metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.6M4(2), C5⋊C8⋊3Q8, C2.8(Q8×F5), C5⋊3(C8⋊4Q8), (C2×Q8).6F5, C10.8(C4×Q8), C4⋊Dic5.8C4, (Q8×C10).7C4, C20⋊C8.6C2, C2.9(Q8.F5), C10.24(C8○D4), Dic5.32(C2×Q8), (Q8×Dic5).16C2, C4.3(C22.F5), Dic5⋊C8.3C2, C10.32(C2×M4(2)), Dic5.71(C4○D4), C22.98(C22×F5), C10.C42.2C2, (C4×Dic5).197C22, (C2×Dic5).358C23, (C4×C5⋊C8).5C2, (C2×C4).42(C2×F5), (C2×C20).27(C2×C4), (C2×C5⋊C8).42C22, C2.11(C2×C22.F5), (C2×C10).87(C22×C4), (C2×Dic5).77(C2×C4), SmallGroup(320,1126)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 266 in 94 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C5, C8 [×5], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×2], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×C8 [×4], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×2], C2×C10, C4×C8, C8⋊C4 [×2], C4⋊C8 [×3], C4×Q8, C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C8⋊4Q8, C4×Dic5, C4×Dic5 [×2], C4⋊Dic5, C4⋊Dic5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], Q8×C10, C4×C5⋊C8, C20⋊C8, C10.C42 [×2], Dic5⋊C8 [×2], Q8×Dic5, C20.6M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, M4(2) [×2], C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C2×M4(2), C8○D4, C2×F5 [×3], C8⋊4Q8, C22.F5 [×2], C22×F5, Q8.F5, Q8×F5, C2×C22.F5, C20.6M4(2)
Generators and relations
G = < a,b,c | a20=b8=1, c2=a10, bab-1=a3, cac-1=a11, cbc-1=b5 >
►Regular action on 320 points
Generators in S320
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)(261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)
(1 160 241 224 103 164 310 47)(2 147 250 227 104 171 319 50)(3 154 259 230 105 178 308 53)(4 141 248 233 106 165 317 56)(5 148 257 236 107 172 306 59)(6 155 246 239 108 179 315 42)(7 142 255 222 109 166 304 45)(8 149 244 225 110 173 313 48)(9 156 253 228 111 180 302 51)(10 143 242 231 112 167 311 54)(11 150 251 234 113 174 320 57)(12 157 260 237 114 161 309 60)(13 144 249 240 115 168 318 43)(14 151 258 223 116 175 307 46)(15 158 247 226 117 162 316 49)(16 145 256 229 118 169 305 52)(17 152 245 232 119 176 314 55)(18 159 254 235 120 163 303 58)(19 146 243 238 101 170 312 41)(20 153 252 221 102 177 301 44)(21 279 89 212 74 186 293 129)(22 266 98 215 75 193 282 132)(23 273 87 218 76 200 291 135)(24 280 96 201 77 187 300 138)(25 267 85 204 78 194 289 121)(26 274 94 207 79 181 298 124)(27 261 83 210 80 188 287 127)(28 268 92 213 61 195 296 130)(29 275 81 216 62 182 285 133)(30 262 90 219 63 189 294 136)(31 269 99 202 64 196 283 139)(32 276 88 205 65 183 292 122)(33 263 97 208 66 190 281 125)(34 270 86 211 67 197 290 128)(35 277 95 214 68 184 299 131)(36 264 84 217 69 191 288 134)(37 271 93 220 70 198 297 137)(38 278 82 203 71 185 286 140)(39 265 91 206 72 192 295 123)(40 272 100 209 73 199 284 126)
(1 275 11 265)(2 266 12 276)(3 277 13 267)(4 268 14 278)(5 279 15 269)(6 270 16 280)(7 261 17 271)(8 272 18 262)(9 263 19 273)(10 274 20 264)(21 226 31 236)(22 237 32 227)(23 228 33 238)(24 239 34 229)(25 230 35 240)(26 221 36 231)(27 232 37 222)(28 223 38 233)(29 234 39 224)(30 225 40 235)(41 76 51 66)(42 67 52 77)(43 78 53 68)(44 69 54 79)(45 80 55 70)(46 71 56 61)(47 62 57 72)(48 73 58 63)(49 64 59 74)(50 75 60 65)(81 174 91 164)(82 165 92 175)(83 176 93 166)(84 167 94 177)(85 178 95 168)(86 169 96 179)(87 180 97 170)(88 171 98 161)(89 162 99 172)(90 173 100 163)(101 200 111 190)(102 191 112 181)(103 182 113 192)(104 193 114 183)(105 184 115 194)(106 195 116 185)(107 186 117 196)(108 197 118 187)(109 188 119 198)(110 199 120 189)(121 308 131 318)(122 319 132 309)(123 310 133 320)(124 301 134 311)(125 312 135 302)(126 303 136 313)(127 314 137 304)(128 305 138 315)(129 316 139 306)(130 307 140 317)(141 296 151 286)(142 287 152 297)(143 298 153 288)(144 289 154 299)(145 300 155 290)(146 291 156 281)(147 282 157 292)(148 293 158 283)(149 284 159 294)(150 295 160 285)(201 246 211 256)(202 257 212 247)(203 248 213 258)(204 259 214 249)(205 250 215 260)(206 241 216 251)(207 252 217 242)(208 243 218 253)(209 254 219 244)(210 245 220 255)
G:=sub<Sym(320)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,160,241,224,103,164,310,47)(2,147,250,227,104,171,319,50)(3,154,259,230,105,178,308,53)(4,141,248,233,106,165,317,56)(5,148,257,236,107,172,306,59)(6,155,246,239,108,179,315,42)(7,142,255,222,109,166,304,45)(8,149,244,225,110,173,313,48)(9,156,253,228,111,180,302,51)(10,143,242,231,112,167,311,54)(11,150,251,234,113,174,320,57)(12,157,260,237,114,161,309,60)(13,144,249,240,115,168,318,43)(14,151,258,223,116,175,307,46)(15,158,247,226,117,162,316,49)(16,145,256,229,118,169,305,52)(17,152,245,232,119,176,314,55)(18,159,254,235,120,163,303,58)(19,146,243,238,101,170,312,41)(20,153,252,221,102,177,301,44)(21,279,89,212,74,186,293,129)(22,266,98,215,75,193,282,132)(23,273,87,218,76,200,291,135)(24,280,96,201,77,187,300,138)(25,267,85,204,78,194,289,121)(26,274,94,207,79,181,298,124)(27,261,83,210,80,188,287,127)(28,268,92,213,61,195,296,130)(29,275,81,216,62,182,285,133)(30,262,90,219,63,189,294,136)(31,269,99,202,64,196,283,139)(32,276,88,205,65,183,292,122)(33,263,97,208,66,190,281,125)(34,270,86,211,67,197,290,128)(35,277,95,214,68,184,299,131)(36,264,84,217,69,191,288,134)(37,271,93,220,70,198,297,137)(38,278,82,203,71,185,286,140)(39,265,91,206,72,192,295,123)(40,272,100,209,73,199,284,126), (1,275,11,265)(2,266,12,276)(3,277,13,267)(4,268,14,278)(5,279,15,269)(6,270,16,280)(7,261,17,271)(8,272,18,262)(9,263,19,273)(10,274,20,264)(21,226,31,236)(22,237,32,227)(23,228,33,238)(24,239,34,229)(25,230,35,240)(26,221,36,231)(27,232,37,222)(28,223,38,233)(29,234,39,224)(30,225,40,235)(41,76,51,66)(42,67,52,77)(43,78,53,68)(44,69,54,79)(45,80,55,70)(46,71,56,61)(47,62,57,72)(48,73,58,63)(49,64,59,74)(50,75,60,65)(81,174,91,164)(82,165,92,175)(83,176,93,166)(84,167,94,177)(85,178,95,168)(86,169,96,179)(87,180,97,170)(88,171,98,161)(89,162,99,172)(90,173,100,163)(101,200,111,190)(102,191,112,181)(103,182,113,192)(104,193,114,183)(105,184,115,194)(106,195,116,185)(107,186,117,196)(108,197,118,187)(109,188,119,198)(110,199,120,189)(121,308,131,318)(122,319,132,309)(123,310,133,320)(124,301,134,311)(125,312,135,302)(126,303,136,313)(127,314,137,304)(128,305,138,315)(129,316,139,306)(130,307,140,317)(141,296,151,286)(142,287,152,297)(143,298,153,288)(144,289,154,299)(145,300,155,290)(146,291,156,281)(147,282,157,292)(148,293,158,283)(149,284,159,294)(150,295,160,285)(201,246,211,256)(202,257,212,247)(203,248,213,258)(204,259,214,249)(205,250,215,260)(206,241,216,251)(207,252,217,242)(208,243,218,253)(209,254,219,244)(210,245,220,255)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320), (1,160,241,224,103,164,310,47)(2,147,250,227,104,171,319,50)(3,154,259,230,105,178,308,53)(4,141,248,233,106,165,317,56)(5,148,257,236,107,172,306,59)(6,155,246,239,108,179,315,42)(7,142,255,222,109,166,304,45)(8,149,244,225,110,173,313,48)(9,156,253,228,111,180,302,51)(10,143,242,231,112,167,311,54)(11,150,251,234,113,174,320,57)(12,157,260,237,114,161,309,60)(13,144,249,240,115,168,318,43)(14,151,258,223,116,175,307,46)(15,158,247,226,117,162,316,49)(16,145,256,229,118,169,305,52)(17,152,245,232,119,176,314,55)(18,159,254,235,120,163,303,58)(19,146,243,238,101,170,312,41)(20,153,252,221,102,177,301,44)(21,279,89,212,74,186,293,129)(22,266,98,215,75,193,282,132)(23,273,87,218,76,200,291,135)(24,280,96,201,77,187,300,138)(25,267,85,204,78,194,289,121)(26,274,94,207,79,181,298,124)(27,261,83,210,80,188,287,127)(28,268,92,213,61,195,296,130)(29,275,81,216,62,182,285,133)(30,262,90,219,63,189,294,136)(31,269,99,202,64,196,283,139)(32,276,88,205,65,183,292,122)(33,263,97,208,66,190,281,125)(34,270,86,211,67,197,290,128)(35,277,95,214,68,184,299,131)(36,264,84,217,69,191,288,134)(37,271,93,220,70,198,297,137)(38,278,82,203,71,185,286,140)(39,265,91,206,72,192,295,123)(40,272,100,209,73,199,284,126), (1,275,11,265)(2,266,12,276)(3,277,13,267)(4,268,14,278)(5,279,15,269)(6,270,16,280)(7,261,17,271)(8,272,18,262)(9,263,19,273)(10,274,20,264)(21,226,31,236)(22,237,32,227)(23,228,33,238)(24,239,34,229)(25,230,35,240)(26,221,36,231)(27,232,37,222)(28,223,38,233)(29,234,39,224)(30,225,40,235)(41,76,51,66)(42,67,52,77)(43,78,53,68)(44,69,54,79)(45,80,55,70)(46,71,56,61)(47,62,57,72)(48,73,58,63)(49,64,59,74)(50,75,60,65)(81,174,91,164)(82,165,92,175)(83,176,93,166)(84,167,94,177)(85,178,95,168)(86,169,96,179)(87,180,97,170)(88,171,98,161)(89,162,99,172)(90,173,100,163)(101,200,111,190)(102,191,112,181)(103,182,113,192)(104,193,114,183)(105,184,115,194)(106,195,116,185)(107,186,117,196)(108,197,118,187)(109,188,119,198)(110,199,120,189)(121,308,131,318)(122,319,132,309)(123,310,133,320)(124,301,134,311)(125,312,135,302)(126,303,136,313)(127,314,137,304)(128,305,138,315)(129,316,139,306)(130,307,140,317)(141,296,151,286)(142,287,152,297)(143,298,153,288)(144,289,154,299)(145,300,155,290)(146,291,156,281)(147,282,157,292)(148,293,158,283)(149,284,159,294)(150,295,160,285)(201,246,211,256)(202,257,212,247)(203,248,213,258)(204,259,214,249)(205,250,215,260)(206,241,216,251)(207,252,217,242)(208,243,218,253)(209,254,219,244)(210,245,220,255) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260),(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)], [(1,160,241,224,103,164,310,47),(2,147,250,227,104,171,319,50),(3,154,259,230,105,178,308,53),(4,141,248,233,106,165,317,56),(5,148,257,236,107,172,306,59),(6,155,246,239,108,179,315,42),(7,142,255,222,109,166,304,45),(8,149,244,225,110,173,313,48),(9,156,253,228,111,180,302,51),(10,143,242,231,112,167,311,54),(11,150,251,234,113,174,320,57),(12,157,260,237,114,161,309,60),(13,144,249,240,115,168,318,43),(14,151,258,223,116,175,307,46),(15,158,247,226,117,162,316,49),(16,145,256,229,118,169,305,52),(17,152,245,232,119,176,314,55),(18,159,254,235,120,163,303,58),(19,146,243,238,101,170,312,41),(20,153,252,221,102,177,301,44),(21,279,89,212,74,186,293,129),(22,266,98,215,75,193,282,132),(23,273,87,218,76,200,291,135),(24,280,96,201,77,187,300,138),(25,267,85,204,78,194,289,121),(26,274,94,207,79,181,298,124),(27,261,83,210,80,188,287,127),(28,268,92,213,61,195,296,130),(29,275,81,216,62,182,285,133),(30,262,90,219,63,189,294,136),(31,269,99,202,64,196,283,139),(32,276,88,205,65,183,292,122),(33,263,97,208,66,190,281,125),(34,270,86,211,67,197,290,128),(35,277,95,214,68,184,299,131),(36,264,84,217,69,191,288,134),(37,271,93,220,70,198,297,137),(38,278,82,203,71,185,286,140),(39,265,91,206,72,192,295,123),(40,272,100,209,73,199,284,126)], [(1,275,11,265),(2,266,12,276),(3,277,13,267),(4,268,14,278),(5,279,15,269),(6,270,16,280),(7,261,17,271),(8,272,18,262),(9,263,19,273),(10,274,20,264),(21,226,31,236),(22,237,32,227),(23,228,33,238),(24,239,34,229),(25,230,35,240),(26,221,36,231),(27,232,37,222),(28,223,38,233),(29,234,39,224),(30,225,40,235),(41,76,51,66),(42,67,52,77),(43,78,53,68),(44,69,54,79),(45,80,55,70),(46,71,56,61),(47,62,57,72),(48,73,58,63),(49,64,59,74),(50,75,60,65),(81,174,91,164),(82,165,92,175),(83,176,93,166),(84,167,94,177),(85,178,95,168),(86,169,96,179),(87,180,97,170),(88,171,98,161),(89,162,99,172),(90,173,100,163),(101,200,111,190),(102,191,112,181),(103,182,113,192),(104,193,114,183),(105,184,115,194),(106,195,116,185),(107,186,117,196),(108,197,118,187),(109,188,119,198),(110,199,120,189),(121,308,131,318),(122,319,132,309),(123,310,133,320),(124,301,134,311),(125,312,135,302),(126,303,136,313),(127,314,137,304),(128,305,138,315),(129,316,139,306),(130,307,140,317),(141,296,151,286),(142,287,152,297),(143,298,153,288),(144,289,154,299),(145,300,155,290),(146,291,156,281),(147,282,157,292),(148,293,158,283),(149,284,159,294),(150,295,160,285),(201,246,211,256),(202,257,212,247),(203,248,213,258),(204,259,214,249),(205,250,215,260),(206,241,216,251),(207,252,217,242),(208,243,218,253),(209,254,219,244),(210,245,220,255)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 22 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 8 | 27 | 40 |
| 0 | 0 | 0 | 0 | 25 | 7 | 36 | 38 |
| 0 | 0 | 0 | 0 | 34 | 5 | 3 | 24 |
| 0 | 0 | 0 | 0 | 1 | 32 | 2 | 33 |
| 40 | 39 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[22,19,0,0,0,0,0,0,16,19,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,39,9,0,0,0,0,0,0,0,0,39,25,34,1,0,0,0,0,8,7,5,32,0,0,0,0,27,36,3,2,0,0,0,0,40,38,24,33],[40,0,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,39,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | Q8 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | C22.F5 | Q8.F5 | Q8×F5 |
| kernel | C20.6M4(2) | C4×C5⋊C8 | C20⋊C8 | C10.C42 | Dic5⋊C8 | Q8×Dic5 | C4⋊Dic5 | Q8×C10 | C5⋊C8 | Dic5 | C20 | C10 | C2×Q8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 6 | 2 | 2 | 2 | 4 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_{20}._6M_{4(2)} % in TeX
G:=Group("C20.6M4(2)"); // GroupNames label
G:=SmallGroup(320,1126);
// by ID
G=gap.SmallGroup(320,1126);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^3,c*a*c^-1=a^11,c*b*c^-1=b^5>;
// generators/relations
×
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ℤ
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