direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C42.C2, C4.9(Q8×C10), C20.98(C2×Q8), (C2×C20).80Q8, (C2×C42).18C10, C42.88(C2×C10), C22.18(Q8×C10), C10.58(C22×Q8), (C4×C20).372C22, (C2×C20).660C23, (C2×C10).347C24, C22.21(C23×C10), C23.71(C22×C10), (C22×C10).469C23, (C22×C20).509C22, C2.4(Q8×C2×C10), (C2×C4×C20).41C2, (C2×C4⋊C4).18C10, (C10×C4⋊C4).47C2, (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
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
Generators and relations
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 >
►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 48 136)(2 149 49 137)(3 150 50 138)(4 141 41 139)(5 142 42 140)(6 143 43 131)(7 144 44 132)(8 145 45 133)(9 146 46 134)(10 147 47 135)(11 251 21 263)(12 252 22 264)(13 253 23 265)(14 254 24 266)(15 255 25 267)(16 256 26 268)(17 257 27 269)(18 258 28 270)(19 259 29 261)(20 260 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)(51 119 63 123)(52 120 64 124)(53 111 65 125)(54 112 66 126)(55 113 67 127)(56 114 68 128)(57 115 69 129)(58 116 70 130)(59 117 61 121)(60 118 62 122)(71 173 85 185)(72 174 86 186)(73 175 87 187)(74 176 88 188)(75 177 89 189)(76 178 90 190)(77 179 81 181)(78 180 82 182)(79 171 83 183)(80 172 84 184)(91 159 103 163)(92 160 104 164)(93 151 105 165)(94 152 106 166)(95 153 107 167)(96 154 108 168)(97 155 109 169)(98 156 110 170)(99 157 101 161)(100 158 102 162)(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)(211 296 223 308)(212 297 224 309)(213 298 225 310)(214 299 226 301)(215 300 227 302)(216 291 228 303)(217 292 229 304)(218 293 230 305)(219 294 221 306)(220 295 222 307)
(1 79 63 96)(2 80 64 97)(3 71 65 98)(4 72 66 99)(5 73 67 100)(6 74 68 91)(7 75 69 92)(8 76 70 93)(9 77 61 94)(10 78 62 95)(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)(21 274 319 291)(22 275 320 292)(23 276 311 293)(24 277 312 294)(25 278 313 295)(26 279 314 296)(27 280 315 297)(28 271 316 298)(29 272 317 299)(30 273 318 300)(41 86 54 101)(42 87 55 102)(43 88 56 103)(44 89 57 104)(45 90 58 105)(46 81 59 106)(47 82 60 107)(48 83 51 108)(49 84 52 109)(50 85 53 110)(111 170 138 185)(112 161 139 186)(113 162 140 187)(114 163 131 188)(115 164 132 189)(116 165 133 190)(117 166 134 181)(118 167 135 182)(119 168 136 183)(120 169 137 184)(121 152 146 179)(122 153 147 180)(123 154 148 171)(124 155 149 172)(125 156 150 173)(126 157 141 174)(127 158 142 175)(128 159 143 176)(129 160 144 177)(130 151 145 178)(191 236 218 253)(192 237 219 254)(193 238 220 255)(194 239 211 256)(195 240 212 257)(196 231 213 258)(197 232 214 259)(198 233 215 260)(199 234 216 251)(200 235 217 252)(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 63 216)(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)(41 206 54 221)(42 207 55 222)(43 208 56 223)(44 209 57 224)(45 210 58 225)(46 201 59 226)(47 202 60 227)(48 203 51 228)(49 204 52 229)(50 205 53 230)(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), (1,148,48,136)(2,149,49,137)(3,150,50,138)(4,141,41,139)(5,142,42,140)(6,143,43,131)(7,144,44,132)(8,145,45,133)(9,146,46,134)(10,147,47,135)(11,251,21,263)(12,252,22,264)(13,253,23,265)(14,254,24,266)(15,255,25,267)(16,256,26,268)(17,257,27,269)(18,258,28,270)(19,259,29,261)(20,260,30,262)(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)(51,119,63,123)(52,120,64,124)(53,111,65,125)(54,112,66,126)(55,113,67,127)(56,114,68,128)(57,115,69,129)(58,116,70,130)(59,117,61,121)(60,118,62,122)(71,173,85,185)(72,174,86,186)(73,175,87,187)(74,176,88,188)(75,177,89,189)(76,178,90,190)(77,179,81,181)(78,180,82,182)(79,171,83,183)(80,172,84,184)(91,159,103,163)(92,160,104,164)(93,151,105,165)(94,152,106,166)(95,153,107,167)(96,154,108,168)(97,155,109,169)(98,156,110,170)(99,157,101,161)(100,158,102,162)(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)(211,296,223,308)(212,297,224,309)(213,298,225,310)(214,299,226,301)(215,300,227,302)(216,291,228,303)(217,292,229,304)(218,293,230,305)(219,294,221,306)(220,295,222,307), (1,79,63,96)(2,80,64,97)(3,71,65,98)(4,72,66,99)(5,73,67,100)(6,74,68,91)(7,75,69,92)(8,76,70,93)(9,77,61,94)(10,78,62,95)(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)(21,274,319,291)(22,275,320,292)(23,276,311,293)(24,277,312,294)(25,278,313,295)(26,279,314,296)(27,280,315,297)(28,271,316,298)(29,272,317,299)(30,273,318,300)(41,86,54,101)(42,87,55,102)(43,88,56,103)(44,89,57,104)(45,90,58,105)(46,81,59,106)(47,82,60,107)(48,83,51,108)(49,84,52,109)(50,85,53,110)(111,170,138,185)(112,161,139,186)(113,162,140,187)(114,163,131,188)(115,164,132,189)(116,165,133,190)(117,166,134,181)(118,167,135,182)(119,168,136,183)(120,169,137,184)(121,152,146,179)(122,153,147,180)(123,154,148,171)(124,155,149,172)(125,156,150,173)(126,157,141,174)(127,158,142,175)(128,159,143,176)(129,160,144,177)(130,151,145,178)(191,236,218,253)(192,237,219,254)(193,238,220,255)(194,239,211,256)(195,240,212,257)(196,231,213,258)(197,232,214,259)(198,233,215,260)(199,234,216,251)(200,235,217,252)(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), (1,199,63,216)(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)(41,206,54,221)(42,207,55,222)(43,208,56,223)(44,209,57,224)(45,210,58,225)(46,201,59,226)(47,202,60,227)(48,203,51,228)(49,204,52,229)(50,205,53,230)(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), (1,148,48,136)(2,149,49,137)(3,150,50,138)(4,141,41,139)(5,142,42,140)(6,143,43,131)(7,144,44,132)(8,145,45,133)(9,146,46,134)(10,147,47,135)(11,251,21,263)(12,252,22,264)(13,253,23,265)(14,254,24,266)(15,255,25,267)(16,256,26,268)(17,257,27,269)(18,258,28,270)(19,259,29,261)(20,260,30,262)(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)(51,119,63,123)(52,120,64,124)(53,111,65,125)(54,112,66,126)(55,113,67,127)(56,114,68,128)(57,115,69,129)(58,116,70,130)(59,117,61,121)(60,118,62,122)(71,173,85,185)(72,174,86,186)(73,175,87,187)(74,176,88,188)(75,177,89,189)(76,178,90,190)(77,179,81,181)(78,180,82,182)(79,171,83,183)(80,172,84,184)(91,159,103,163)(92,160,104,164)(93,151,105,165)(94,152,106,166)(95,153,107,167)(96,154,108,168)(97,155,109,169)(98,156,110,170)(99,157,101,161)(100,158,102,162)(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)(211,296,223,308)(212,297,224,309)(213,298,225,310)(214,299,226,301)(215,300,227,302)(216,291,228,303)(217,292,229,304)(218,293,230,305)(219,294,221,306)(220,295,222,307), (1,79,63,96)(2,80,64,97)(3,71,65,98)(4,72,66,99)(5,73,67,100)(6,74,68,91)(7,75,69,92)(8,76,70,93)(9,77,61,94)(10,78,62,95)(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)(21,274,319,291)(22,275,320,292)(23,276,311,293)(24,277,312,294)(25,278,313,295)(26,279,314,296)(27,280,315,297)(28,271,316,298)(29,272,317,299)(30,273,318,300)(41,86,54,101)(42,87,55,102)(43,88,56,103)(44,89,57,104)(45,90,58,105)(46,81,59,106)(47,82,60,107)(48,83,51,108)(49,84,52,109)(50,85,53,110)(111,170,138,185)(112,161,139,186)(113,162,140,187)(114,163,131,188)(115,164,132,189)(116,165,133,190)(117,166,134,181)(118,167,135,182)(119,168,136,183)(120,169,137,184)(121,152,146,179)(122,153,147,180)(123,154,148,171)(124,155,149,172)(125,156,150,173)(126,157,141,174)(127,158,142,175)(128,159,143,176)(129,160,144,177)(130,151,145,178)(191,236,218,253)(192,237,219,254)(193,238,220,255)(194,239,211,256)(195,240,212,257)(196,231,213,258)(197,232,214,259)(198,233,215,260)(199,234,216,251)(200,235,217,252)(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), (1,199,63,216)(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)(41,206,54,221)(42,207,55,222)(43,208,56,223)(44,209,57,224)(45,210,58,225)(46,201,59,226)(47,202,60,227)(48,203,51,228)(49,204,52,229)(50,205,53,230)(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)], [(1,148,48,136),(2,149,49,137),(3,150,50,138),(4,141,41,139),(5,142,42,140),(6,143,43,131),(7,144,44,132),(8,145,45,133),(9,146,46,134),(10,147,47,135),(11,251,21,263),(12,252,22,264),(13,253,23,265),(14,254,24,266),(15,255,25,267),(16,256,26,268),(17,257,27,269),(18,258,28,270),(19,259,29,261),(20,260,30,262),(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),(51,119,63,123),(52,120,64,124),(53,111,65,125),(54,112,66,126),(55,113,67,127),(56,114,68,128),(57,115,69,129),(58,116,70,130),(59,117,61,121),(60,118,62,122),(71,173,85,185),(72,174,86,186),(73,175,87,187),(74,176,88,188),(75,177,89,189),(76,178,90,190),(77,179,81,181),(78,180,82,182),(79,171,83,183),(80,172,84,184),(91,159,103,163),(92,160,104,164),(93,151,105,165),(94,152,106,166),(95,153,107,167),(96,154,108,168),(97,155,109,169),(98,156,110,170),(99,157,101,161),(100,158,102,162),(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),(211,296,223,308),(212,297,224,309),(213,298,225,310),(214,299,226,301),(215,300,227,302),(216,291,228,303),(217,292,229,304),(218,293,230,305),(219,294,221,306),(220,295,222,307)], [(1,79,63,96),(2,80,64,97),(3,71,65,98),(4,72,66,99),(5,73,67,100),(6,74,68,91),(7,75,69,92),(8,76,70,93),(9,77,61,94),(10,78,62,95),(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),(21,274,319,291),(22,275,320,292),(23,276,311,293),(24,277,312,294),(25,278,313,295),(26,279,314,296),(27,280,315,297),(28,271,316,298),(29,272,317,299),(30,273,318,300),(41,86,54,101),(42,87,55,102),(43,88,56,103),(44,89,57,104),(45,90,58,105),(46,81,59,106),(47,82,60,107),(48,83,51,108),(49,84,52,109),(50,85,53,110),(111,170,138,185),(112,161,139,186),(113,162,140,187),(114,163,131,188),(115,164,132,189),(116,165,133,190),(117,166,134,181),(118,167,135,182),(119,168,136,183),(120,169,137,184),(121,152,146,179),(122,153,147,180),(123,154,148,171),(124,155,149,172),(125,156,150,173),(126,157,141,174),(127,158,142,175),(128,159,143,176),(129,160,144,177),(130,151,145,178),(191,236,218,253),(192,237,219,254),(193,238,220,255),(194,239,211,256),(195,240,212,257),(196,231,213,258),(197,232,214,259),(198,233,215,260),(199,234,216,251),(200,235,217,252),(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)], [(1,199,63,216),(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),(41,206,54,221),(42,207,55,222),(43,208,56,223),(44,209,57,224),(45,210,58,225),(46,201,59,226),(47,202,60,227),(48,203,51,228),(49,204,52,229),(50,205,53,230),(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)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 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 | 39 |
| 0 | 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 39 | 28 | 0 | 0 |
| 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 32 | 40 |
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] >;
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AV | 20AW | ··· | 20CB |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | Q8 | C4○D4 | C5×Q8 | C5×C4○D4 |
| kernel | C10×C42.C2 | C2×C4×C20 | C10×C4⋊C4 | C5×C42.C2 | C2×C42.C2 | C2×C42 | C2×C4⋊C4 | C42.C2 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 1 | 6 | 8 | 4 | 4 | 24 | 32 | 4 | 8 | 16 | 32 |
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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ℤ
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