metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.147D10, C10.272- 1+4, C4⋊C4.110D10, C42.C2.7D5, (C4×C20).223C22, (C2×C10).230C24, (C2×C20).187C23, C5⋊(C22.58C24), C4.Dic10.14C2, C20.6Q8.12C2, C4⋊Dic5.237C22, Dic5.Q8.3C2, C22.251(C23×D5), (C4×Dic5).146C22, (C2×Dic5).120C23, C10.D4.85C22, C2.56(D4.10D10), C2.28(Q8.10D10), (C5×C42.C2).6C2, (C5×C4⋊C4).185C22, (C2×C4).202(C22×D5), SmallGroup(320,1358)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.147D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 470 in 172 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, C2×C4, C2×C4, C10, C10, C42, C42, C4⋊C4, C4⋊C4, Dic5, C20, C2×C10, C42.C2, C42.C2, C2×Dic5, C2×C20, C2×C20, C22.58C24, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C20.6Q8, Dic5.Q8, C4.Dic10, C5×C42.C2, C42.147D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2- 1+4, C22×D5, C22.58C24, C23×D5, Q8.10D10, D4.10D10, C42.147D10
►Regular action on 320 points
Generators in S320
(1 83 11 93)(2 229 12 239)(3 85 13 95)(4 231 14 221)(5 87 15 97)(6 233 16 223)(7 89 17 99)(8 235 18 225)(9 91 19 81)(10 237 20 227)(21 282 31 292)(22 128 32 138)(23 284 33 294)(24 130 34 140)(25 286 35 296)(26 132 36 122)(27 288 37 298)(28 134 38 124)(29 290 39 300)(30 136 40 126)(41 105 51 115)(42 78 52 68)(43 107 53 117)(44 80 54 70)(45 109 55 119)(46 62 56 72)(47 111 57 101)(48 64 58 74)(49 113 59 103)(50 66 60 76)(61 270 71 280)(63 272 73 262)(65 274 75 264)(67 276 77 266)(69 278 79 268)(82 181 92 191)(84 183 94 193)(86 185 96 195)(88 187 98 197)(90 189 100 199)(102 263 112 273)(104 265 114 275)(106 267 116 277)(108 269 118 279)(110 271 120 261)(121 304 131 314)(123 306 133 316)(125 308 135 318)(127 310 137 320)(129 312 139 302)(141 251 151 241)(142 179 152 169)(143 253 153 243)(144 161 154 171)(145 255 155 245)(146 163 156 173)(147 257 157 247)(148 165 158 175)(149 259 159 249)(150 167 160 177)(162 213 172 203)(164 215 174 205)(166 217 176 207)(168 219 178 209)(170 201 180 211)(182 238 192 228)(184 240 194 230)(186 222 196 232)(188 224 198 234)(190 226 200 236)(202 254 212 244)(204 256 214 246)(206 258 216 248)(208 260 218 250)(210 242 220 252)(281 309 291 319)(283 311 293 301)(285 313 295 303)(287 315 297 305)(289 317 299 307)
(1 206 182 158)(2 159 183 207)(3 208 184 160)(4 141 185 209)(5 210 186 142)(6 143 187 211)(7 212 188 144)(8 145 189 213)(9 214 190 146)(10 147 191 215)(11 216 192 148)(12 149 193 217)(13 218 194 150)(14 151 195 219)(15 220 196 152)(16 153 197 201)(17 202 198 154)(18 155 199 203)(19 204 200 156)(20 157 181 205)(21 265 310 60)(22 41 311 266)(23 267 312 42)(24 43 313 268)(25 269 314 44)(26 45 315 270)(27 271 316 46)(28 47 317 272)(29 273 318 48)(30 49 319 274)(31 275 320 50)(32 51 301 276)(33 277 302 52)(34 53 303 278)(35 279 304 54)(36 55 305 280)(37 261 306 56)(38 57 307 262)(39 263 308 58)(40 59 309 264)(61 122 119 287)(62 288 120 123)(63 124 101 289)(64 290 102 125)(65 126 103 291)(66 292 104 127)(67 128 105 293)(68 294 106 129)(69 130 107 295)(70 296 108 131)(71 132 109 297)(72 298 110 133)(73 134 111 299)(74 300 112 135)(75 136 113 281)(76 282 114 137)(77 138 115 283)(78 284 116 139)(79 140 117 285)(80 286 118 121)(81 256 236 173)(82 174 237 257)(83 258 238 175)(84 176 239 259)(85 260 240 177)(86 178 221 241)(87 242 222 179)(88 180 223 243)(89 244 224 161)(90 162 225 245)(91 246 226 163)(92 164 227 247)(93 248 228 165)(94 166 229 249)(95 250 230 167)(96 168 231 251)(97 252 232 169)(98 170 233 253)(99 254 234 171)(100 172 235 255)
(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)
(1 131 192 286)(2 140 193 295)(3 129 194 284)(4 138 195 293)(5 127 196 282)(6 136 197 291)(7 125 198 300)(8 134 199 289)(9 123 200 298)(10 132 181 287)(11 121 182 296)(12 130 183 285)(13 139 184 294)(14 128 185 283)(15 137 186 292)(16 126 187 281)(17 135 188 290)(18 124 189 299)(19 133 190 288)(20 122 191 297)(21 87 320 232)(22 96 301 221)(23 85 302 230)(24 94 303 239)(25 83 304 228)(26 92 305 237)(27 81 306 226)(28 90 307 235)(29 99 308 224)(30 88 309 233)(31 97 310 222)(32 86 311 231)(33 95 312 240)(34 84 313 229)(35 93 314 238)(36 82 315 227)(37 91 316 236)(38 100 317 225)(39 89 318 234)(40 98 319 223)(41 241 276 168)(42 250 277 177)(43 259 278 166)(44 248 279 175)(45 257 280 164)(46 246 261 173)(47 255 262 162)(48 244 263 171)(49 253 264 180)(50 242 265 169)(51 251 266 178)(52 260 267 167)(53 249 268 176)(54 258 269 165)(55 247 270 174)(56 256 271 163)(57 245 272 172)(58 254 273 161)(59 243 274 170)(60 252 275 179)(61 205 109 147)(62 214 110 156)(63 203 111 145)(64 212 112 154)(65 201 113 143)(66 210 114 152)(67 219 115 141)(68 208 116 150)(69 217 117 159)(70 206 118 148)(71 215 119 157)(72 204 120 146)(73 213 101 155)(74 202 102 144)(75 211 103 153)(76 220 104 142)(77 209 105 151)(78 218 106 160)(79 207 107 149)(80 216 108 158)
G:=sub<Sym(320)| (1,83,11,93)(2,229,12,239)(3,85,13,95)(4,231,14,221)(5,87,15,97)(6,233,16,223)(7,89,17,99)(8,235,18,225)(9,91,19,81)(10,237,20,227)(21,282,31,292)(22,128,32,138)(23,284,33,294)(24,130,34,140)(25,286,35,296)(26,132,36,122)(27,288,37,298)(28,134,38,124)(29,290,39,300)(30,136,40,126)(41,105,51,115)(42,78,52,68)(43,107,53,117)(44,80,54,70)(45,109,55,119)(46,62,56,72)(47,111,57,101)(48,64,58,74)(49,113,59,103)(50,66,60,76)(61,270,71,280)(63,272,73,262)(65,274,75,264)(67,276,77,266)(69,278,79,268)(82,181,92,191)(84,183,94,193)(86,185,96,195)(88,187,98,197)(90,189,100,199)(102,263,112,273)(104,265,114,275)(106,267,116,277)(108,269,118,279)(110,271,120,261)(121,304,131,314)(123,306,133,316)(125,308,135,318)(127,310,137,320)(129,312,139,302)(141,251,151,241)(142,179,152,169)(143,253,153,243)(144,161,154,171)(145,255,155,245)(146,163,156,173)(147,257,157,247)(148,165,158,175)(149,259,159,249)(150,167,160,177)(162,213,172,203)(164,215,174,205)(166,217,176,207)(168,219,178,209)(170,201,180,211)(182,238,192,228)(184,240,194,230)(186,222,196,232)(188,224,198,234)(190,226,200,236)(202,254,212,244)(204,256,214,246)(206,258,216,248)(208,260,218,250)(210,242,220,252)(281,309,291,319)(283,311,293,301)(285,313,295,303)(287,315,297,305)(289,317,299,307), (1,206,182,158)(2,159,183,207)(3,208,184,160)(4,141,185,209)(5,210,186,142)(6,143,187,211)(7,212,188,144)(8,145,189,213)(9,214,190,146)(10,147,191,215)(11,216,192,148)(12,149,193,217)(13,218,194,150)(14,151,195,219)(15,220,196,152)(16,153,197,201)(17,202,198,154)(18,155,199,203)(19,204,200,156)(20,157,181,205)(21,265,310,60)(22,41,311,266)(23,267,312,42)(24,43,313,268)(25,269,314,44)(26,45,315,270)(27,271,316,46)(28,47,317,272)(29,273,318,48)(30,49,319,274)(31,275,320,50)(32,51,301,276)(33,277,302,52)(34,53,303,278)(35,279,304,54)(36,55,305,280)(37,261,306,56)(38,57,307,262)(39,263,308,58)(40,59,309,264)(61,122,119,287)(62,288,120,123)(63,124,101,289)(64,290,102,125)(65,126,103,291)(66,292,104,127)(67,128,105,293)(68,294,106,129)(69,130,107,295)(70,296,108,131)(71,132,109,297)(72,298,110,133)(73,134,111,299)(74,300,112,135)(75,136,113,281)(76,282,114,137)(77,138,115,283)(78,284,116,139)(79,140,117,285)(80,286,118,121)(81,256,236,173)(82,174,237,257)(83,258,238,175)(84,176,239,259)(85,260,240,177)(86,178,221,241)(87,242,222,179)(88,180,223,243)(89,244,224,161)(90,162,225,245)(91,246,226,163)(92,164,227,247)(93,248,228,165)(94,166,229,249)(95,250,230,167)(96,168,231,251)(97,252,232,169)(98,170,233,253)(99,254,234,171)(100,172,235,255), (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), (1,131,192,286)(2,140,193,295)(3,129,194,284)(4,138,195,293)(5,127,196,282)(6,136,197,291)(7,125,198,300)(8,134,199,289)(9,123,200,298)(10,132,181,287)(11,121,182,296)(12,130,183,285)(13,139,184,294)(14,128,185,283)(15,137,186,292)(16,126,187,281)(17,135,188,290)(18,124,189,299)(19,133,190,288)(20,122,191,297)(21,87,320,232)(22,96,301,221)(23,85,302,230)(24,94,303,239)(25,83,304,228)(26,92,305,237)(27,81,306,226)(28,90,307,235)(29,99,308,224)(30,88,309,233)(31,97,310,222)(32,86,311,231)(33,95,312,240)(34,84,313,229)(35,93,314,238)(36,82,315,227)(37,91,316,236)(38,100,317,225)(39,89,318,234)(40,98,319,223)(41,241,276,168)(42,250,277,177)(43,259,278,166)(44,248,279,175)(45,257,280,164)(46,246,261,173)(47,255,262,162)(48,244,263,171)(49,253,264,180)(50,242,265,169)(51,251,266,178)(52,260,267,167)(53,249,268,176)(54,258,269,165)(55,247,270,174)(56,256,271,163)(57,245,272,172)(58,254,273,161)(59,243,274,170)(60,252,275,179)(61,205,109,147)(62,214,110,156)(63,203,111,145)(64,212,112,154)(65,201,113,143)(66,210,114,152)(67,219,115,141)(68,208,116,150)(69,217,117,159)(70,206,118,148)(71,215,119,157)(72,204,120,146)(73,213,101,155)(74,202,102,144)(75,211,103,153)(76,220,104,142)(77,209,105,151)(78,218,106,160)(79,207,107,149)(80,216,108,158)>;
G:=Group( (1,83,11,93)(2,229,12,239)(3,85,13,95)(4,231,14,221)(5,87,15,97)(6,233,16,223)(7,89,17,99)(8,235,18,225)(9,91,19,81)(10,237,20,227)(21,282,31,292)(22,128,32,138)(23,284,33,294)(24,130,34,140)(25,286,35,296)(26,132,36,122)(27,288,37,298)(28,134,38,124)(29,290,39,300)(30,136,40,126)(41,105,51,115)(42,78,52,68)(43,107,53,117)(44,80,54,70)(45,109,55,119)(46,62,56,72)(47,111,57,101)(48,64,58,74)(49,113,59,103)(50,66,60,76)(61,270,71,280)(63,272,73,262)(65,274,75,264)(67,276,77,266)(69,278,79,268)(82,181,92,191)(84,183,94,193)(86,185,96,195)(88,187,98,197)(90,189,100,199)(102,263,112,273)(104,265,114,275)(106,267,116,277)(108,269,118,279)(110,271,120,261)(121,304,131,314)(123,306,133,316)(125,308,135,318)(127,310,137,320)(129,312,139,302)(141,251,151,241)(142,179,152,169)(143,253,153,243)(144,161,154,171)(145,255,155,245)(146,163,156,173)(147,257,157,247)(148,165,158,175)(149,259,159,249)(150,167,160,177)(162,213,172,203)(164,215,174,205)(166,217,176,207)(168,219,178,209)(170,201,180,211)(182,238,192,228)(184,240,194,230)(186,222,196,232)(188,224,198,234)(190,226,200,236)(202,254,212,244)(204,256,214,246)(206,258,216,248)(208,260,218,250)(210,242,220,252)(281,309,291,319)(283,311,293,301)(285,313,295,303)(287,315,297,305)(289,317,299,307), (1,206,182,158)(2,159,183,207)(3,208,184,160)(4,141,185,209)(5,210,186,142)(6,143,187,211)(7,212,188,144)(8,145,189,213)(9,214,190,146)(10,147,191,215)(11,216,192,148)(12,149,193,217)(13,218,194,150)(14,151,195,219)(15,220,196,152)(16,153,197,201)(17,202,198,154)(18,155,199,203)(19,204,200,156)(20,157,181,205)(21,265,310,60)(22,41,311,266)(23,267,312,42)(24,43,313,268)(25,269,314,44)(26,45,315,270)(27,271,316,46)(28,47,317,272)(29,273,318,48)(30,49,319,274)(31,275,320,50)(32,51,301,276)(33,277,302,52)(34,53,303,278)(35,279,304,54)(36,55,305,280)(37,261,306,56)(38,57,307,262)(39,263,308,58)(40,59,309,264)(61,122,119,287)(62,288,120,123)(63,124,101,289)(64,290,102,125)(65,126,103,291)(66,292,104,127)(67,128,105,293)(68,294,106,129)(69,130,107,295)(70,296,108,131)(71,132,109,297)(72,298,110,133)(73,134,111,299)(74,300,112,135)(75,136,113,281)(76,282,114,137)(77,138,115,283)(78,284,116,139)(79,140,117,285)(80,286,118,121)(81,256,236,173)(82,174,237,257)(83,258,238,175)(84,176,239,259)(85,260,240,177)(86,178,221,241)(87,242,222,179)(88,180,223,243)(89,244,224,161)(90,162,225,245)(91,246,226,163)(92,164,227,247)(93,248,228,165)(94,166,229,249)(95,250,230,167)(96,168,231,251)(97,252,232,169)(98,170,233,253)(99,254,234,171)(100,172,235,255), (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), (1,131,192,286)(2,140,193,295)(3,129,194,284)(4,138,195,293)(5,127,196,282)(6,136,197,291)(7,125,198,300)(8,134,199,289)(9,123,200,298)(10,132,181,287)(11,121,182,296)(12,130,183,285)(13,139,184,294)(14,128,185,283)(15,137,186,292)(16,126,187,281)(17,135,188,290)(18,124,189,299)(19,133,190,288)(20,122,191,297)(21,87,320,232)(22,96,301,221)(23,85,302,230)(24,94,303,239)(25,83,304,228)(26,92,305,237)(27,81,306,226)(28,90,307,235)(29,99,308,224)(30,88,309,233)(31,97,310,222)(32,86,311,231)(33,95,312,240)(34,84,313,229)(35,93,314,238)(36,82,315,227)(37,91,316,236)(38,100,317,225)(39,89,318,234)(40,98,319,223)(41,241,276,168)(42,250,277,177)(43,259,278,166)(44,248,279,175)(45,257,280,164)(46,246,261,173)(47,255,262,162)(48,244,263,171)(49,253,264,180)(50,242,265,169)(51,251,266,178)(52,260,267,167)(53,249,268,176)(54,258,269,165)(55,247,270,174)(56,256,271,163)(57,245,272,172)(58,254,273,161)(59,243,274,170)(60,252,275,179)(61,205,109,147)(62,214,110,156)(63,203,111,145)(64,212,112,154)(65,201,113,143)(66,210,114,152)(67,219,115,141)(68,208,116,150)(69,217,117,159)(70,206,118,148)(71,215,119,157)(72,204,120,146)(73,213,101,155)(74,202,102,144)(75,211,103,153)(76,220,104,142)(77,209,105,151)(78,218,106,160)(79,207,107,149)(80,216,108,158) );
G=PermutationGroup([[(1,83,11,93),(2,229,12,239),(3,85,13,95),(4,231,14,221),(5,87,15,97),(6,233,16,223),(7,89,17,99),(8,235,18,225),(9,91,19,81),(10,237,20,227),(21,282,31,292),(22,128,32,138),(23,284,33,294),(24,130,34,140),(25,286,35,296),(26,132,36,122),(27,288,37,298),(28,134,38,124),(29,290,39,300),(30,136,40,126),(41,105,51,115),(42,78,52,68),(43,107,53,117),(44,80,54,70),(45,109,55,119),(46,62,56,72),(47,111,57,101),(48,64,58,74),(49,113,59,103),(50,66,60,76),(61,270,71,280),(63,272,73,262),(65,274,75,264),(67,276,77,266),(69,278,79,268),(82,181,92,191),(84,183,94,193),(86,185,96,195),(88,187,98,197),(90,189,100,199),(102,263,112,273),(104,265,114,275),(106,267,116,277),(108,269,118,279),(110,271,120,261),(121,304,131,314),(123,306,133,316),(125,308,135,318),(127,310,137,320),(129,312,139,302),(141,251,151,241),(142,179,152,169),(143,253,153,243),(144,161,154,171),(145,255,155,245),(146,163,156,173),(147,257,157,247),(148,165,158,175),(149,259,159,249),(150,167,160,177),(162,213,172,203),(164,215,174,205),(166,217,176,207),(168,219,178,209),(170,201,180,211),(182,238,192,228),(184,240,194,230),(186,222,196,232),(188,224,198,234),(190,226,200,236),(202,254,212,244),(204,256,214,246),(206,258,216,248),(208,260,218,250),(210,242,220,252),(281,309,291,319),(283,311,293,301),(285,313,295,303),(287,315,297,305),(289,317,299,307)], [(1,206,182,158),(2,159,183,207),(3,208,184,160),(4,141,185,209),(5,210,186,142),(6,143,187,211),(7,212,188,144),(8,145,189,213),(9,214,190,146),(10,147,191,215),(11,216,192,148),(12,149,193,217),(13,218,194,150),(14,151,195,219),(15,220,196,152),(16,153,197,201),(17,202,198,154),(18,155,199,203),(19,204,200,156),(20,157,181,205),(21,265,310,60),(22,41,311,266),(23,267,312,42),(24,43,313,268),(25,269,314,44),(26,45,315,270),(27,271,316,46),(28,47,317,272),(29,273,318,48),(30,49,319,274),(31,275,320,50),(32,51,301,276),(33,277,302,52),(34,53,303,278),(35,279,304,54),(36,55,305,280),(37,261,306,56),(38,57,307,262),(39,263,308,58),(40,59,309,264),(61,122,119,287),(62,288,120,123),(63,124,101,289),(64,290,102,125),(65,126,103,291),(66,292,104,127),(67,128,105,293),(68,294,106,129),(69,130,107,295),(70,296,108,131),(71,132,109,297),(72,298,110,133),(73,134,111,299),(74,300,112,135),(75,136,113,281),(76,282,114,137),(77,138,115,283),(78,284,116,139),(79,140,117,285),(80,286,118,121),(81,256,236,173),(82,174,237,257),(83,258,238,175),(84,176,239,259),(85,260,240,177),(86,178,221,241),(87,242,222,179),(88,180,223,243),(89,244,224,161),(90,162,225,245),(91,246,226,163),(92,164,227,247),(93,248,228,165),(94,166,229,249),(95,250,230,167),(96,168,231,251),(97,252,232,169),(98,170,233,253),(99,254,234,171),(100,172,235,255)], [(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)], [(1,131,192,286),(2,140,193,295),(3,129,194,284),(4,138,195,293),(5,127,196,282),(6,136,197,291),(7,125,198,300),(8,134,199,289),(9,123,200,298),(10,132,181,287),(11,121,182,296),(12,130,183,285),(13,139,184,294),(14,128,185,283),(15,137,186,292),(16,126,187,281),(17,135,188,290),(18,124,189,299),(19,133,190,288),(20,122,191,297),(21,87,320,232),(22,96,301,221),(23,85,302,230),(24,94,303,239),(25,83,304,228),(26,92,305,237),(27,81,306,226),(28,90,307,235),(29,99,308,224),(30,88,309,233),(31,97,310,222),(32,86,311,231),(33,95,312,240),(34,84,313,229),(35,93,314,238),(36,82,315,227),(37,91,316,236),(38,100,317,225),(39,89,318,234),(40,98,319,223),(41,241,276,168),(42,250,277,177),(43,259,278,166),(44,248,279,175),(45,257,280,164),(46,246,261,173),(47,255,262,162),(48,244,263,171),(49,253,264,180),(50,242,265,169),(51,251,266,178),(52,260,267,167),(53,249,268,176),(54,258,269,165),(55,247,270,174),(56,256,271,163),(57,245,272,172),(58,254,273,161),(59,243,274,170),(60,252,275,179),(61,205,109,147),(62,214,110,156),(63,203,111,145),(64,212,112,154),(65,201,113,143),(66,210,114,152),(67,219,115,141),(68,208,116,150),(69,217,117,159),(70,206,118,148),(71,215,119,157),(72,204,120,146),(73,213,101,155),(74,202,102,144),(75,211,103,153),(76,220,104,142),(77,209,105,151),(78,218,106,160),(79,207,107,149),(80,216,108,158)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4G | 4H | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | 2- 1+4 | Q8.10D10 | D4.10D10 |
| kernel | C42.147D10 | C20.6Q8 | Dic5.Q8 | C4.Dic10 | C5×C42.C2 | C42.C2 | C42 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 2 | 2 | 12 | 3 | 4 | 8 |
Matrix representation of C42.147D10 ►in GL8(𝔽41)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 39 |
| 0 | 0 | 0 | 0 | 39 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 | 0 | 0 |
| 24 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 34 | 34 | 8 | 33 | 0 | 0 | 0 | 0 |
| 7 | 1 | 8 | 23 | 0 | 0 | 0 | 0 |
| 8 | 33 | 7 | 7 | 0 | 0 | 0 | 0 |
| 8 | 23 | 34 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 35 | 10 | 10 |
| 0 | 0 | 0 | 0 | 6 | 11 | 31 | 29 |
| 0 | 0 | 0 | 0 | 10 | 10 | 35 | 6 |
| 0 | 0 | 0 | 0 | 31 | 29 | 35 | 30 |
| 23 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 23 | 20 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 18 |
| 0 | 0 | 0 | 0 | 23 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 18 | 0 | 0 |
G:=sub<GL(8,GF(41))| [0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,2,28,0,0,0,0,0,0,13,39,0,0],[24,40,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,24,40,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[34,7,8,8,0,0,0,0,34,1,33,23,0,0,0,0,8,8,7,34,0,0,0,0,33,23,7,40,0,0,0,0,0,0,0,0,6,6,10,31,0,0,0,0,35,11,10,29,0,0,0,0,10,31,35,35,0,0,0,0,10,29,6,30],[23,28,0,0,0,0,0,0,25,18,0,0,0,0,0,0,0,0,18,13,0,0,0,0,0,0,16,23,0,0,0,0,0,0,0,0,0,0,23,35,0,0,0,0,0,0,20,18,0,0,0,0,23,35,0,0,0,0,0,0,20,18,0,0] >;
C42.147D10 in GAP, Magma, Sage, TeX
C_4^2._{147}D_{10} % in TeX
G:=Group("C4^2.147D10"); // GroupNames label
G:=SmallGroup(320,1358);
// by ID
G=gap.SmallGroup(320,1358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,758,555,100,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations