metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.5M4(2), C5⋊C8⋊2Q8, C2.5(Q8×F5), C5⋊2(C8⋊4Q8), C4⋊C4.10F5, C10.3(C4×Q8), C20⋊C8.4C2, C4.3(C4.F5), C10.13(C8○D4), Dic5.30(C2×Q8), C10.D4.4C4, C2.14(D4.F5), (C2×Dic10).12C4, C10.16(C2×M4(2)), Dic5.69(C4○D4), Dic5⋊3Q8.17C2, C22.81(C22×F5), C10.C42.4C2, (C2×Dic5).336C23, (C4×Dic5).193C22, (C4×C5⋊C8).3C2, (C5×C4⋊C4).13C4, (C2×C4).29(C2×F5), C2.10(C2×C4.F5), (C2×C20).87(C2×C4), (C2×C5⋊C8).32C22, (C2×C10).47(C22×C4), (C2×Dic5).62(C2×C4), SmallGroup(320,1047)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 282 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, C4⋊C4 [×2], 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], Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C8⋊4Q8, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4, C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×Dic10, C4×C5⋊C8, C20⋊C8, C20⋊C8 [×2], C10.C42 [×2], Dic5⋊3Q8, C20.M4(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, C4.F5 [×2], C22×F5, C2×C4.F5, D4.F5, Q8×F5, C20.M4(2)
Generators and relations
G = < a,b,c | a20=b8=1, c2=a10, bab-1=a7, cac-1=a-1, 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 57 173 85 298 72 185 307)(2 60 162 92 299 75 194 314)(3 43 171 99 300 78 183 301)(4 46 180 86 281 61 192 308)(5 49 169 93 282 64 181 315)(6 52 178 100 283 67 190 302)(7 55 167 87 284 70 199 309)(8 58 176 94 285 73 188 316)(9 41 165 81 286 76 197 303)(10 44 174 88 287 79 186 310)(11 47 163 95 288 62 195 317)(12 50 172 82 289 65 184 304)(13 53 161 89 290 68 193 311)(14 56 170 96 291 71 182 318)(15 59 179 83 292 74 191 305)(16 42 168 90 293 77 200 312)(17 45 177 97 294 80 189 319)(18 48 166 84 295 63 198 306)(19 51 175 91 296 66 187 313)(20 54 164 98 297 69 196 320)(21 272 253 144 135 219 225 103)(22 275 242 151 136 202 234 110)(23 278 251 158 137 205 223 117)(24 261 260 145 138 208 232 104)(25 264 249 152 139 211 221 111)(26 267 258 159 140 214 230 118)(27 270 247 146 121 217 239 105)(28 273 256 153 122 220 228 112)(29 276 245 160 123 203 237 119)(30 279 254 147 124 206 226 106)(31 262 243 154 125 209 235 113)(32 265 252 141 126 212 224 120)(33 268 241 148 127 215 233 107)(34 271 250 155 128 218 222 114)(35 274 259 142 129 201 231 101)(36 277 248 149 130 204 240 108)(37 280 257 156 131 207 229 115)(38 263 246 143 132 210 238 102)(39 266 255 150 133 213 227 109)(40 269 244 157 134 216 236 116)
(1 119 11 109)(2 118 12 108)(3 117 13 107)(4 116 14 106)(5 115 15 105)(6 114 16 104)(7 113 17 103)(8 112 18 102)(9 111 19 101)(10 110 20 120)(21 70 31 80)(22 69 32 79)(23 68 33 78)(24 67 34 77)(25 66 35 76)(26 65 36 75)(27 64 37 74)(28 63 38 73)(29 62 39 72)(30 61 40 71)(41 139 51 129)(42 138 52 128)(43 137 53 127)(44 136 54 126)(45 135 55 125)(46 134 56 124)(47 133 57 123)(48 132 58 122)(49 131 59 121)(50 130 60 140)(81 221 91 231)(82 240 92 230)(83 239 93 229)(84 238 94 228)(85 237 95 227)(86 236 96 226)(87 235 97 225)(88 234 98 224)(89 233 99 223)(90 232 100 222)(141 287 151 297)(142 286 152 296)(143 285 153 295)(144 284 154 294)(145 283 155 293)(146 282 156 292)(147 281 157 291)(148 300 158 290)(149 299 159 289)(150 298 160 288)(161 268 171 278)(162 267 172 277)(163 266 173 276)(164 265 174 275)(165 264 175 274)(166 263 176 273)(167 262 177 272)(168 261 178 271)(169 280 179 270)(170 279 180 269)(181 207 191 217)(182 206 192 216)(183 205 193 215)(184 204 194 214)(185 203 195 213)(186 202 196 212)(187 201 197 211)(188 220 198 210)(189 219 199 209)(190 218 200 208)(241 301 251 311)(242 320 252 310)(243 319 253 309)(244 318 254 308)(245 317 255 307)(246 316 256 306)(247 315 257 305)(248 314 258 304)(249 313 259 303)(250 312 260 302)
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), (1,57,173,85,298,72,185,307)(2,60,162,92,299,75,194,314)(3,43,171,99,300,78,183,301)(4,46,180,86,281,61,192,308)(5,49,169,93,282,64,181,315)(6,52,178,100,283,67,190,302)(7,55,167,87,284,70,199,309)(8,58,176,94,285,73,188,316)(9,41,165,81,286,76,197,303)(10,44,174,88,287,79,186,310)(11,47,163,95,288,62,195,317)(12,50,172,82,289,65,184,304)(13,53,161,89,290,68,193,311)(14,56,170,96,291,71,182,318)(15,59,179,83,292,74,191,305)(16,42,168,90,293,77,200,312)(17,45,177,97,294,80,189,319)(18,48,166,84,295,63,198,306)(19,51,175,91,296,66,187,313)(20,54,164,98,297,69,196,320)(21,272,253,144,135,219,225,103)(22,275,242,151,136,202,234,110)(23,278,251,158,137,205,223,117)(24,261,260,145,138,208,232,104)(25,264,249,152,139,211,221,111)(26,267,258,159,140,214,230,118)(27,270,247,146,121,217,239,105)(28,273,256,153,122,220,228,112)(29,276,245,160,123,203,237,119)(30,279,254,147,124,206,226,106)(31,262,243,154,125,209,235,113)(32,265,252,141,126,212,224,120)(33,268,241,148,127,215,233,107)(34,271,250,155,128,218,222,114)(35,274,259,142,129,201,231,101)(36,277,248,149,130,204,240,108)(37,280,257,156,131,207,229,115)(38,263,246,143,132,210,238,102)(39,266,255,150,133,213,227,109)(40,269,244,157,134,216,236,116), (1,119,11,109)(2,118,12,108)(3,117,13,107)(4,116,14,106)(5,115,15,105)(6,114,16,104)(7,113,17,103)(8,112,18,102)(9,111,19,101)(10,110,20,120)(21,70,31,80)(22,69,32,79)(23,68,33,78)(24,67,34,77)(25,66,35,76)(26,65,36,75)(27,64,37,74)(28,63,38,73)(29,62,39,72)(30,61,40,71)(41,139,51,129)(42,138,52,128)(43,137,53,127)(44,136,54,126)(45,135,55,125)(46,134,56,124)(47,133,57,123)(48,132,58,122)(49,131,59,121)(50,130,60,140)(81,221,91,231)(82,240,92,230)(83,239,93,229)(84,238,94,228)(85,237,95,227)(86,236,96,226)(87,235,97,225)(88,234,98,224)(89,233,99,223)(90,232,100,222)(141,287,151,297)(142,286,152,296)(143,285,153,295)(144,284,154,294)(145,283,155,293)(146,282,156,292)(147,281,157,291)(148,300,158,290)(149,299,159,289)(150,298,160,288)(161,268,171,278)(162,267,172,277)(163,266,173,276)(164,265,174,275)(165,264,175,274)(166,263,176,273)(167,262,177,272)(168,261,178,271)(169,280,179,270)(170,279,180,269)(181,207,191,217)(182,206,192,216)(183,205,193,215)(184,204,194,214)(185,203,195,213)(186,202,196,212)(187,201,197,211)(188,220,198,210)(189,219,199,209)(190,218,200,208)(241,301,251,311)(242,320,252,310)(243,319,253,309)(244,318,254,308)(245,317,255,307)(246,316,256,306)(247,315,257,305)(248,314,258,304)(249,313,259,303)(250,312,260,302)>;
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), (1,57,173,85,298,72,185,307)(2,60,162,92,299,75,194,314)(3,43,171,99,300,78,183,301)(4,46,180,86,281,61,192,308)(5,49,169,93,282,64,181,315)(6,52,178,100,283,67,190,302)(7,55,167,87,284,70,199,309)(8,58,176,94,285,73,188,316)(9,41,165,81,286,76,197,303)(10,44,174,88,287,79,186,310)(11,47,163,95,288,62,195,317)(12,50,172,82,289,65,184,304)(13,53,161,89,290,68,193,311)(14,56,170,96,291,71,182,318)(15,59,179,83,292,74,191,305)(16,42,168,90,293,77,200,312)(17,45,177,97,294,80,189,319)(18,48,166,84,295,63,198,306)(19,51,175,91,296,66,187,313)(20,54,164,98,297,69,196,320)(21,272,253,144,135,219,225,103)(22,275,242,151,136,202,234,110)(23,278,251,158,137,205,223,117)(24,261,260,145,138,208,232,104)(25,264,249,152,139,211,221,111)(26,267,258,159,140,214,230,118)(27,270,247,146,121,217,239,105)(28,273,256,153,122,220,228,112)(29,276,245,160,123,203,237,119)(30,279,254,147,124,206,226,106)(31,262,243,154,125,209,235,113)(32,265,252,141,126,212,224,120)(33,268,241,148,127,215,233,107)(34,271,250,155,128,218,222,114)(35,274,259,142,129,201,231,101)(36,277,248,149,130,204,240,108)(37,280,257,156,131,207,229,115)(38,263,246,143,132,210,238,102)(39,266,255,150,133,213,227,109)(40,269,244,157,134,216,236,116), (1,119,11,109)(2,118,12,108)(3,117,13,107)(4,116,14,106)(5,115,15,105)(6,114,16,104)(7,113,17,103)(8,112,18,102)(9,111,19,101)(10,110,20,120)(21,70,31,80)(22,69,32,79)(23,68,33,78)(24,67,34,77)(25,66,35,76)(26,65,36,75)(27,64,37,74)(28,63,38,73)(29,62,39,72)(30,61,40,71)(41,139,51,129)(42,138,52,128)(43,137,53,127)(44,136,54,126)(45,135,55,125)(46,134,56,124)(47,133,57,123)(48,132,58,122)(49,131,59,121)(50,130,60,140)(81,221,91,231)(82,240,92,230)(83,239,93,229)(84,238,94,228)(85,237,95,227)(86,236,96,226)(87,235,97,225)(88,234,98,224)(89,233,99,223)(90,232,100,222)(141,287,151,297)(142,286,152,296)(143,285,153,295)(144,284,154,294)(145,283,155,293)(146,282,156,292)(147,281,157,291)(148,300,158,290)(149,299,159,289)(150,298,160,288)(161,268,171,278)(162,267,172,277)(163,266,173,276)(164,265,174,275)(165,264,175,274)(166,263,176,273)(167,262,177,272)(168,261,178,271)(169,280,179,270)(170,279,180,269)(181,207,191,217)(182,206,192,216)(183,205,193,215)(184,204,194,214)(185,203,195,213)(186,202,196,212)(187,201,197,211)(188,220,198,210)(189,219,199,209)(190,218,200,208)(241,301,251,311)(242,320,252,310)(243,319,253,309)(244,318,254,308)(245,317,255,307)(246,316,256,306)(247,315,257,305)(248,314,258,304)(249,313,259,303)(250,312,260,302) );
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)], [(1,57,173,85,298,72,185,307),(2,60,162,92,299,75,194,314),(3,43,171,99,300,78,183,301),(4,46,180,86,281,61,192,308),(5,49,169,93,282,64,181,315),(6,52,178,100,283,67,190,302),(7,55,167,87,284,70,199,309),(8,58,176,94,285,73,188,316),(9,41,165,81,286,76,197,303),(10,44,174,88,287,79,186,310),(11,47,163,95,288,62,195,317),(12,50,172,82,289,65,184,304),(13,53,161,89,290,68,193,311),(14,56,170,96,291,71,182,318),(15,59,179,83,292,74,191,305),(16,42,168,90,293,77,200,312),(17,45,177,97,294,80,189,319),(18,48,166,84,295,63,198,306),(19,51,175,91,296,66,187,313),(20,54,164,98,297,69,196,320),(21,272,253,144,135,219,225,103),(22,275,242,151,136,202,234,110),(23,278,251,158,137,205,223,117),(24,261,260,145,138,208,232,104),(25,264,249,152,139,211,221,111),(26,267,258,159,140,214,230,118),(27,270,247,146,121,217,239,105),(28,273,256,153,122,220,228,112),(29,276,245,160,123,203,237,119),(30,279,254,147,124,206,226,106),(31,262,243,154,125,209,235,113),(32,265,252,141,126,212,224,120),(33,268,241,148,127,215,233,107),(34,271,250,155,128,218,222,114),(35,274,259,142,129,201,231,101),(36,277,248,149,130,204,240,108),(37,280,257,156,131,207,229,115),(38,263,246,143,132,210,238,102),(39,266,255,150,133,213,227,109),(40,269,244,157,134,216,236,116)], [(1,119,11,109),(2,118,12,108),(3,117,13,107),(4,116,14,106),(5,115,15,105),(6,114,16,104),(7,113,17,103),(8,112,18,102),(9,111,19,101),(10,110,20,120),(21,70,31,80),(22,69,32,79),(23,68,33,78),(24,67,34,77),(25,66,35,76),(26,65,36,75),(27,64,37,74),(28,63,38,73),(29,62,39,72),(30,61,40,71),(41,139,51,129),(42,138,52,128),(43,137,53,127),(44,136,54,126),(45,135,55,125),(46,134,56,124),(47,133,57,123),(48,132,58,122),(49,131,59,121),(50,130,60,140),(81,221,91,231),(82,240,92,230),(83,239,93,229),(84,238,94,228),(85,237,95,227),(86,236,96,226),(87,235,97,225),(88,234,98,224),(89,233,99,223),(90,232,100,222),(141,287,151,297),(142,286,152,296),(143,285,153,295),(144,284,154,294),(145,283,155,293),(146,282,156,292),(147,281,157,291),(148,300,158,290),(149,299,159,289),(150,298,160,288),(161,268,171,278),(162,267,172,277),(163,266,173,276),(164,265,174,275),(165,264,175,274),(166,263,176,273),(167,262,177,272),(168,261,178,271),(169,280,179,270),(170,279,180,269),(181,207,191,217),(182,206,192,216),(183,205,193,215),(184,204,194,214),(185,203,195,213),(186,202,196,212),(187,201,197,211),(188,220,198,210),(189,219,199,209),(190,218,200,208),(241,301,251,311),(242,320,252,310),(243,319,253,309),(244,318,254,308),(245,317,255,307),(246,316,256,306),(247,315,257,305),(248,314,258,304),(249,313,259,303),(250,312,260,302)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 26 | 26 | 0 | 0 | 0 | 0 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 10 | 16 | 0 |
| 0 | 0 | 38 | 10 | 16 | 0 |
| 0 | 0 | 37 | 11 | 16 | 0 |
| 0 | 0 | 14 | 8 | 21 | 18 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 34 | 5 | 2 |
| 0 | 0 | 4 | 18 | 24 | 22 |
| 0 | 0 | 32 | 1 | 38 | 17 |
| 0 | 0 | 23 | 5 | 14 | 36 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 14 | 3 | 11 |
| 0 | 0 | 5 | 4 | 15 | 4 |
| 0 | 0 | 23 | 14 | 20 | 1 |
| 0 | 0 | 40 | 39 | 19 | 40 |
G:=sub<GL(6,GF(41))| [26,26,0,0,0,0,26,15,0,0,0,0,0,0,37,38,37,14,0,0,10,10,11,8,0,0,16,16,16,21,0,0,0,0,0,18],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,31,4,32,23,0,0,34,18,1,5,0,0,5,24,38,14,0,0,2,22,17,36],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,18,5,23,40,0,0,14,4,14,39,0,0,3,15,20,19,0,0,11,4,1,40] >;
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 | C4 | C4 | C4 | Q8 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | C4.F5 | D4.F5 | Q8×F5 |
| kernel | C20.M4(2) | C4×C5⋊C8 | C20⋊C8 | C10.C42 | Dic5⋊3Q8 | C10.D4 | C5×C4⋊C4 | C2×Dic10 | C5⋊C8 | Dic5 | C20 | C10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 3 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_{20}.M_{4(2)} % in TeX
G:=Group("C20.M4(2)"); // GroupNames label
G:=SmallGroup(320,1047);
// by ID
G=gap.SmallGroup(320,1047);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,268,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations
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