metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.14D10, (C2×C4).24D20, C8⋊C4.6D5, (C2×C20).35D4, (C2×C8).155D10, (C4×C20).2C22, C20⋊2Q8.7C2, C22.96(C2×D20), C4⋊Dic5.8C22, C20.6Q8.3C2, C20.221(C4○D4), C4.105(C4○D20), (C2×C40).310C22, (C2×C20).730C23, C2.7(C8.D10), C10.7(C4.4D4), C10.2(C8.C22), C20.44D4.15C2, C2.12(C4.D20), (C2×Dic10).8C22, C5⋊1(C42.30C22), (C5×C8⋊C4).10C2, (C2×C10).113(C2×D4), (C2×C4).674(C22×D5), SmallGroup(320,330)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.14D10
G = < a,b,c,d | a4=b4=1, c10=a2b-1, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=bc9 >
Subgroups: 350 in 90 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C42.30C22, C10.D4, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C20.44D4, C5×C8⋊C4, C20⋊2Q8, C20.6Q8, C42.14D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8.C22, D20, C22×D5, C42.30C22, C2×D20, C4○D20, C4.D20, C8.D10, C42.14D10
►Regular action on 320 points
Generators in S320
(1 292 66 215)(2 313 67 236)(3 294 68 217)(4 315 69 238)(5 296 70 219)(6 317 71 240)(7 298 72 221)(8 319 73 202)(9 300 74 223)(10 281 75 204)(11 302 76 225)(12 283 77 206)(13 304 78 227)(14 285 79 208)(15 306 80 229)(16 287 41 210)(17 308 42 231)(18 289 43 212)(19 310 44 233)(20 291 45 214)(21 312 46 235)(22 293 47 216)(23 314 48 237)(24 295 49 218)(25 316 50 239)(26 297 51 220)(27 318 52 201)(28 299 53 222)(29 320 54 203)(30 301 55 224)(31 282 56 205)(32 303 57 226)(33 284 58 207)(34 305 59 228)(35 286 60 209)(36 307 61 230)(37 288 62 211)(38 309 63 232)(39 290 64 213)(40 311 65 234)(81 266 187 128)(82 247 188 149)(83 268 189 130)(84 249 190 151)(85 270 191 132)(86 251 192 153)(87 272 193 134)(88 253 194 155)(89 274 195 136)(90 255 196 157)(91 276 197 138)(92 257 198 159)(93 278 199 140)(94 259 200 121)(95 280 161 142)(96 261 162 123)(97 242 163 144)(98 263 164 125)(99 244 165 146)(100 265 166 127)(101 246 167 148)(102 267 168 129)(103 248 169 150)(104 269 170 131)(105 250 171 152)(106 271 172 133)(107 252 173 154)(108 273 174 135)(109 254 175 156)(110 275 176 137)(111 256 177 158)(112 277 178 139)(113 258 179 160)(114 279 180 141)(115 260 181 122)(116 241 182 143)(117 262 183 124)(118 243 184 145)(119 264 185 126)(120 245 186 147)
(1 56 21 76)(2 57 22 77)(3 58 23 78)(4 59 24 79)(5 60 25 80)(6 61 26 41)(7 62 27 42)(8 63 28 43)(9 64 29 44)(10 65 30 45)(11 66 31 46)(12 67 32 47)(13 68 33 48)(14 69 34 49)(15 70 35 50)(16 71 36 51)(17 72 37 52)(18 73 38 53)(19 74 39 54)(20 75 40 55)(81 177 101 197)(82 178 102 198)(83 179 103 199)(84 180 104 200)(85 181 105 161)(86 182 106 162)(87 183 107 163)(88 184 108 164)(89 185 109 165)(90 186 110 166)(91 187 111 167)(92 188 112 168)(93 189 113 169)(94 190 114 170)(95 191 115 171)(96 192 116 172)(97 193 117 173)(98 194 118 174)(99 195 119 175)(100 196 120 176)(121 249 141 269)(122 250 142 270)(123 251 143 271)(124 252 144 272)(125 253 145 273)(126 254 146 274)(127 255 147 275)(128 256 148 276)(129 257 149 277)(130 258 150 278)(131 259 151 279)(132 260 152 280)(133 261 153 241)(134 262 154 242)(135 263 155 243)(136 264 156 244)(137 265 157 245)(138 266 158 246)(139 267 159 247)(140 268 160 248)(201 308 221 288)(202 309 222 289)(203 310 223 290)(204 311 224 291)(205 312 225 292)(206 313 226 293)(207 314 227 294)(208 315 228 295)(209 316 229 296)(210 317 230 297)(211 318 231 298)(212 319 232 299)(213 320 233 300)(214 281 234 301)(215 282 235 302)(216 283 236 303)(217 284 237 304)(218 285 238 305)(219 286 239 306)(220 287 240 307)
(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)
(1 165 21 185)(2 98 22 118)(3 163 23 183)(4 96 24 116)(5 161 25 181)(6 94 26 114)(7 199 27 179)(8 92 28 112)(9 197 29 177)(10 90 30 110)(11 195 31 175)(12 88 32 108)(13 193 33 173)(14 86 34 106)(15 191 35 171)(16 84 36 104)(17 189 37 169)(18 82 38 102)(19 187 39 167)(20 120 40 100)(41 190 61 170)(42 83 62 103)(43 188 63 168)(44 81 64 101)(45 186 65 166)(46 119 66 99)(47 184 67 164)(48 117 68 97)(49 182 69 162)(50 115 70 95)(51 180 71 200)(52 113 72 93)(53 178 73 198)(54 111 74 91)(55 176 75 196)(56 109 76 89)(57 174 77 194)(58 107 78 87)(59 172 79 192)(60 105 80 85)(121 297 141 317)(122 219 142 239)(123 295 143 315)(124 217 144 237)(125 293 145 313)(126 215 146 235)(127 291 147 311)(128 213 148 233)(129 289 149 309)(130 211 150 231)(131 287 151 307)(132 209 152 229)(133 285 153 305)(134 207 154 227)(135 283 155 303)(136 205 156 225)(137 281 157 301)(138 203 158 223)(139 319 159 299)(140 201 160 221)(202 257 222 277)(204 255 224 275)(206 253 226 273)(208 251 228 271)(210 249 230 269)(212 247 232 267)(214 245 234 265)(216 243 236 263)(218 241 238 261)(220 279 240 259)(242 314 262 294)(244 312 264 292)(246 310 266 290)(248 308 268 288)(250 306 270 286)(252 304 272 284)(254 302 274 282)(256 300 276 320)(258 298 278 318)(260 296 280 316)
G:=sub<Sym(320)| (1,292,66,215)(2,313,67,236)(3,294,68,217)(4,315,69,238)(5,296,70,219)(6,317,71,240)(7,298,72,221)(8,319,73,202)(9,300,74,223)(10,281,75,204)(11,302,76,225)(12,283,77,206)(13,304,78,227)(14,285,79,208)(15,306,80,229)(16,287,41,210)(17,308,42,231)(18,289,43,212)(19,310,44,233)(20,291,45,214)(21,312,46,235)(22,293,47,216)(23,314,48,237)(24,295,49,218)(25,316,50,239)(26,297,51,220)(27,318,52,201)(28,299,53,222)(29,320,54,203)(30,301,55,224)(31,282,56,205)(32,303,57,226)(33,284,58,207)(34,305,59,228)(35,286,60,209)(36,307,61,230)(37,288,62,211)(38,309,63,232)(39,290,64,213)(40,311,65,234)(81,266,187,128)(82,247,188,149)(83,268,189,130)(84,249,190,151)(85,270,191,132)(86,251,192,153)(87,272,193,134)(88,253,194,155)(89,274,195,136)(90,255,196,157)(91,276,197,138)(92,257,198,159)(93,278,199,140)(94,259,200,121)(95,280,161,142)(96,261,162,123)(97,242,163,144)(98,263,164,125)(99,244,165,146)(100,265,166,127)(101,246,167,148)(102,267,168,129)(103,248,169,150)(104,269,170,131)(105,250,171,152)(106,271,172,133)(107,252,173,154)(108,273,174,135)(109,254,175,156)(110,275,176,137)(111,256,177,158)(112,277,178,139)(113,258,179,160)(114,279,180,141)(115,260,181,122)(116,241,182,143)(117,262,183,124)(118,243,184,145)(119,264,185,126)(120,245,186,147), (1,56,21,76)(2,57,22,77)(3,58,23,78)(4,59,24,79)(5,60,25,80)(6,61,26,41)(7,62,27,42)(8,63,28,43)(9,64,29,44)(10,65,30,45)(11,66,31,46)(12,67,32,47)(13,68,33,48)(14,69,34,49)(15,70,35,50)(16,71,36,51)(17,72,37,52)(18,73,38,53)(19,74,39,54)(20,75,40,55)(81,177,101,197)(82,178,102,198)(83,179,103,199)(84,180,104,200)(85,181,105,161)(86,182,106,162)(87,183,107,163)(88,184,108,164)(89,185,109,165)(90,186,110,166)(91,187,111,167)(92,188,112,168)(93,189,113,169)(94,190,114,170)(95,191,115,171)(96,192,116,172)(97,193,117,173)(98,194,118,174)(99,195,119,175)(100,196,120,176)(121,249,141,269)(122,250,142,270)(123,251,143,271)(124,252,144,272)(125,253,145,273)(126,254,146,274)(127,255,147,275)(128,256,148,276)(129,257,149,277)(130,258,150,278)(131,259,151,279)(132,260,152,280)(133,261,153,241)(134,262,154,242)(135,263,155,243)(136,264,156,244)(137,265,157,245)(138,266,158,246)(139,267,159,247)(140,268,160,248)(201,308,221,288)(202,309,222,289)(203,310,223,290)(204,311,224,291)(205,312,225,292)(206,313,226,293)(207,314,227,294)(208,315,228,295)(209,316,229,296)(210,317,230,297)(211,318,231,298)(212,319,232,299)(213,320,233,300)(214,281,234,301)(215,282,235,302)(216,283,236,303)(217,284,237,304)(218,285,238,305)(219,286,239,306)(220,287,240,307), (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), (1,165,21,185)(2,98,22,118)(3,163,23,183)(4,96,24,116)(5,161,25,181)(6,94,26,114)(7,199,27,179)(8,92,28,112)(9,197,29,177)(10,90,30,110)(11,195,31,175)(12,88,32,108)(13,193,33,173)(14,86,34,106)(15,191,35,171)(16,84,36,104)(17,189,37,169)(18,82,38,102)(19,187,39,167)(20,120,40,100)(41,190,61,170)(42,83,62,103)(43,188,63,168)(44,81,64,101)(45,186,65,166)(46,119,66,99)(47,184,67,164)(48,117,68,97)(49,182,69,162)(50,115,70,95)(51,180,71,200)(52,113,72,93)(53,178,73,198)(54,111,74,91)(55,176,75,196)(56,109,76,89)(57,174,77,194)(58,107,78,87)(59,172,79,192)(60,105,80,85)(121,297,141,317)(122,219,142,239)(123,295,143,315)(124,217,144,237)(125,293,145,313)(126,215,146,235)(127,291,147,311)(128,213,148,233)(129,289,149,309)(130,211,150,231)(131,287,151,307)(132,209,152,229)(133,285,153,305)(134,207,154,227)(135,283,155,303)(136,205,156,225)(137,281,157,301)(138,203,158,223)(139,319,159,299)(140,201,160,221)(202,257,222,277)(204,255,224,275)(206,253,226,273)(208,251,228,271)(210,249,230,269)(212,247,232,267)(214,245,234,265)(216,243,236,263)(218,241,238,261)(220,279,240,259)(242,314,262,294)(244,312,264,292)(246,310,266,290)(248,308,268,288)(250,306,270,286)(252,304,272,284)(254,302,274,282)(256,300,276,320)(258,298,278,318)(260,296,280,316)>;
G:=Group( (1,292,66,215)(2,313,67,236)(3,294,68,217)(4,315,69,238)(5,296,70,219)(6,317,71,240)(7,298,72,221)(8,319,73,202)(9,300,74,223)(10,281,75,204)(11,302,76,225)(12,283,77,206)(13,304,78,227)(14,285,79,208)(15,306,80,229)(16,287,41,210)(17,308,42,231)(18,289,43,212)(19,310,44,233)(20,291,45,214)(21,312,46,235)(22,293,47,216)(23,314,48,237)(24,295,49,218)(25,316,50,239)(26,297,51,220)(27,318,52,201)(28,299,53,222)(29,320,54,203)(30,301,55,224)(31,282,56,205)(32,303,57,226)(33,284,58,207)(34,305,59,228)(35,286,60,209)(36,307,61,230)(37,288,62,211)(38,309,63,232)(39,290,64,213)(40,311,65,234)(81,266,187,128)(82,247,188,149)(83,268,189,130)(84,249,190,151)(85,270,191,132)(86,251,192,153)(87,272,193,134)(88,253,194,155)(89,274,195,136)(90,255,196,157)(91,276,197,138)(92,257,198,159)(93,278,199,140)(94,259,200,121)(95,280,161,142)(96,261,162,123)(97,242,163,144)(98,263,164,125)(99,244,165,146)(100,265,166,127)(101,246,167,148)(102,267,168,129)(103,248,169,150)(104,269,170,131)(105,250,171,152)(106,271,172,133)(107,252,173,154)(108,273,174,135)(109,254,175,156)(110,275,176,137)(111,256,177,158)(112,277,178,139)(113,258,179,160)(114,279,180,141)(115,260,181,122)(116,241,182,143)(117,262,183,124)(118,243,184,145)(119,264,185,126)(120,245,186,147), (1,56,21,76)(2,57,22,77)(3,58,23,78)(4,59,24,79)(5,60,25,80)(6,61,26,41)(7,62,27,42)(8,63,28,43)(9,64,29,44)(10,65,30,45)(11,66,31,46)(12,67,32,47)(13,68,33,48)(14,69,34,49)(15,70,35,50)(16,71,36,51)(17,72,37,52)(18,73,38,53)(19,74,39,54)(20,75,40,55)(81,177,101,197)(82,178,102,198)(83,179,103,199)(84,180,104,200)(85,181,105,161)(86,182,106,162)(87,183,107,163)(88,184,108,164)(89,185,109,165)(90,186,110,166)(91,187,111,167)(92,188,112,168)(93,189,113,169)(94,190,114,170)(95,191,115,171)(96,192,116,172)(97,193,117,173)(98,194,118,174)(99,195,119,175)(100,196,120,176)(121,249,141,269)(122,250,142,270)(123,251,143,271)(124,252,144,272)(125,253,145,273)(126,254,146,274)(127,255,147,275)(128,256,148,276)(129,257,149,277)(130,258,150,278)(131,259,151,279)(132,260,152,280)(133,261,153,241)(134,262,154,242)(135,263,155,243)(136,264,156,244)(137,265,157,245)(138,266,158,246)(139,267,159,247)(140,268,160,248)(201,308,221,288)(202,309,222,289)(203,310,223,290)(204,311,224,291)(205,312,225,292)(206,313,226,293)(207,314,227,294)(208,315,228,295)(209,316,229,296)(210,317,230,297)(211,318,231,298)(212,319,232,299)(213,320,233,300)(214,281,234,301)(215,282,235,302)(216,283,236,303)(217,284,237,304)(218,285,238,305)(219,286,239,306)(220,287,240,307), (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), (1,165,21,185)(2,98,22,118)(3,163,23,183)(4,96,24,116)(5,161,25,181)(6,94,26,114)(7,199,27,179)(8,92,28,112)(9,197,29,177)(10,90,30,110)(11,195,31,175)(12,88,32,108)(13,193,33,173)(14,86,34,106)(15,191,35,171)(16,84,36,104)(17,189,37,169)(18,82,38,102)(19,187,39,167)(20,120,40,100)(41,190,61,170)(42,83,62,103)(43,188,63,168)(44,81,64,101)(45,186,65,166)(46,119,66,99)(47,184,67,164)(48,117,68,97)(49,182,69,162)(50,115,70,95)(51,180,71,200)(52,113,72,93)(53,178,73,198)(54,111,74,91)(55,176,75,196)(56,109,76,89)(57,174,77,194)(58,107,78,87)(59,172,79,192)(60,105,80,85)(121,297,141,317)(122,219,142,239)(123,295,143,315)(124,217,144,237)(125,293,145,313)(126,215,146,235)(127,291,147,311)(128,213,148,233)(129,289,149,309)(130,211,150,231)(131,287,151,307)(132,209,152,229)(133,285,153,305)(134,207,154,227)(135,283,155,303)(136,205,156,225)(137,281,157,301)(138,203,158,223)(139,319,159,299)(140,201,160,221)(202,257,222,277)(204,255,224,275)(206,253,226,273)(208,251,228,271)(210,249,230,269)(212,247,232,267)(214,245,234,265)(216,243,236,263)(218,241,238,261)(220,279,240,259)(242,314,262,294)(244,312,264,292)(246,310,266,290)(248,308,268,288)(250,306,270,286)(252,304,272,284)(254,302,274,282)(256,300,276,320)(258,298,278,318)(260,296,280,316) );
G=PermutationGroup([[(1,292,66,215),(2,313,67,236),(3,294,68,217),(4,315,69,238),(5,296,70,219),(6,317,71,240),(7,298,72,221),(8,319,73,202),(9,300,74,223),(10,281,75,204),(11,302,76,225),(12,283,77,206),(13,304,78,227),(14,285,79,208),(15,306,80,229),(16,287,41,210),(17,308,42,231),(18,289,43,212),(19,310,44,233),(20,291,45,214),(21,312,46,235),(22,293,47,216),(23,314,48,237),(24,295,49,218),(25,316,50,239),(26,297,51,220),(27,318,52,201),(28,299,53,222),(29,320,54,203),(30,301,55,224),(31,282,56,205),(32,303,57,226),(33,284,58,207),(34,305,59,228),(35,286,60,209),(36,307,61,230),(37,288,62,211),(38,309,63,232),(39,290,64,213),(40,311,65,234),(81,266,187,128),(82,247,188,149),(83,268,189,130),(84,249,190,151),(85,270,191,132),(86,251,192,153),(87,272,193,134),(88,253,194,155),(89,274,195,136),(90,255,196,157),(91,276,197,138),(92,257,198,159),(93,278,199,140),(94,259,200,121),(95,280,161,142),(96,261,162,123),(97,242,163,144),(98,263,164,125),(99,244,165,146),(100,265,166,127),(101,246,167,148),(102,267,168,129),(103,248,169,150),(104,269,170,131),(105,250,171,152),(106,271,172,133),(107,252,173,154),(108,273,174,135),(109,254,175,156),(110,275,176,137),(111,256,177,158),(112,277,178,139),(113,258,179,160),(114,279,180,141),(115,260,181,122),(116,241,182,143),(117,262,183,124),(118,243,184,145),(119,264,185,126),(120,245,186,147)], [(1,56,21,76),(2,57,22,77),(3,58,23,78),(4,59,24,79),(5,60,25,80),(6,61,26,41),(7,62,27,42),(8,63,28,43),(9,64,29,44),(10,65,30,45),(11,66,31,46),(12,67,32,47),(13,68,33,48),(14,69,34,49),(15,70,35,50),(16,71,36,51),(17,72,37,52),(18,73,38,53),(19,74,39,54),(20,75,40,55),(81,177,101,197),(82,178,102,198),(83,179,103,199),(84,180,104,200),(85,181,105,161),(86,182,106,162),(87,183,107,163),(88,184,108,164),(89,185,109,165),(90,186,110,166),(91,187,111,167),(92,188,112,168),(93,189,113,169),(94,190,114,170),(95,191,115,171),(96,192,116,172),(97,193,117,173),(98,194,118,174),(99,195,119,175),(100,196,120,176),(121,249,141,269),(122,250,142,270),(123,251,143,271),(124,252,144,272),(125,253,145,273),(126,254,146,274),(127,255,147,275),(128,256,148,276),(129,257,149,277),(130,258,150,278),(131,259,151,279),(132,260,152,280),(133,261,153,241),(134,262,154,242),(135,263,155,243),(136,264,156,244),(137,265,157,245),(138,266,158,246),(139,267,159,247),(140,268,160,248),(201,308,221,288),(202,309,222,289),(203,310,223,290),(204,311,224,291),(205,312,225,292),(206,313,226,293),(207,314,227,294),(208,315,228,295),(209,316,229,296),(210,317,230,297),(211,318,231,298),(212,319,232,299),(213,320,233,300),(214,281,234,301),(215,282,235,302),(216,283,236,303),(217,284,237,304),(218,285,238,305),(219,286,239,306),(220,287,240,307)], [(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)], [(1,165,21,185),(2,98,22,118),(3,163,23,183),(4,96,24,116),(5,161,25,181),(6,94,26,114),(7,199,27,179),(8,92,28,112),(9,197,29,177),(10,90,30,110),(11,195,31,175),(12,88,32,108),(13,193,33,173),(14,86,34,106),(15,191,35,171),(16,84,36,104),(17,189,37,169),(18,82,38,102),(19,187,39,167),(20,120,40,100),(41,190,61,170),(42,83,62,103),(43,188,63,168),(44,81,64,101),(45,186,65,166),(46,119,66,99),(47,184,67,164),(48,117,68,97),(49,182,69,162),(50,115,70,95),(51,180,71,200),(52,113,72,93),(53,178,73,198),(54,111,74,91),(55,176,75,196),(56,109,76,89),(57,174,77,194),(58,107,78,87),(59,172,79,192),(60,105,80,85),(121,297,141,317),(122,219,142,239),(123,295,143,315),(124,217,144,237),(125,293,145,313),(126,215,146,235),(127,291,147,311),(128,213,148,233),(129,289,149,309),(130,211,150,231),(131,287,151,307),(132,209,152,229),(133,285,153,305),(134,207,154,227),(135,283,155,303),(136,205,156,225),(137,281,157,301),(138,203,158,223),(139,319,159,299),(140,201,160,221),(202,257,222,277),(204,255,224,275),(206,253,226,273),(208,251,228,271),(210,249,230,269),(212,247,232,267),(214,245,234,265),(216,243,236,263),(218,241,238,261),(220,279,240,259),(242,314,262,294),(244,312,264,292),(246,310,266,290),(248,308,268,288),(250,306,270,286),(252,304,272,284),(254,302,274,282),(256,300,276,320),(258,298,278,318),(260,296,280,316)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 40 | 40 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C4○D20 | C8.C22 | C8.D10 |
| kernel | C42.14D10 | C20.44D4 | C5×C8⋊C4 | C20⋊2Q8 | C20.6Q8 | C2×C20 | C8⋊C4 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 16 | 2 | 8 |
Matrix representation of C42.14D10 ►in GL6(𝔽41)
| 39 | 32 | 0 | 0 | 0 | 0 |
| 37 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 27 | 4 | 28 |
| 0 | 0 | 14 | 38 | 13 | 0 |
| 0 | 0 | 30 | 8 | 5 | 14 |
| 0 | 0 | 22 | 23 | 29 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 32 |
| 0 | 0 | 0 | 0 | 37 | 2 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 28 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 20 | 26 | 35 |
| 0 | 0 | 21 | 31 | 26 | 21 |
| 0 | 0 | 0 | 19 | 13 | 21 |
| 0 | 0 | 22 | 9 | 23 | 31 |
| 36 | 26 | 0 | 0 | 0 | 0 |
| 18 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 10 | 28 | 1 |
| 0 | 0 | 17 | 1 | 32 | 12 |
| 0 | 0 | 4 | 27 | 36 | 19 |
| 0 | 0 | 3 | 23 | 12 | 5 |
G:=sub<GL(6,GF(41))| [39,37,0,0,0,0,32,2,0,0,0,0,0,0,22,14,30,22,0,0,27,38,8,23,0,0,4,13,5,29,0,0,28,0,14,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,39,37,0,0,0,0,32,2],[0,28,0,0,0,0,22,13,0,0,0,0,0,0,7,21,0,22,0,0,20,31,19,9,0,0,26,26,13,23,0,0,35,21,21,31],[36,18,0,0,0,0,26,5,0,0,0,0,0,0,40,17,4,3,0,0,10,1,27,23,0,0,28,32,36,12,0,0,1,12,19,5] >;
C42.14D10 in GAP, Magma, Sage, TeX
C_4^2._{14}D_{10} % in TeX
G:=Group("C4^2.14D10"); // GroupNames label
G:=SmallGroup(320,330);
// by ID
G=gap.SmallGroup(320,330);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,387,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^-1,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations