metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.3M4(2), C5⋊C8⋊1Q8, C4⋊C4.8F5, C2.4(Q8×F5), C5⋊1(C8⋊4Q8), C10.2(C4×Q8), C20⋊C8.11C2, C10.11(C8○D4), Dic5.29(C2×Q8), (C2×Dic10).8C4, C2.12(D4.F5), Dic5⋊C8.5C2, C10.D4.10C4, C10.15(C2×M4(2)), Dic5.68(C4○D4), Dic5⋊3Q8.16C2, C22.79(C22×F5), C2.14(D5⋊M4(2)), C10.C42.3C2, (C2×Dic5).334C23, (C4×Dic5).250C22, (C4×C5⋊C8).9C2, (C5×C4⋊C4).11C4, (C2×C4).27(C2×F5), (C2×C20).85(C2×C4), (C2×C5⋊C8).31C22, (C2×C10).45(C22×C4), (C2×Dic5).60(C2×C4), SmallGroup(320,1045)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 282 in 94 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C4 [×9], C22, C5, C8 [×5], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×Q8, Dic5 [×2], Dic5 [×2], Dic5 [×2], C20 [×3], C2×C10, C4×C8, C8⋊C4 [×2], C4⋊C8 [×3], C4×Q8, C5⋊C8 [×2], C5⋊C8 [×3], Dic10 [×2], C2×Dic5 [×4], C2×C20 [×3], C8⋊4Q8, C4×Dic5 [×3], C10.D4 [×2], C5×C4⋊C4, C2×C5⋊C8 [×4], C2×Dic10, C4×C5⋊C8, C20⋊C8, C10.C42 [×2], Dic5⋊C8 [×2], Dic5⋊3Q8, Dic5.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, C22×F5, D5⋊M4(2), D4.F5, Q8×F5, Dic5.M4(2)
Generators and relations
G = < a,b,c,d | a10=c8=1, b2=d2=a5, bab-1=a-1, cac-1=a3, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5c5 >
►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)
(1 111 6 116)(2 120 7 115)(3 119 8 114)(4 118 9 113)(5 117 10 112)(11 221 16 226)(12 230 17 225)(13 229 18 224)(14 228 19 223)(15 227 20 222)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 105 36 110)(32 104 37 109)(33 103 38 108)(34 102 39 107)(35 101 40 106)(41 98 46 93)(42 97 47 92)(43 96 48 91)(44 95 49 100)(45 94 50 99)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(61 168 66 163)(62 167 67 162)(63 166 68 161)(64 165 69 170)(65 164 70 169)(71 145 76 150)(72 144 77 149)(73 143 78 148)(74 142 79 147)(75 141 80 146)(81 138 86 133)(82 137 87 132)(83 136 88 131)(84 135 89 140)(85 134 90 139)(171 274 176 279)(172 273 177 278)(173 272 178 277)(174 271 179 276)(175 280 180 275)(181 289 186 284)(182 288 187 283)(183 287 188 282)(184 286 189 281)(185 285 190 290)(191 266 196 261)(192 265 197 270)(193 264 198 269)(194 263 199 268)(195 262 200 267)(201 257 206 252)(202 256 207 251)(203 255 208 260)(204 254 209 259)(205 253 210 258)(211 314 216 319)(212 313 217 318)(213 312 218 317)(214 311 219 316)(215 320 220 315)(231 306 236 301)(232 305 237 310)(233 304 238 309)(234 303 239 308)(235 302 240 307)(241 297 246 292)(242 296 247 291)(243 295 248 300)(244 294 249 299)(245 293 250 298)
(1 238 36 224 25 241 50 212)(2 235 35 227 26 248 49 215)(3 232 34 230 27 245 48 218)(4 239 33 223 28 242 47 211)(5 236 32 226 29 249 46 214)(6 233 31 229 30 246 45 217)(7 240 40 222 21 243 44 220)(8 237 39 225 22 250 43 213)(9 234 38 228 23 247 42 216)(10 231 37 221 24 244 41 219)(11 126 299 98 311 112 301 109)(12 123 298 91 312 119 310 102)(13 130 297 94 313 116 309 105)(14 127 296 97 314 113 308 108)(15 124 295 100 315 120 307 101)(16 121 294 93 316 117 306 104)(17 128 293 96 317 114 305 107)(18 125 292 99 318 111 304 110)(19 122 291 92 319 118 303 103)(20 129 300 95 320 115 302 106)(51 192 79 190 62 205 83 178)(52 199 78 183 63 202 82 171)(53 196 77 186 64 209 81 174)(54 193 76 189 65 206 90 177)(55 200 75 182 66 203 89 180)(56 197 74 185 67 210 88 173)(57 194 73 188 68 207 87 176)(58 191 72 181 69 204 86 179)(59 198 71 184 70 201 85 172)(60 195 80 187 61 208 84 175)(131 277 159 265 142 290 162 253)(132 274 158 268 143 287 161 256)(133 271 157 261 144 284 170 259)(134 278 156 264 145 281 169 252)(135 275 155 267 146 288 168 255)(136 272 154 270 147 285 167 258)(137 279 153 263 148 282 166 251)(138 276 152 266 149 289 165 254)(139 273 151 269 150 286 164 257)(140 280 160 262 141 283 163 260)
(1 59 6 54)(2 60 7 55)(3 51 8 56)(4 52 9 57)(5 53 10 58)(11 271 16 276)(12 272 17 277)(13 273 18 278)(14 274 19 279)(15 275 20 280)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(41 86 46 81)(42 87 47 82)(43 88 48 83)(44 89 49 84)(45 90 50 85)(91 131 96 136)(92 132 97 137)(93 133 98 138)(94 134 99 139)(95 135 100 140)(101 141 106 146)(102 142 107 147)(103 143 108 148)(104 144 109 149)(105 145 110 150)(111 151 116 156)(112 152 117 157)(113 153 118 158)(114 154 119 159)(115 155 120 160)(121 170 126 165)(122 161 127 166)(123 162 128 167)(124 163 129 168)(125 164 130 169)(171 223 176 228)(172 224 177 229)(173 225 178 230)(174 226 179 221)(175 227 180 222)(181 219 186 214)(182 220 187 215)(183 211 188 216)(184 212 189 217)(185 213 190 218)(191 244 196 249)(192 245 197 250)(193 246 198 241)(194 247 199 242)(195 248 200 243)(201 238 206 233)(202 239 207 234)(203 240 208 235)(204 231 209 236)(205 232 210 237)(251 308 256 303)(252 309 257 304)(253 310 258 305)(254 301 259 306)(255 302 260 307)(261 294 266 299)(262 295 267 300)(263 296 268 291)(264 297 269 292)(265 298 270 293)(281 313 286 318)(282 314 287 319)(283 315 288 320)(284 316 289 311)(285 317 290 312)
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), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(11,221,16,226)(12,230,17,225)(13,229,18,224)(14,228,19,223)(15,227,20,222)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,105,36,110)(32,104,37,109)(33,103,38,108)(34,102,39,107)(35,101,40,106)(41,98,46,93)(42,97,47,92)(43,96,48,91)(44,95,49,100)(45,94,50,99)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(61,168,66,163)(62,167,67,162)(63,166,68,161)(64,165,69,170)(65,164,70,169)(71,145,76,150)(72,144,77,149)(73,143,78,148)(74,142,79,147)(75,141,80,146)(81,138,86,133)(82,137,87,132)(83,136,88,131)(84,135,89,140)(85,134,90,139)(171,274,176,279)(172,273,177,278)(173,272,178,277)(174,271,179,276)(175,280,180,275)(181,289,186,284)(182,288,187,283)(183,287,188,282)(184,286,189,281)(185,285,190,290)(191,266,196,261)(192,265,197,270)(193,264,198,269)(194,263,199,268)(195,262,200,267)(201,257,206,252)(202,256,207,251)(203,255,208,260)(204,254,209,259)(205,253,210,258)(211,314,216,319)(212,313,217,318)(213,312,218,317)(214,311,219,316)(215,320,220,315)(231,306,236,301)(232,305,237,310)(233,304,238,309)(234,303,239,308)(235,302,240,307)(241,297,246,292)(242,296,247,291)(243,295,248,300)(244,294,249,299)(245,293,250,298), (1,238,36,224,25,241,50,212)(2,235,35,227,26,248,49,215)(3,232,34,230,27,245,48,218)(4,239,33,223,28,242,47,211)(5,236,32,226,29,249,46,214)(6,233,31,229,30,246,45,217)(7,240,40,222,21,243,44,220)(8,237,39,225,22,250,43,213)(9,234,38,228,23,247,42,216)(10,231,37,221,24,244,41,219)(11,126,299,98,311,112,301,109)(12,123,298,91,312,119,310,102)(13,130,297,94,313,116,309,105)(14,127,296,97,314,113,308,108)(15,124,295,100,315,120,307,101)(16,121,294,93,316,117,306,104)(17,128,293,96,317,114,305,107)(18,125,292,99,318,111,304,110)(19,122,291,92,319,118,303,103)(20,129,300,95,320,115,302,106)(51,192,79,190,62,205,83,178)(52,199,78,183,63,202,82,171)(53,196,77,186,64,209,81,174)(54,193,76,189,65,206,90,177)(55,200,75,182,66,203,89,180)(56,197,74,185,67,210,88,173)(57,194,73,188,68,207,87,176)(58,191,72,181,69,204,86,179)(59,198,71,184,70,201,85,172)(60,195,80,187,61,208,84,175)(131,277,159,265,142,290,162,253)(132,274,158,268,143,287,161,256)(133,271,157,261,144,284,170,259)(134,278,156,264,145,281,169,252)(135,275,155,267,146,288,168,255)(136,272,154,270,147,285,167,258)(137,279,153,263,148,282,166,251)(138,276,152,266,149,289,165,254)(139,273,151,269,150,286,164,257)(140,280,160,262,141,283,163,260), (1,59,6,54)(2,60,7,55)(3,51,8,56)(4,52,9,57)(5,53,10,58)(11,271,16,276)(12,272,17,277)(13,273,18,278)(14,274,19,279)(15,275,20,280)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(41,86,46,81)(42,87,47,82)(43,88,48,83)(44,89,49,84)(45,90,50,85)(91,131,96,136)(92,132,97,137)(93,133,98,138)(94,134,99,139)(95,135,100,140)(101,141,106,146)(102,142,107,147)(103,143,108,148)(104,144,109,149)(105,145,110,150)(111,151,116,156)(112,152,117,157)(113,153,118,158)(114,154,119,159)(115,155,120,160)(121,170,126,165)(122,161,127,166)(123,162,128,167)(124,163,129,168)(125,164,130,169)(171,223,176,228)(172,224,177,229)(173,225,178,230)(174,226,179,221)(175,227,180,222)(181,219,186,214)(182,220,187,215)(183,211,188,216)(184,212,189,217)(185,213,190,218)(191,244,196,249)(192,245,197,250)(193,246,198,241)(194,247,199,242)(195,248,200,243)(201,238,206,233)(202,239,207,234)(203,240,208,235)(204,231,209,236)(205,232,210,237)(251,308,256,303)(252,309,257,304)(253,310,258,305)(254,301,259,306)(255,302,260,307)(261,294,266,299)(262,295,267,300)(263,296,268,291)(264,297,269,292)(265,298,270,293)(281,313,286,318)(282,314,287,319)(283,315,288,320)(284,316,289,311)(285,317,290,312)>;
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), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(11,221,16,226)(12,230,17,225)(13,229,18,224)(14,228,19,223)(15,227,20,222)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,105,36,110)(32,104,37,109)(33,103,38,108)(34,102,39,107)(35,101,40,106)(41,98,46,93)(42,97,47,92)(43,96,48,91)(44,95,49,100)(45,94,50,99)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(61,168,66,163)(62,167,67,162)(63,166,68,161)(64,165,69,170)(65,164,70,169)(71,145,76,150)(72,144,77,149)(73,143,78,148)(74,142,79,147)(75,141,80,146)(81,138,86,133)(82,137,87,132)(83,136,88,131)(84,135,89,140)(85,134,90,139)(171,274,176,279)(172,273,177,278)(173,272,178,277)(174,271,179,276)(175,280,180,275)(181,289,186,284)(182,288,187,283)(183,287,188,282)(184,286,189,281)(185,285,190,290)(191,266,196,261)(192,265,197,270)(193,264,198,269)(194,263,199,268)(195,262,200,267)(201,257,206,252)(202,256,207,251)(203,255,208,260)(204,254,209,259)(205,253,210,258)(211,314,216,319)(212,313,217,318)(213,312,218,317)(214,311,219,316)(215,320,220,315)(231,306,236,301)(232,305,237,310)(233,304,238,309)(234,303,239,308)(235,302,240,307)(241,297,246,292)(242,296,247,291)(243,295,248,300)(244,294,249,299)(245,293,250,298), (1,238,36,224,25,241,50,212)(2,235,35,227,26,248,49,215)(3,232,34,230,27,245,48,218)(4,239,33,223,28,242,47,211)(5,236,32,226,29,249,46,214)(6,233,31,229,30,246,45,217)(7,240,40,222,21,243,44,220)(8,237,39,225,22,250,43,213)(9,234,38,228,23,247,42,216)(10,231,37,221,24,244,41,219)(11,126,299,98,311,112,301,109)(12,123,298,91,312,119,310,102)(13,130,297,94,313,116,309,105)(14,127,296,97,314,113,308,108)(15,124,295,100,315,120,307,101)(16,121,294,93,316,117,306,104)(17,128,293,96,317,114,305,107)(18,125,292,99,318,111,304,110)(19,122,291,92,319,118,303,103)(20,129,300,95,320,115,302,106)(51,192,79,190,62,205,83,178)(52,199,78,183,63,202,82,171)(53,196,77,186,64,209,81,174)(54,193,76,189,65,206,90,177)(55,200,75,182,66,203,89,180)(56,197,74,185,67,210,88,173)(57,194,73,188,68,207,87,176)(58,191,72,181,69,204,86,179)(59,198,71,184,70,201,85,172)(60,195,80,187,61,208,84,175)(131,277,159,265,142,290,162,253)(132,274,158,268,143,287,161,256)(133,271,157,261,144,284,170,259)(134,278,156,264,145,281,169,252)(135,275,155,267,146,288,168,255)(136,272,154,270,147,285,167,258)(137,279,153,263,148,282,166,251)(138,276,152,266,149,289,165,254)(139,273,151,269,150,286,164,257)(140,280,160,262,141,283,163,260), (1,59,6,54)(2,60,7,55)(3,51,8,56)(4,52,9,57)(5,53,10,58)(11,271,16,276)(12,272,17,277)(13,273,18,278)(14,274,19,279)(15,275,20,280)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(41,86,46,81)(42,87,47,82)(43,88,48,83)(44,89,49,84)(45,90,50,85)(91,131,96,136)(92,132,97,137)(93,133,98,138)(94,134,99,139)(95,135,100,140)(101,141,106,146)(102,142,107,147)(103,143,108,148)(104,144,109,149)(105,145,110,150)(111,151,116,156)(112,152,117,157)(113,153,118,158)(114,154,119,159)(115,155,120,160)(121,170,126,165)(122,161,127,166)(123,162,128,167)(124,163,129,168)(125,164,130,169)(171,223,176,228)(172,224,177,229)(173,225,178,230)(174,226,179,221)(175,227,180,222)(181,219,186,214)(182,220,187,215)(183,211,188,216)(184,212,189,217)(185,213,190,218)(191,244,196,249)(192,245,197,250)(193,246,198,241)(194,247,199,242)(195,248,200,243)(201,238,206,233)(202,239,207,234)(203,240,208,235)(204,231,209,236)(205,232,210,237)(251,308,256,303)(252,309,257,304)(253,310,258,305)(254,301,259,306)(255,302,260,307)(261,294,266,299)(262,295,267,300)(263,296,268,291)(264,297,269,292)(265,298,270,293)(281,313,286,318)(282,314,287,319)(283,315,288,320)(284,316,289,311)(285,317,290,312) );
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)], [(1,111,6,116),(2,120,7,115),(3,119,8,114),(4,118,9,113),(5,117,10,112),(11,221,16,226),(12,230,17,225),(13,229,18,224),(14,228,19,223),(15,227,20,222),(21,129,26,124),(22,128,27,123),(23,127,28,122),(24,126,29,121),(25,125,30,130),(31,105,36,110),(32,104,37,109),(33,103,38,108),(34,102,39,107),(35,101,40,106),(41,98,46,93),(42,97,47,92),(43,96,48,91),(44,95,49,100),(45,94,50,99),(51,154,56,159),(52,153,57,158),(53,152,58,157),(54,151,59,156),(55,160,60,155),(61,168,66,163),(62,167,67,162),(63,166,68,161),(64,165,69,170),(65,164,70,169),(71,145,76,150),(72,144,77,149),(73,143,78,148),(74,142,79,147),(75,141,80,146),(81,138,86,133),(82,137,87,132),(83,136,88,131),(84,135,89,140),(85,134,90,139),(171,274,176,279),(172,273,177,278),(173,272,178,277),(174,271,179,276),(175,280,180,275),(181,289,186,284),(182,288,187,283),(183,287,188,282),(184,286,189,281),(185,285,190,290),(191,266,196,261),(192,265,197,270),(193,264,198,269),(194,263,199,268),(195,262,200,267),(201,257,206,252),(202,256,207,251),(203,255,208,260),(204,254,209,259),(205,253,210,258),(211,314,216,319),(212,313,217,318),(213,312,218,317),(214,311,219,316),(215,320,220,315),(231,306,236,301),(232,305,237,310),(233,304,238,309),(234,303,239,308),(235,302,240,307),(241,297,246,292),(242,296,247,291),(243,295,248,300),(244,294,249,299),(245,293,250,298)], [(1,238,36,224,25,241,50,212),(2,235,35,227,26,248,49,215),(3,232,34,230,27,245,48,218),(4,239,33,223,28,242,47,211),(5,236,32,226,29,249,46,214),(6,233,31,229,30,246,45,217),(7,240,40,222,21,243,44,220),(8,237,39,225,22,250,43,213),(9,234,38,228,23,247,42,216),(10,231,37,221,24,244,41,219),(11,126,299,98,311,112,301,109),(12,123,298,91,312,119,310,102),(13,130,297,94,313,116,309,105),(14,127,296,97,314,113,308,108),(15,124,295,100,315,120,307,101),(16,121,294,93,316,117,306,104),(17,128,293,96,317,114,305,107),(18,125,292,99,318,111,304,110),(19,122,291,92,319,118,303,103),(20,129,300,95,320,115,302,106),(51,192,79,190,62,205,83,178),(52,199,78,183,63,202,82,171),(53,196,77,186,64,209,81,174),(54,193,76,189,65,206,90,177),(55,200,75,182,66,203,89,180),(56,197,74,185,67,210,88,173),(57,194,73,188,68,207,87,176),(58,191,72,181,69,204,86,179),(59,198,71,184,70,201,85,172),(60,195,80,187,61,208,84,175),(131,277,159,265,142,290,162,253),(132,274,158,268,143,287,161,256),(133,271,157,261,144,284,170,259),(134,278,156,264,145,281,169,252),(135,275,155,267,146,288,168,255),(136,272,154,270,147,285,167,258),(137,279,153,263,148,282,166,251),(138,276,152,266,149,289,165,254),(139,273,151,269,150,286,164,257),(140,280,160,262,141,283,163,260)], [(1,59,6,54),(2,60,7,55),(3,51,8,56),(4,52,9,57),(5,53,10,58),(11,271,16,276),(12,272,17,277),(13,273,18,278),(14,274,19,279),(15,275,20,280),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75),(41,86,46,81),(42,87,47,82),(43,88,48,83),(44,89,49,84),(45,90,50,85),(91,131,96,136),(92,132,97,137),(93,133,98,138),(94,134,99,139),(95,135,100,140),(101,141,106,146),(102,142,107,147),(103,143,108,148),(104,144,109,149),(105,145,110,150),(111,151,116,156),(112,152,117,157),(113,153,118,158),(114,154,119,159),(115,155,120,160),(121,170,126,165),(122,161,127,166),(123,162,128,167),(124,163,129,168),(125,164,130,169),(171,223,176,228),(172,224,177,229),(173,225,178,230),(174,226,179,221),(175,227,180,222),(181,219,186,214),(182,220,187,215),(183,211,188,216),(184,212,189,217),(185,213,190,218),(191,244,196,249),(192,245,197,250),(193,246,198,241),(194,247,199,242),(195,248,200,243),(201,238,206,233),(202,239,207,234),(203,240,208,235),(204,231,209,236),(205,232,210,237),(251,308,256,303),(252,309,257,304),(253,310,258,305),(254,301,259,306),(255,302,260,307),(261,294,266,299),(262,295,267,300),(263,296,268,291),(264,297,269,292),(265,298,270,293),(281,313,286,318),(282,314,287,319),(283,315,288,320),(284,316,289,311),(285,317,290,312)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 1 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 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 | 19 | 3 | 0 | 38 |
| 0 | 0 | 0 | 0 | 22 | 3 | 38 | 19 |
| 0 | 0 | 0 | 0 | 22 | 0 | 19 | 38 |
| 0 | 0 | 0 | 0 | 19 | 22 | 38 | 0 |
| 24 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 27 | 3 | 27 |
| 0 | 0 | 0 | 0 | 36 | 13 | 36 | 19 |
| 0 | 0 | 0 | 0 | 28 | 5 | 22 | 22 |
| 0 | 0 | 0 | 0 | 14 | 8 | 8 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,30,0,0,0,0,0,0,30,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,19,22,22,19,0,0,0,0,3,3,0,22,0,0,0,0,0,38,19,38,0,0,0,0,38,19,38,0],[24,32,0,0,0,0,0,0,32,17,0,0,0,0,0,0,0,0,9,20,0,0,0,0,0,0,21,32,0,0,0,0,0,0,0,0,33,36,28,14,0,0,0,0,27,13,5,8,0,0,0,0,3,36,22,8,0,0,0,0,27,19,22,14],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
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 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | - | + | + | - | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | M4(2) | C4○D4 | C8○D4 | F5 | C2×F5 | D5⋊M4(2) | D4.F5 | Q8×F5 |
| kernel | Dic5.M4(2) | C4×C5⋊C8 | C20⋊C8 | C10.C42 | Dic5⋊C8 | Dic5⋊3Q8 | C10.D4 | C5×C4⋊C4 | C2×Dic10 | C5⋊C8 | Dic5 | Dic5 | C10 | C4⋊C4 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
Dic_5.M_{4(2)} % in TeX
G:=Group("Dic5.M4(2)"); // GroupNames label
G:=SmallGroup(320,1045);
// by ID
G=gap.SmallGroup(320,1045);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,344,1094,219,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=1,b^2=d^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^5>;
// generators/relations
×
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ℤ
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