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