2×C10Q8×C10C2×Q8⋊C4Q8⋊C4C2×C4⋊C4C22×C8C22×Q8C2×Q8C2×C20C22×C10C2×C10C2×C10C2×C4C23C22C22
# reps1411184164443231441241616

Matrix representation of C10×Q8⋊C4 in GL5(𝔽41)

400000
023000
002300
000230
000023
,
10000
040000
004000
0004039
00011
,
400000
004000
040000
0002114
0002720
,
90000
017100
0402400
0003126
0002310

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,21,27,0,0,0,14,20],[9,0,0,0,0,0,17,40,0,0,0,1,24,0,0,0,0,0,31,23,0,0,0,26,10] >;

C10×Q8⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C10xQ8:C4");
// GroupNames label

G:=SmallGroup(320,916);
// by ID

G=gap.SmallGroup(320,916);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,7004,3511,172]);
// 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,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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