direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊Q8, C40⋊7Q8, C8⋊(C5×Q8), C4.8(Q8×C10), C4⋊Q8.12C10, C20.97(C2×Q8), C8⋊C4.2C10, C4.Q8.3C10, C2.D8.8C10, (C2×C20).345D4, C10.43(C4⋊Q8), C42.31(C2×C10), C42.C2.5C10, (C4×C20).273C22, (C2×C40).277C22, (C2×C20).957C23, C22.122(D4×C10), C10.148(C8⋊C22), C10.148(C8.C22), C2.9(C5×C4⋊Q8), (C2×C4).46(C5×D4), (C5×C4⋊Q8).27C2, (C5×C4.Q8).8C2, (C5×C8⋊C4).6C2, C4⋊C4.26(C2×C10), (C2×C8).29(C2×C10), C2.23(C5×C8⋊C22), (C5×C2.D8).17C2, (C2×C10).678(C2×D4), C2.23(C5×C8.C22), (C5×C4⋊C4).246C22, (C5×C42.C2).12C2, (C2×C4).132(C22×C10), SmallGroup(320,1002)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 146 in 90 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C22, C5, C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×Q8, C20 [×2], C20 [×6], C2×C10, C8⋊C4, C4.Q8 [×2], C2.D8 [×2], C42.C2, C4⋊Q8, C40 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C8⋊Q8, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4 [×3], C2×C40 [×2], Q8×C10, C5×C8⋊C4, C5×C4.Q8 [×2], C5×C2.D8 [×2], C5×C42.C2, C5×C4⋊Q8, C5×C8⋊Q8
Quotients:
C1, C2 [×7], C22 [×7], C5, D4 [×2], Q8 [×4], C23, C10 [×7], C2×D4, C2×Q8 [×2], C2×C10 [×7], C4⋊Q8, C8⋊C22, C8.C22, C5×D4 [×2], C5×Q8 [×4], C22×C10, C8⋊Q8, D4×C10, Q8×C10 [×2], C5×C4⋊Q8, C5×C8⋊C22, C5×C8.C22, C5×C8⋊Q8
Generators and relations
G = < a,b,c,d | a5=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=c-1 >
►Regular action on 320 points
Generators in S320
(1 16 55 57 17)(2 9 56 58 18)(3 10 49 59 19)(4 11 50 60 20)(5 12 51 61 21)(6 13 52 62 22)(7 14 53 63 23)(8 15 54 64 24)(25 88 73 33 65)(26 81 74 34 66)(27 82 75 35 67)(28 83 76 36 68)(29 84 77 37 69)(30 85 78 38 70)(31 86 79 39 71)(32 87 80 40 72)(41 317 274 309 266)(42 318 275 310 267)(43 319 276 311 268)(44 320 277 312 269)(45 313 278 305 270)(46 314 279 306 271)(47 315 280 307 272)(48 316 273 308 265)(89 125 138 97 130)(90 126 139 98 131)(91 127 140 99 132)(92 128 141 100 133)(93 121 142 101 134)(94 122 143 102 135)(95 123 144 103 136)(96 124 137 104 129)(105 168 154 113 146)(106 161 155 114 147)(107 162 156 115 148)(108 163 157 116 149)(109 164 158 117 150)(110 165 159 118 151)(111 166 160 119 152)(112 167 153 120 145)(169 208 219 177 211)(170 201 220 178 212)(171 202 221 179 213)(172 203 222 180 214)(173 204 223 181 215)(174 205 224 182 216)(175 206 217 183 209)(176 207 218 184 210)(185 248 235 193 227)(186 241 236 194 228)(187 242 237 195 229)(188 243 238 196 230)(189 244 239 197 231)(190 245 240 198 232)(191 246 233 199 225)(192 247 234 200 226)(249 285 299 264 291)(250 286 300 257 292)(251 287 301 258 293)(252 288 302 259 294)(253 281 303 260 295)(254 282 304 261 296)(255 283 297 262 289)(256 284 298 263 290)
(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)
(1 106 31 96)(2 111 32 93)(3 108 25 90)(4 105 26 95)(5 110 27 92)(6 107 28 89)(7 112 29 94)(8 109 30 91)(9 166 87 121)(10 163 88 126)(11 168 81 123)(12 165 82 128)(13 162 83 125)(14 167 84 122)(15 164 85 127)(16 161 86 124)(17 147 71 129)(18 152 72 134)(19 149 65 131)(20 146 66 136)(21 151 67 133)(22 148 68 130)(23 145 69 135)(24 150 70 132)(33 98 59 116)(34 103 60 113)(35 100 61 118)(36 97 62 115)(37 102 63 120)(38 99 64 117)(39 104 57 114)(40 101 58 119)(41 202 251 246)(42 207 252 243)(43 204 253 248)(44 201 254 245)(45 206 255 242)(46 203 256 247)(47 208 249 244)(48 205 250 241)(49 157 73 139)(50 154 74 144)(51 159 75 141)(52 156 76 138)(53 153 77 143)(54 158 78 140)(55 155 79 137)(56 160 80 142)(169 291 189 272)(170 296 190 269)(171 293 191 266)(172 290 192 271)(173 295 185 268)(174 292 186 265)(175 289 187 270)(176 294 188 267)(177 299 197 280)(178 304 198 277)(179 301 199 274)(180 298 200 279)(181 303 193 276)(182 300 194 273)(183 297 195 278)(184 302 196 275)(209 262 229 305)(210 259 230 310)(211 264 231 307)(212 261 232 312)(213 258 225 309)(214 263 226 306)(215 260 227 311)(216 257 228 308)(217 283 237 313)(218 288 238 318)(219 285 239 315)(220 282 240 320)(221 287 233 317)(222 284 234 314)(223 281 235 319)(224 286 236 316)
(1 187 31 175)(2 190 32 170)(3 185 25 173)(4 188 26 176)(5 191 27 171)(6 186 28 174)(7 189 29 169)(8 192 30 172)(9 245 87 201)(10 248 88 204)(11 243 81 207)(12 246 82 202)(13 241 83 205)(14 244 84 208)(15 247 85 203)(16 242 86 206)(17 229 71 209)(18 232 72 212)(19 227 65 215)(20 230 66 210)(21 225 67 213)(22 228 68 216)(23 231 69 211)(24 226 70 214)(33 181 59 193)(34 184 60 196)(35 179 61 199)(36 182 62 194)(37 177 63 197)(38 180 64 200)(39 183 57 195)(40 178 58 198)(41 165 251 128)(42 168 252 123)(43 163 253 126)(44 166 254 121)(45 161 255 124)(46 164 256 127)(47 167 249 122)(48 162 250 125)(49 235 73 223)(50 238 74 218)(51 233 75 221)(52 236 76 224)(53 239 77 219)(54 234 78 222)(55 237 79 217)(56 240 80 220)(89 265 107 292)(90 268 108 295)(91 271 109 290)(92 266 110 293)(93 269 111 296)(94 272 112 291)(95 267 105 294)(96 270 106 289)(97 273 115 300)(98 276 116 303)(99 279 117 298)(100 274 118 301)(101 277 119 304)(102 280 120 299)(103 275 113 302)(104 278 114 297)(129 305 147 262)(130 308 148 257)(131 311 149 260)(132 306 150 263)(133 309 151 258)(134 312 152 261)(135 307 145 264)(136 310 146 259)(137 313 155 283)(138 316 156 286)(139 319 157 281)(140 314 158 284)(141 317 159 287)(142 320 160 282)(143 315 153 285)(144 318 154 288)
G:=sub<Sym(320)| (1,16,55,57,17)(2,9,56,58,18)(3,10,49,59,19)(4,11,50,60,20)(5,12,51,61,21)(6,13,52,62,22)(7,14,53,63,23)(8,15,54,64,24)(25,88,73,33,65)(26,81,74,34,66)(27,82,75,35,67)(28,83,76,36,68)(29,84,77,37,69)(30,85,78,38,70)(31,86,79,39,71)(32,87,80,40,72)(41,317,274,309,266)(42,318,275,310,267)(43,319,276,311,268)(44,320,277,312,269)(45,313,278,305,270)(46,314,279,306,271)(47,315,280,307,272)(48,316,273,308,265)(89,125,138,97,130)(90,126,139,98,131)(91,127,140,99,132)(92,128,141,100,133)(93,121,142,101,134)(94,122,143,102,135)(95,123,144,103,136)(96,124,137,104,129)(105,168,154,113,146)(106,161,155,114,147)(107,162,156,115,148)(108,163,157,116,149)(109,164,158,117,150)(110,165,159,118,151)(111,166,160,119,152)(112,167,153,120,145)(169,208,219,177,211)(170,201,220,178,212)(171,202,221,179,213)(172,203,222,180,214)(173,204,223,181,215)(174,205,224,182,216)(175,206,217,183,209)(176,207,218,184,210)(185,248,235,193,227)(186,241,236,194,228)(187,242,237,195,229)(188,243,238,196,230)(189,244,239,197,231)(190,245,240,198,232)(191,246,233,199,225)(192,247,234,200,226)(249,285,299,264,291)(250,286,300,257,292)(251,287,301,258,293)(252,288,302,259,294)(253,281,303,260,295)(254,282,304,261,296)(255,283,297,262,289)(256,284,298,263,290), (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), (1,106,31,96)(2,111,32,93)(3,108,25,90)(4,105,26,95)(5,110,27,92)(6,107,28,89)(7,112,29,94)(8,109,30,91)(9,166,87,121)(10,163,88,126)(11,168,81,123)(12,165,82,128)(13,162,83,125)(14,167,84,122)(15,164,85,127)(16,161,86,124)(17,147,71,129)(18,152,72,134)(19,149,65,131)(20,146,66,136)(21,151,67,133)(22,148,68,130)(23,145,69,135)(24,150,70,132)(33,98,59,116)(34,103,60,113)(35,100,61,118)(36,97,62,115)(37,102,63,120)(38,99,64,117)(39,104,57,114)(40,101,58,119)(41,202,251,246)(42,207,252,243)(43,204,253,248)(44,201,254,245)(45,206,255,242)(46,203,256,247)(47,208,249,244)(48,205,250,241)(49,157,73,139)(50,154,74,144)(51,159,75,141)(52,156,76,138)(53,153,77,143)(54,158,78,140)(55,155,79,137)(56,160,80,142)(169,291,189,272)(170,296,190,269)(171,293,191,266)(172,290,192,271)(173,295,185,268)(174,292,186,265)(175,289,187,270)(176,294,188,267)(177,299,197,280)(178,304,198,277)(179,301,199,274)(180,298,200,279)(181,303,193,276)(182,300,194,273)(183,297,195,278)(184,302,196,275)(209,262,229,305)(210,259,230,310)(211,264,231,307)(212,261,232,312)(213,258,225,309)(214,263,226,306)(215,260,227,311)(216,257,228,308)(217,283,237,313)(218,288,238,318)(219,285,239,315)(220,282,240,320)(221,287,233,317)(222,284,234,314)(223,281,235,319)(224,286,236,316), (1,187,31,175)(2,190,32,170)(3,185,25,173)(4,188,26,176)(5,191,27,171)(6,186,28,174)(7,189,29,169)(8,192,30,172)(9,245,87,201)(10,248,88,204)(11,243,81,207)(12,246,82,202)(13,241,83,205)(14,244,84,208)(15,247,85,203)(16,242,86,206)(17,229,71,209)(18,232,72,212)(19,227,65,215)(20,230,66,210)(21,225,67,213)(22,228,68,216)(23,231,69,211)(24,226,70,214)(33,181,59,193)(34,184,60,196)(35,179,61,199)(36,182,62,194)(37,177,63,197)(38,180,64,200)(39,183,57,195)(40,178,58,198)(41,165,251,128)(42,168,252,123)(43,163,253,126)(44,166,254,121)(45,161,255,124)(46,164,256,127)(47,167,249,122)(48,162,250,125)(49,235,73,223)(50,238,74,218)(51,233,75,221)(52,236,76,224)(53,239,77,219)(54,234,78,222)(55,237,79,217)(56,240,80,220)(89,265,107,292)(90,268,108,295)(91,271,109,290)(92,266,110,293)(93,269,111,296)(94,272,112,291)(95,267,105,294)(96,270,106,289)(97,273,115,300)(98,276,116,303)(99,279,117,298)(100,274,118,301)(101,277,119,304)(102,280,120,299)(103,275,113,302)(104,278,114,297)(129,305,147,262)(130,308,148,257)(131,311,149,260)(132,306,150,263)(133,309,151,258)(134,312,152,261)(135,307,145,264)(136,310,146,259)(137,313,155,283)(138,316,156,286)(139,319,157,281)(140,314,158,284)(141,317,159,287)(142,320,160,282)(143,315,153,285)(144,318,154,288)>;
G:=Group( (1,16,55,57,17)(2,9,56,58,18)(3,10,49,59,19)(4,11,50,60,20)(5,12,51,61,21)(6,13,52,62,22)(7,14,53,63,23)(8,15,54,64,24)(25,88,73,33,65)(26,81,74,34,66)(27,82,75,35,67)(28,83,76,36,68)(29,84,77,37,69)(30,85,78,38,70)(31,86,79,39,71)(32,87,80,40,72)(41,317,274,309,266)(42,318,275,310,267)(43,319,276,311,268)(44,320,277,312,269)(45,313,278,305,270)(46,314,279,306,271)(47,315,280,307,272)(48,316,273,308,265)(89,125,138,97,130)(90,126,139,98,131)(91,127,140,99,132)(92,128,141,100,133)(93,121,142,101,134)(94,122,143,102,135)(95,123,144,103,136)(96,124,137,104,129)(105,168,154,113,146)(106,161,155,114,147)(107,162,156,115,148)(108,163,157,116,149)(109,164,158,117,150)(110,165,159,118,151)(111,166,160,119,152)(112,167,153,120,145)(169,208,219,177,211)(170,201,220,178,212)(171,202,221,179,213)(172,203,222,180,214)(173,204,223,181,215)(174,205,224,182,216)(175,206,217,183,209)(176,207,218,184,210)(185,248,235,193,227)(186,241,236,194,228)(187,242,237,195,229)(188,243,238,196,230)(189,244,239,197,231)(190,245,240,198,232)(191,246,233,199,225)(192,247,234,200,226)(249,285,299,264,291)(250,286,300,257,292)(251,287,301,258,293)(252,288,302,259,294)(253,281,303,260,295)(254,282,304,261,296)(255,283,297,262,289)(256,284,298,263,290), (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), (1,106,31,96)(2,111,32,93)(3,108,25,90)(4,105,26,95)(5,110,27,92)(6,107,28,89)(7,112,29,94)(8,109,30,91)(9,166,87,121)(10,163,88,126)(11,168,81,123)(12,165,82,128)(13,162,83,125)(14,167,84,122)(15,164,85,127)(16,161,86,124)(17,147,71,129)(18,152,72,134)(19,149,65,131)(20,146,66,136)(21,151,67,133)(22,148,68,130)(23,145,69,135)(24,150,70,132)(33,98,59,116)(34,103,60,113)(35,100,61,118)(36,97,62,115)(37,102,63,120)(38,99,64,117)(39,104,57,114)(40,101,58,119)(41,202,251,246)(42,207,252,243)(43,204,253,248)(44,201,254,245)(45,206,255,242)(46,203,256,247)(47,208,249,244)(48,205,250,241)(49,157,73,139)(50,154,74,144)(51,159,75,141)(52,156,76,138)(53,153,77,143)(54,158,78,140)(55,155,79,137)(56,160,80,142)(169,291,189,272)(170,296,190,269)(171,293,191,266)(172,290,192,271)(173,295,185,268)(174,292,186,265)(175,289,187,270)(176,294,188,267)(177,299,197,280)(178,304,198,277)(179,301,199,274)(180,298,200,279)(181,303,193,276)(182,300,194,273)(183,297,195,278)(184,302,196,275)(209,262,229,305)(210,259,230,310)(211,264,231,307)(212,261,232,312)(213,258,225,309)(214,263,226,306)(215,260,227,311)(216,257,228,308)(217,283,237,313)(218,288,238,318)(219,285,239,315)(220,282,240,320)(221,287,233,317)(222,284,234,314)(223,281,235,319)(224,286,236,316), (1,187,31,175)(2,190,32,170)(3,185,25,173)(4,188,26,176)(5,191,27,171)(6,186,28,174)(7,189,29,169)(8,192,30,172)(9,245,87,201)(10,248,88,204)(11,243,81,207)(12,246,82,202)(13,241,83,205)(14,244,84,208)(15,247,85,203)(16,242,86,206)(17,229,71,209)(18,232,72,212)(19,227,65,215)(20,230,66,210)(21,225,67,213)(22,228,68,216)(23,231,69,211)(24,226,70,214)(33,181,59,193)(34,184,60,196)(35,179,61,199)(36,182,62,194)(37,177,63,197)(38,180,64,200)(39,183,57,195)(40,178,58,198)(41,165,251,128)(42,168,252,123)(43,163,253,126)(44,166,254,121)(45,161,255,124)(46,164,256,127)(47,167,249,122)(48,162,250,125)(49,235,73,223)(50,238,74,218)(51,233,75,221)(52,236,76,224)(53,239,77,219)(54,234,78,222)(55,237,79,217)(56,240,80,220)(89,265,107,292)(90,268,108,295)(91,271,109,290)(92,266,110,293)(93,269,111,296)(94,272,112,291)(95,267,105,294)(96,270,106,289)(97,273,115,300)(98,276,116,303)(99,279,117,298)(100,274,118,301)(101,277,119,304)(102,280,120,299)(103,275,113,302)(104,278,114,297)(129,305,147,262)(130,308,148,257)(131,311,149,260)(132,306,150,263)(133,309,151,258)(134,312,152,261)(135,307,145,264)(136,310,146,259)(137,313,155,283)(138,316,156,286)(139,319,157,281)(140,314,158,284)(141,317,159,287)(142,320,160,282)(143,315,153,285)(144,318,154,288) );
G=PermutationGroup([(1,16,55,57,17),(2,9,56,58,18),(3,10,49,59,19),(4,11,50,60,20),(5,12,51,61,21),(6,13,52,62,22),(7,14,53,63,23),(8,15,54,64,24),(25,88,73,33,65),(26,81,74,34,66),(27,82,75,35,67),(28,83,76,36,68),(29,84,77,37,69),(30,85,78,38,70),(31,86,79,39,71),(32,87,80,40,72),(41,317,274,309,266),(42,318,275,310,267),(43,319,276,311,268),(44,320,277,312,269),(45,313,278,305,270),(46,314,279,306,271),(47,315,280,307,272),(48,316,273,308,265),(89,125,138,97,130),(90,126,139,98,131),(91,127,140,99,132),(92,128,141,100,133),(93,121,142,101,134),(94,122,143,102,135),(95,123,144,103,136),(96,124,137,104,129),(105,168,154,113,146),(106,161,155,114,147),(107,162,156,115,148),(108,163,157,116,149),(109,164,158,117,150),(110,165,159,118,151),(111,166,160,119,152),(112,167,153,120,145),(169,208,219,177,211),(170,201,220,178,212),(171,202,221,179,213),(172,203,222,180,214),(173,204,223,181,215),(174,205,224,182,216),(175,206,217,183,209),(176,207,218,184,210),(185,248,235,193,227),(186,241,236,194,228),(187,242,237,195,229),(188,243,238,196,230),(189,244,239,197,231),(190,245,240,198,232),(191,246,233,199,225),(192,247,234,200,226),(249,285,299,264,291),(250,286,300,257,292),(251,287,301,258,293),(252,288,302,259,294),(253,281,303,260,295),(254,282,304,261,296),(255,283,297,262,289),(256,284,298,263,290)], [(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)], [(1,106,31,96),(2,111,32,93),(3,108,25,90),(4,105,26,95),(5,110,27,92),(6,107,28,89),(7,112,29,94),(8,109,30,91),(9,166,87,121),(10,163,88,126),(11,168,81,123),(12,165,82,128),(13,162,83,125),(14,167,84,122),(15,164,85,127),(16,161,86,124),(17,147,71,129),(18,152,72,134),(19,149,65,131),(20,146,66,136),(21,151,67,133),(22,148,68,130),(23,145,69,135),(24,150,70,132),(33,98,59,116),(34,103,60,113),(35,100,61,118),(36,97,62,115),(37,102,63,120),(38,99,64,117),(39,104,57,114),(40,101,58,119),(41,202,251,246),(42,207,252,243),(43,204,253,248),(44,201,254,245),(45,206,255,242),(46,203,256,247),(47,208,249,244),(48,205,250,241),(49,157,73,139),(50,154,74,144),(51,159,75,141),(52,156,76,138),(53,153,77,143),(54,158,78,140),(55,155,79,137),(56,160,80,142),(169,291,189,272),(170,296,190,269),(171,293,191,266),(172,290,192,271),(173,295,185,268),(174,292,186,265),(175,289,187,270),(176,294,188,267),(177,299,197,280),(178,304,198,277),(179,301,199,274),(180,298,200,279),(181,303,193,276),(182,300,194,273),(183,297,195,278),(184,302,196,275),(209,262,229,305),(210,259,230,310),(211,264,231,307),(212,261,232,312),(213,258,225,309),(214,263,226,306),(215,260,227,311),(216,257,228,308),(217,283,237,313),(218,288,238,318),(219,285,239,315),(220,282,240,320),(221,287,233,317),(222,284,234,314),(223,281,235,319),(224,286,236,316)], [(1,187,31,175),(2,190,32,170),(3,185,25,173),(4,188,26,176),(5,191,27,171),(6,186,28,174),(7,189,29,169),(8,192,30,172),(9,245,87,201),(10,248,88,204),(11,243,81,207),(12,246,82,202),(13,241,83,205),(14,244,84,208),(15,247,85,203),(16,242,86,206),(17,229,71,209),(18,232,72,212),(19,227,65,215),(20,230,66,210),(21,225,67,213),(22,228,68,216),(23,231,69,211),(24,226,70,214),(33,181,59,193),(34,184,60,196),(35,179,61,199),(36,182,62,194),(37,177,63,197),(38,180,64,200),(39,183,57,195),(40,178,58,198),(41,165,251,128),(42,168,252,123),(43,163,253,126),(44,166,254,121),(45,161,255,124),(46,164,256,127),(47,167,249,122),(48,162,250,125),(49,235,73,223),(50,238,74,218),(51,233,75,221),(52,236,76,224),(53,239,77,219),(54,234,78,222),(55,237,79,217),(56,240,80,220),(89,265,107,292),(90,268,108,295),(91,271,109,290),(92,266,110,293),(93,269,111,296),(94,272,112,291),(95,267,105,294),(96,270,106,289),(97,273,115,300),(98,276,116,303),(99,279,117,298),(100,274,118,301),(101,277,119,304),(102,280,120,299),(103,275,113,302),(104,278,114,297),(129,305,147,262),(130,308,148,257),(131,311,149,260),(132,306,150,263),(133,309,151,258),(134,312,152,261),(135,307,145,264),(136,310,146,259),(137,313,155,283),(138,316,156,286),(139,319,157,281),(140,314,158,284),(141,317,159,287),(142,320,160,282),(143,315,153,285),(144,318,154,288)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 13 | 39 | 1 |
| 0 | 0 | 14 | 30 | 20 | 40 |
| 0 | 0 | 1 | 38 | 24 | 28 |
| 0 | 0 | 22 | 39 | 27 | 11 |
| 31 | 22 | 0 | 0 | 0 | 0 |
| 1 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 17 | 10 | 30 |
| 0 | 0 | 7 | 22 | 25 | 31 |
| 0 | 0 | 31 | 11 | 36 | 27 |
| 0 | 0 | 16 | 10 | 29 | 5 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,1,0,0,0,0,0,0,1,0,0],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,17,14,1,22,0,0,13,30,38,39,0,0,39,20,24,27,0,0,1,40,28,11],[31,1,0,0,0,0,22,10,0,0,0,0,0,0,19,7,31,16,0,0,17,22,11,10,0,0,10,25,36,29,0,0,30,31,27,5] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | Q8 | D4 | C5×Q8 | C5×D4 | C8⋊C22 | C8.C22 | C5×C8⋊C22 | C5×C8.C22 |
| kernel | C5×C8⋊Q8 | C5×C8⋊C4 | C5×C4.Q8 | C5×C2.D8 | C5×C42.C2 | C5×C4⋊Q8 | C8⋊Q8 | C8⋊C4 | C4.Q8 | C2.D8 | C42.C2 | C4⋊Q8 | C40 | C2×C20 | C8 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 2 | 16 | 8 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_5\times C_8\rtimes Q_8
% in TeX
G:=Group("C5xC8:Q8"); // GroupNames label
G:=SmallGroup(320,1002);
// by ID
G=gap.SmallGroup(320,1002);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,288,1766,1731,436,10085,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀