metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.4F5, C20.18C42, C10.5M5(2), C5⋊C16⋊6C4, C5⋊2C8.8C8, C10.6(C4×C8), (C2×C20).3C8, C5⋊2(C16⋊5C4), C4.16(C4×F5), (C4×C20).15C4, C20.19(C2×C8), C4.14(D5⋊C8), C2.2(C20.C8), C2.4(C4×C5⋊C8), (C2×C5⋊C16).2C2, (C2×C4).2(C5⋊C8), C22.9(C2×C5⋊C8), (C4×C5⋊2C8).21C2, (C2×C5⋊2C8).31C4, (C2×C10).25(C2×C8), C5⋊2C8.35(C2×C4), (C2×C4).149(C2×F5), (C2×C20).155(C2×C4), (C2×C5⋊2C8).342C22, SmallGroup(320,197)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.4F5
G = < a,b,c,d | a4=b4=c5=1, d4=b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c3 >
Smallest permutation representation of C42.4F5
►Regular action on 320 points
Generators in S320
(1 303 115 133)(2 296 116 142)(3 289 117 135)(4 298 118 144)(5 291 119 137)(6 300 120 130)(7 293 121 139)(8 302 122 132)(9 295 123 141)(10 304 124 134)(11 297 125 143)(12 290 126 136)(13 299 127 129)(14 292 128 138)(15 301 113 131)(16 294 114 140)(17 62 80 199)(18 55 65 208)(19 64 66 201)(20 57 67 194)(21 50 68 203)(22 59 69 196)(23 52 70 205)(24 61 71 198)(25 54 72 207)(26 63 73 200)(27 56 74 193)(28 49 75 202)(29 58 76 195)(30 51 77 204)(31 60 78 197)(32 53 79 206)(33 215 169 101)(34 224 170 110)(35 217 171 103)(36 210 172 112)(37 219 173 105)(38 212 174 98)(39 221 175 107)(40 214 176 100)(41 223 161 109)(42 216 162 102)(43 209 163 111)(44 218 164 104)(45 211 165 97)(46 220 166 106)(47 213 167 99)(48 222 168 108)(81 190 227 249)(82 183 228 242)(83 192 229 251)(84 185 230 244)(85 178 231 253)(86 187 232 246)(87 180 233 255)(88 189 234 248)(89 182 235 241)(90 191 236 250)(91 184 237 243)(92 177 238 252)(93 186 239 245)(94 179 240 254)(95 188 225 247)(96 181 226 256)(145 270 305 288)(146 263 306 281)(147 272 307 274)(148 265 308 283)(149 258 309 276)(150 267 310 285)(151 260 311 278)(152 269 312 287)(153 262 313 280)(154 271 314 273)(155 264 315 282)(156 257 316 275)(157 266 317 284)(158 259 318 277)(159 268 319 286)(160 261 320 279)
(1 5 9 13)(2 6 10 14)(3 7 11 15)(4 8 12 16)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)(33 37 41 45)(34 38 42 46)(35 39 43 47)(36 40 44 48)(49 53 57 61)(50 54 58 62)(51 55 59 63)(52 56 60 64)(65 69 73 77)(66 70 74 78)(67 71 75 79)(68 72 76 80)(81 85 89 93)(82 86 90 94)(83 87 91 95)(84 88 92 96)(97 101 105 109)(98 102 106 110)(99 103 107 111)(100 104 108 112)(113 117 121 125)(114 118 122 126)(115 119 123 127)(116 120 124 128)(129 133 137 141)(130 134 138 142)(131 135 139 143)(132 136 140 144)(145 149 153 157)(146 150 154 158)(147 151 155 159)(148 152 156 160)(161 165 169 173)(162 166 170 174)(163 167 171 175)(164 168 172 176)(177 181 185 189)(178 182 186 190)(179 183 187 191)(180 184 188 192)(193 197 201 205)(194 198 202 206)(195 199 203 207)(196 200 204 208)(209 213 217 221)(210 214 218 222)(211 215 219 223)(212 216 220 224)(225 229 233 237)(226 230 234 238)(227 231 235 239)(228 232 236 240)(241 245 249 253)(242 246 250 254)(243 247 251 255)(244 248 252 256)(257 261 265 269)(258 262 266 270)(259 263 267 271)(260 264 268 272)(273 277 281 285)(274 278 282 286)(275 279 283 287)(276 280 284 288)(289 293 297 301)(290 294 298 302)(291 295 299 303)(292 296 300 304)(305 309 313 317)(306 310 314 318)(307 311 315 319)(308 312 316 320)
(1 283 57 229 33)(2 230 284 34 58)(3 35 231 59 285)(4 60 36 286 232)(5 287 61 233 37)(6 234 288 38 62)(7 39 235 63 273)(8 64 40 274 236)(9 275 49 237 41)(10 238 276 42 50)(11 43 239 51 277)(12 52 44 278 240)(13 279 53 225 45)(14 226 280 46 54)(15 47 227 55 281)(16 56 48 282 228)(17 130 189 305 98)(18 306 131 99 190)(19 100 307 191 132)(20 192 101 133 308)(21 134 177 309 102)(22 310 135 103 178)(23 104 311 179 136)(24 180 105 137 312)(25 138 181 313 106)(26 314 139 107 182)(27 108 315 183 140)(28 184 109 141 316)(29 142 185 317 110)(30 318 143 111 186)(31 112 319 187 144)(32 188 97 129 320)(65 146 301 213 249)(66 214 147 250 302)(67 251 215 303 148)(68 304 252 149 216)(69 150 289 217 253)(70 218 151 254 290)(71 255 219 291 152)(72 292 256 153 220)(73 154 293 221 241)(74 222 155 242 294)(75 243 223 295 156)(76 296 244 157 224)(77 158 297 209 245)(78 210 159 246 298)(79 247 211 299 160)(80 300 248 145 212)(81 208 263 113 167)(82 114 193 168 264)(83 169 115 265 194)(84 266 170 195 116)(85 196 267 117 171)(86 118 197 172 268)(87 173 119 269 198)(88 270 174 199 120)(89 200 271 121 175)(90 122 201 176 272)(91 161 123 257 202)(92 258 162 203 124)(93 204 259 125 163)(94 126 205 164 260)(95 165 127 261 206)(96 262 166 207 128)
(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)
G:=sub<Sym(320)| (1,303,115,133)(2,296,116,142)(3,289,117,135)(4,298,118,144)(5,291,119,137)(6,300,120,130)(7,293,121,139)(8,302,122,132)(9,295,123,141)(10,304,124,134)(11,297,125,143)(12,290,126,136)(13,299,127,129)(14,292,128,138)(15,301,113,131)(16,294,114,140)(17,62,80,199)(18,55,65,208)(19,64,66,201)(20,57,67,194)(21,50,68,203)(22,59,69,196)(23,52,70,205)(24,61,71,198)(25,54,72,207)(26,63,73,200)(27,56,74,193)(28,49,75,202)(29,58,76,195)(30,51,77,204)(31,60,78,197)(32,53,79,206)(33,215,169,101)(34,224,170,110)(35,217,171,103)(36,210,172,112)(37,219,173,105)(38,212,174,98)(39,221,175,107)(40,214,176,100)(41,223,161,109)(42,216,162,102)(43,209,163,111)(44,218,164,104)(45,211,165,97)(46,220,166,106)(47,213,167,99)(48,222,168,108)(81,190,227,249)(82,183,228,242)(83,192,229,251)(84,185,230,244)(85,178,231,253)(86,187,232,246)(87,180,233,255)(88,189,234,248)(89,182,235,241)(90,191,236,250)(91,184,237,243)(92,177,238,252)(93,186,239,245)(94,179,240,254)(95,188,225,247)(96,181,226,256)(145,270,305,288)(146,263,306,281)(147,272,307,274)(148,265,308,283)(149,258,309,276)(150,267,310,285)(151,260,311,278)(152,269,312,287)(153,262,313,280)(154,271,314,273)(155,264,315,282)(156,257,316,275)(157,266,317,284)(158,259,318,277)(159,268,319,286)(160,261,320,279), (1,5,9,13)(2,6,10,14)(3,7,11,15)(4,8,12,16)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32)(33,37,41,45)(34,38,42,46)(35,39,43,47)(36,40,44,48)(49,53,57,61)(50,54,58,62)(51,55,59,63)(52,56,60,64)(65,69,73,77)(66,70,74,78)(67,71,75,79)(68,72,76,80)(81,85,89,93)(82,86,90,94)(83,87,91,95)(84,88,92,96)(97,101,105,109)(98,102,106,110)(99,103,107,111)(100,104,108,112)(113,117,121,125)(114,118,122,126)(115,119,123,127)(116,120,124,128)(129,133,137,141)(130,134,138,142)(131,135,139,143)(132,136,140,144)(145,149,153,157)(146,150,154,158)(147,151,155,159)(148,152,156,160)(161,165,169,173)(162,166,170,174)(163,167,171,175)(164,168,172,176)(177,181,185,189)(178,182,186,190)(179,183,187,191)(180,184,188,192)(193,197,201,205)(194,198,202,206)(195,199,203,207)(196,200,204,208)(209,213,217,221)(210,214,218,222)(211,215,219,223)(212,216,220,224)(225,229,233,237)(226,230,234,238)(227,231,235,239)(228,232,236,240)(241,245,249,253)(242,246,250,254)(243,247,251,255)(244,248,252,256)(257,261,265,269)(258,262,266,270)(259,263,267,271)(260,264,268,272)(273,277,281,285)(274,278,282,286)(275,279,283,287)(276,280,284,288)(289,293,297,301)(290,294,298,302)(291,295,299,303)(292,296,300,304)(305,309,313,317)(306,310,314,318)(307,311,315,319)(308,312,316,320), (1,283,57,229,33)(2,230,284,34,58)(3,35,231,59,285)(4,60,36,286,232)(5,287,61,233,37)(6,234,288,38,62)(7,39,235,63,273)(8,64,40,274,236)(9,275,49,237,41)(10,238,276,42,50)(11,43,239,51,277)(12,52,44,278,240)(13,279,53,225,45)(14,226,280,46,54)(15,47,227,55,281)(16,56,48,282,228)(17,130,189,305,98)(18,306,131,99,190)(19,100,307,191,132)(20,192,101,133,308)(21,134,177,309,102)(22,310,135,103,178)(23,104,311,179,136)(24,180,105,137,312)(25,138,181,313,106)(26,314,139,107,182)(27,108,315,183,140)(28,184,109,141,316)(29,142,185,317,110)(30,318,143,111,186)(31,112,319,187,144)(32,188,97,129,320)(65,146,301,213,249)(66,214,147,250,302)(67,251,215,303,148)(68,304,252,149,216)(69,150,289,217,253)(70,218,151,254,290)(71,255,219,291,152)(72,292,256,153,220)(73,154,293,221,241)(74,222,155,242,294)(75,243,223,295,156)(76,296,244,157,224)(77,158,297,209,245)(78,210,159,246,298)(79,247,211,299,160)(80,300,248,145,212)(81,208,263,113,167)(82,114,193,168,264)(83,169,115,265,194)(84,266,170,195,116)(85,196,267,117,171)(86,118,197,172,268)(87,173,119,269,198)(88,270,174,199,120)(89,200,271,121,175)(90,122,201,176,272)(91,161,123,257,202)(92,258,162,203,124)(93,204,259,125,163)(94,126,205,164,260)(95,165,127,261,206)(96,262,166,207,128), (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)>;
G:=Group( (1,303,115,133)(2,296,116,142)(3,289,117,135)(4,298,118,144)(5,291,119,137)(6,300,120,130)(7,293,121,139)(8,302,122,132)(9,295,123,141)(10,304,124,134)(11,297,125,143)(12,290,126,136)(13,299,127,129)(14,292,128,138)(15,301,113,131)(16,294,114,140)(17,62,80,199)(18,55,65,208)(19,64,66,201)(20,57,67,194)(21,50,68,203)(22,59,69,196)(23,52,70,205)(24,61,71,198)(25,54,72,207)(26,63,73,200)(27,56,74,193)(28,49,75,202)(29,58,76,195)(30,51,77,204)(31,60,78,197)(32,53,79,206)(33,215,169,101)(34,224,170,110)(35,217,171,103)(36,210,172,112)(37,219,173,105)(38,212,174,98)(39,221,175,107)(40,214,176,100)(41,223,161,109)(42,216,162,102)(43,209,163,111)(44,218,164,104)(45,211,165,97)(46,220,166,106)(47,213,167,99)(48,222,168,108)(81,190,227,249)(82,183,228,242)(83,192,229,251)(84,185,230,244)(85,178,231,253)(86,187,232,246)(87,180,233,255)(88,189,234,248)(89,182,235,241)(90,191,236,250)(91,184,237,243)(92,177,238,252)(93,186,239,245)(94,179,240,254)(95,188,225,247)(96,181,226,256)(145,270,305,288)(146,263,306,281)(147,272,307,274)(148,265,308,283)(149,258,309,276)(150,267,310,285)(151,260,311,278)(152,269,312,287)(153,262,313,280)(154,271,314,273)(155,264,315,282)(156,257,316,275)(157,266,317,284)(158,259,318,277)(159,268,319,286)(160,261,320,279), (1,5,9,13)(2,6,10,14)(3,7,11,15)(4,8,12,16)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32)(33,37,41,45)(34,38,42,46)(35,39,43,47)(36,40,44,48)(49,53,57,61)(50,54,58,62)(51,55,59,63)(52,56,60,64)(65,69,73,77)(66,70,74,78)(67,71,75,79)(68,72,76,80)(81,85,89,93)(82,86,90,94)(83,87,91,95)(84,88,92,96)(97,101,105,109)(98,102,106,110)(99,103,107,111)(100,104,108,112)(113,117,121,125)(114,118,122,126)(115,119,123,127)(116,120,124,128)(129,133,137,141)(130,134,138,142)(131,135,139,143)(132,136,140,144)(145,149,153,157)(146,150,154,158)(147,151,155,159)(148,152,156,160)(161,165,169,173)(162,166,170,174)(163,167,171,175)(164,168,172,176)(177,181,185,189)(178,182,186,190)(179,183,187,191)(180,184,188,192)(193,197,201,205)(194,198,202,206)(195,199,203,207)(196,200,204,208)(209,213,217,221)(210,214,218,222)(211,215,219,223)(212,216,220,224)(225,229,233,237)(226,230,234,238)(227,231,235,239)(228,232,236,240)(241,245,249,253)(242,246,250,254)(243,247,251,255)(244,248,252,256)(257,261,265,269)(258,262,266,270)(259,263,267,271)(260,264,268,272)(273,277,281,285)(274,278,282,286)(275,279,283,287)(276,280,284,288)(289,293,297,301)(290,294,298,302)(291,295,299,303)(292,296,300,304)(305,309,313,317)(306,310,314,318)(307,311,315,319)(308,312,316,320), (1,283,57,229,33)(2,230,284,34,58)(3,35,231,59,285)(4,60,36,286,232)(5,287,61,233,37)(6,234,288,38,62)(7,39,235,63,273)(8,64,40,274,236)(9,275,49,237,41)(10,238,276,42,50)(11,43,239,51,277)(12,52,44,278,240)(13,279,53,225,45)(14,226,280,46,54)(15,47,227,55,281)(16,56,48,282,228)(17,130,189,305,98)(18,306,131,99,190)(19,100,307,191,132)(20,192,101,133,308)(21,134,177,309,102)(22,310,135,103,178)(23,104,311,179,136)(24,180,105,137,312)(25,138,181,313,106)(26,314,139,107,182)(27,108,315,183,140)(28,184,109,141,316)(29,142,185,317,110)(30,318,143,111,186)(31,112,319,187,144)(32,188,97,129,320)(65,146,301,213,249)(66,214,147,250,302)(67,251,215,303,148)(68,304,252,149,216)(69,150,289,217,253)(70,218,151,254,290)(71,255,219,291,152)(72,292,256,153,220)(73,154,293,221,241)(74,222,155,242,294)(75,243,223,295,156)(76,296,244,157,224)(77,158,297,209,245)(78,210,159,246,298)(79,247,211,299,160)(80,300,248,145,212)(81,208,263,113,167)(82,114,193,168,264)(83,169,115,265,194)(84,266,170,195,116)(85,196,267,117,171)(86,118,197,172,268)(87,173,119,269,198)(88,270,174,199,120)(89,200,271,121,175)(90,122,201,176,272)(91,161,123,257,202)(92,258,162,203,124)(93,204,259,125,163)(94,126,205,164,260)(95,165,127,261,206)(96,262,166,207,128), (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) );
G=PermutationGroup([[(1,303,115,133),(2,296,116,142),(3,289,117,135),(4,298,118,144),(5,291,119,137),(6,300,120,130),(7,293,121,139),(8,302,122,132),(9,295,123,141),(10,304,124,134),(11,297,125,143),(12,290,126,136),(13,299,127,129),(14,292,128,138),(15,301,113,131),(16,294,114,140),(17,62,80,199),(18,55,65,208),(19,64,66,201),(20,57,67,194),(21,50,68,203),(22,59,69,196),(23,52,70,205),(24,61,71,198),(25,54,72,207),(26,63,73,200),(27,56,74,193),(28,49,75,202),(29,58,76,195),(30,51,77,204),(31,60,78,197),(32,53,79,206),(33,215,169,101),(34,224,170,110),(35,217,171,103),(36,210,172,112),(37,219,173,105),(38,212,174,98),(39,221,175,107),(40,214,176,100),(41,223,161,109),(42,216,162,102),(43,209,163,111),(44,218,164,104),(45,211,165,97),(46,220,166,106),(47,213,167,99),(48,222,168,108),(81,190,227,249),(82,183,228,242),(83,192,229,251),(84,185,230,244),(85,178,231,253),(86,187,232,246),(87,180,233,255),(88,189,234,248),(89,182,235,241),(90,191,236,250),(91,184,237,243),(92,177,238,252),(93,186,239,245),(94,179,240,254),(95,188,225,247),(96,181,226,256),(145,270,305,288),(146,263,306,281),(147,272,307,274),(148,265,308,283),(149,258,309,276),(150,267,310,285),(151,260,311,278),(152,269,312,287),(153,262,313,280),(154,271,314,273),(155,264,315,282),(156,257,316,275),(157,266,317,284),(158,259,318,277),(159,268,319,286),(160,261,320,279)], [(1,5,9,13),(2,6,10,14),(3,7,11,15),(4,8,12,16),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32),(33,37,41,45),(34,38,42,46),(35,39,43,47),(36,40,44,48),(49,53,57,61),(50,54,58,62),(51,55,59,63),(52,56,60,64),(65,69,73,77),(66,70,74,78),(67,71,75,79),(68,72,76,80),(81,85,89,93),(82,86,90,94),(83,87,91,95),(84,88,92,96),(97,101,105,109),(98,102,106,110),(99,103,107,111),(100,104,108,112),(113,117,121,125),(114,118,122,126),(115,119,123,127),(116,120,124,128),(129,133,137,141),(130,134,138,142),(131,135,139,143),(132,136,140,144),(145,149,153,157),(146,150,154,158),(147,151,155,159),(148,152,156,160),(161,165,169,173),(162,166,170,174),(163,167,171,175),(164,168,172,176),(177,181,185,189),(178,182,186,190),(179,183,187,191),(180,184,188,192),(193,197,201,205),(194,198,202,206),(195,199,203,207),(196,200,204,208),(209,213,217,221),(210,214,218,222),(211,215,219,223),(212,216,220,224),(225,229,233,237),(226,230,234,238),(227,231,235,239),(228,232,236,240),(241,245,249,253),(242,246,250,254),(243,247,251,255),(244,248,252,256),(257,261,265,269),(258,262,266,270),(259,263,267,271),(260,264,268,272),(273,277,281,285),(274,278,282,286),(275,279,283,287),(276,280,284,288),(289,293,297,301),(290,294,298,302),(291,295,299,303),(292,296,300,304),(305,309,313,317),(306,310,314,318),(307,311,315,319),(308,312,316,320)], [(1,283,57,229,33),(2,230,284,34,58),(3,35,231,59,285),(4,60,36,286,232),(5,287,61,233,37),(6,234,288,38,62),(7,39,235,63,273),(8,64,40,274,236),(9,275,49,237,41),(10,238,276,42,50),(11,43,239,51,277),(12,52,44,278,240),(13,279,53,225,45),(14,226,280,46,54),(15,47,227,55,281),(16,56,48,282,228),(17,130,189,305,98),(18,306,131,99,190),(19,100,307,191,132),(20,192,101,133,308),(21,134,177,309,102),(22,310,135,103,178),(23,104,311,179,136),(24,180,105,137,312),(25,138,181,313,106),(26,314,139,107,182),(27,108,315,183,140),(28,184,109,141,316),(29,142,185,317,110),(30,318,143,111,186),(31,112,319,187,144),(32,188,97,129,320),(65,146,301,213,249),(66,214,147,250,302),(67,251,215,303,148),(68,304,252,149,216),(69,150,289,217,253),(70,218,151,254,290),(71,255,219,291,152),(72,292,256,153,220),(73,154,293,221,241),(74,222,155,242,294),(75,243,223,295,156),(76,296,244,157,224),(77,158,297,209,245),(78,210,159,246,298),(79,247,211,299,160),(80,300,248,145,212),(81,208,263,113,167),(82,114,193,168,264),(83,169,115,265,194),(84,266,170,195,116),(85,196,267,117,171),(86,118,197,172,268),(87,173,119,269,198),(88,270,174,199,120),(89,200,271,121,175),(90,122,201,176,272),(91,161,123,257,202),(92,258,162,203,124),(93,204,259,125,163),(94,126,205,164,260),(95,165,127,261,206),(96,262,166,207,128)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 16A | ··· | 16P | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | M5(2) | F5 | C5⋊C8 | C2×F5 | D5⋊C8 | C4×F5 | C20.C8 |
| kernel | C42.4F5 | C4×C5⋊2C8 | C2×C5⋊C16 | C5⋊C16 | C2×C5⋊2C8 | C4×C20 | C5⋊2C8 | C2×C20 | C10 | C42 | C2×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C42.4F5 ►in GL6(𝔽241)
| 138 | 5 | 0 | 0 | 0 | 0 |
| 47 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 64 | 232 | 0 | 0 | 0 | 0 |
| 215 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 138 | 164 | 238 | 94 |
| 0 | 0 | 135 | 17 | 86 | 232 |
| 0 | 0 | 224 | 155 | 9 | 229 |
| 0 | 0 | 147 | 152 | 103 | 77 |
G:=sub<GL(6,GF(241))| [138,47,0,0,0,0,5,103,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[64,215,0,0,0,0,232,177,0,0,0,0,0,0,138,135,224,147,0,0,164,17,155,152,0,0,238,86,9,103,0,0,94,232,229,77] >;
C42.4F5 in GAP, Magma, Sage, TeX
C_4^2._4F_5
% in TeX
G:=Group("C4^2.4F5"); // GroupNames label
G:=SmallGroup(320,197);
// by ID
G=gap.SmallGroup(320,197);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,100,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
Export