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G = C10×C42.C2order 320 = 26·5

Direct product of C10 and C42.C2

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

Aliases: C10×C42.C2, C4.9(Q8×C10), (C2×C20).80Q8, C20.98(C2×Q8), (C2×C42).18C10, C42.88(C2×C10), C10.58(C22×Q8), C22.18(Q8×C10), (C2×C10).347C24, (C2×C20).660C23, (C4×C20).372C22, C22.21(C23×C10), C23.71(C22×C10), (C22×C10).469C23, (C22×C20).509C22, C2.4(Q8×C2×C10), (C2×C4×C20).41C2, (C10×C4⋊C4).47C2, (C2×C4⋊C4).18C10, (C2×C4).22(C5×Q8), C4⋊C4.63(C2×C10), C2.10(C10×C4○D4), C10.229(C2×C4○D4), (C2×C10).116(C2×Q8), C22.33(C5×C4○D4), (C5×C4⋊C4).386C22, (C2×C4).15(C22×C10), (C2×C10).233(C4○D4), (C22×C4).101(C2×C10), SmallGroup(320,1529)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C42.C2
C1C2C22C2×C10C2×C20C5×C4⋊C4C5×C42.C2 — C10×C42.C2
C1C22 — C10×C42.C2
C1C22×C10 — C10×C42.C2

Generators and relations for C10×C42.C2
 G = < a,b,c,d | a10=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 274 in 226 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×12], C22, C22 [×6], C5, C2×C4 [×18], C2×C4 [×12], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×24], C22×C4, C22×C4 [×6], C20 [×4], C20 [×12], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×6], C42.C2 [×8], C2×C20 [×18], C2×C20 [×12], C22×C10, C2×C42.C2, C4×C20 [×4], C5×C4⋊C4 [×24], C22×C20, C22×C20 [×6], C2×C4×C20, C10×C4⋊C4 [×6], C5×C42.C2 [×8], C10×C42.C2
Quotients: C1, C2 [×15], C22 [×35], C5, Q8 [×4], C23 [×15], C10 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C2×C10 [×35], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C5×Q8 [×4], C22×C10 [×15], C2×C42.C2, Q8×C10 [×6], C5×C4○D4 [×4], C23×C10, C5×C42.C2 [×4], Q8×C2×C10, C10×C4○D4 [×2], C10×C42.C2

Smallest permutation representation of C10×C42.C2
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)
(1 148 55 140)(2 149 56 131)(3 150 57 132)(4 141 58 133)(5 142 59 134)(6 143 60 135)(7 144 51 136)(8 145 52 137)(9 146 53 138)(10 147 54 139)(11 255 21 263)(12 256 22 264)(13 257 23 265)(14 258 24 266)(15 259 25 267)(16 260 26 268)(17 251 27 269)(18 252 28 270)(19 253 29 261)(20 254 30 262)(31 235 320 249)(32 236 311 250)(33 237 312 241)(34 238 313 242)(35 239 314 243)(36 240 315 244)(37 231 316 245)(38 232 317 246)(39 233 318 247)(40 234 319 248)(41 114 68 128)(42 115 69 129)(43 116 70 130)(44 117 61 121)(45 118 62 122)(46 119 63 123)(47 120 64 124)(48 111 65 125)(49 112 66 126)(50 113 67 127)(71 177 85 185)(72 178 86 186)(73 179 87 187)(74 180 88 188)(75 171 89 189)(76 172 90 190)(77 173 81 181)(78 174 82 182)(79 175 83 183)(80 176 84 184)(91 155 109 169)(92 156 110 170)(93 157 101 161)(94 158 102 162)(95 159 103 163)(96 160 104 164)(97 151 105 165)(98 152 106 166)(99 153 107 167)(100 154 108 168)(191 276 205 290)(192 277 206 281)(193 278 207 282)(194 279 208 283)(195 280 209 284)(196 271 210 285)(197 272 201 286)(198 273 202 287)(199 274 203 288)(200 275 204 289)(211 296 229 304)(212 297 230 305)(213 298 221 306)(214 299 222 307)(215 300 223 308)(216 291 224 309)(217 292 225 310)(218 293 226 301)(219 294 227 302)(220 295 228 303)
(1 79 63 100)(2 80 64 91)(3 71 65 92)(4 72 66 93)(5 73 67 94)(6 74 68 95)(7 75 69 96)(8 76 70 97)(9 77 61 98)(10 78 62 99)(11 288 40 303)(12 289 31 304)(13 290 32 305)(14 281 33 306)(15 282 34 307)(16 283 35 308)(17 284 36 309)(18 285 37 310)(19 286 38 301)(20 287 39 302)(21 274 319 295)(22 275 320 296)(23 276 311 297)(24 277 312 298)(25 278 313 299)(26 279 314 300)(27 280 315 291)(28 271 316 292)(29 272 317 293)(30 273 318 294)(41 103 60 88)(42 104 51 89)(43 105 52 90)(44 106 53 81)(45 107 54 82)(46 108 55 83)(47 109 56 84)(48 110 57 85)(49 101 58 86)(50 102 59 87)(111 170 132 185)(112 161 133 186)(113 162 134 187)(114 163 135 188)(115 164 136 189)(116 165 137 190)(117 166 138 181)(118 167 139 182)(119 168 140 183)(120 169 131 184)(121 152 146 173)(122 153 147 174)(123 154 148 175)(124 155 149 176)(125 156 150 177)(126 157 141 178)(127 158 142 179)(128 159 143 180)(129 160 144 171)(130 151 145 172)(191 236 212 257)(192 237 213 258)(193 238 214 259)(194 239 215 260)(195 240 216 251)(196 231 217 252)(197 232 218 253)(198 233 219 254)(199 234 220 255)(200 235 211 256)(201 246 226 261)(202 247 227 262)(203 248 228 263)(204 249 229 264)(205 250 230 265)(206 241 221 266)(207 242 222 267)(208 243 223 268)(209 244 224 269)(210 245 225 270)
(1 199 63 220)(2 200 64 211)(3 191 65 212)(4 192 66 213)(5 193 67 214)(6 194 68 215)(7 195 69 216)(8 196 70 217)(9 197 61 218)(10 198 62 219)(11 154 40 175)(12 155 31 176)(13 156 32 177)(14 157 33 178)(15 158 34 179)(16 159 35 180)(17 160 36 171)(18 151 37 172)(19 152 38 173)(20 153 39 174)(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 60 208)(42 224 51 209)(43 225 52 210)(44 226 53 201)(45 227 54 202)(46 228 55 203)(47 229 56 204)(48 230 57 205)(49 221 58 206)(50 222 59 207)(71 250 92 265)(72 241 93 266)(73 242 94 267)(74 243 95 268)(75 244 96 269)(76 245 97 270)(77 246 98 261)(78 247 99 262)(79 248 100 263)(80 249 91 264)(81 232 106 253)(82 233 107 254)(83 234 108 255)(84 235 109 256)(85 236 110 257)(86 237 101 258)(87 238 102 259)(88 239 103 260)(89 240 104 251)(90 231 105 252)(111 290 132 305)(112 281 133 306)(113 282 134 307)(114 283 135 308)(115 284 136 309)(116 285 137 310)(117 286 138 301)(118 287 139 302)(119 288 140 303)(120 289 131 304)(121 272 146 293)(122 273 147 294)(123 274 148 295)(124 275 149 296)(125 276 150 297)(126 277 141 298)(127 278 142 299)(128 279 143 300)(129 280 144 291)(130 271 145 292)

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

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

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

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

dim111111112222
type++++-
imageC1C2C2C2C5C10C10C10Q8C4○D4C5×Q8C5×C4○D4
kernelC10×C42.C2C2×C4×C20C10×C4⋊C4C5×C42.C2C2×C42.C2C2×C42C2×C4⋊C4C42.C2C2×C20C2×C10C2×C4C22
# reps1168442432481632

Matrix representation of C10×C42.C2 in GL5(𝔽41)

400000
018000
001800
000100
000010
,
400000
004000
040000
0003239
00009
,
400000
09000
00900
0003239
00009
,
10000
0392800
013200
000123
0003240

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,10,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,39,9],[40,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,39,9],[1,0,0,0,0,0,39,13,0,0,0,28,2,0,0,0,0,0,1,32,0,0,0,23,40] >;

C10×C42.C2 in GAP, Magma, Sage, TeX

C_{10}\times C_4^2.C_2
% in TeX

G:=Group("C10xC4^2.C2");
// GroupNames label

G:=SmallGroup(320,1529);
// by ID

G=gap.SmallGroup(320,1529);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,436]);
// 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*c^2,d*c*d^-1=b^2*c>;
// generators/relations

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