direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊2Q8, C40⋊12Q8, C20.43D8, C20.21Q16, C8⋊2(C5×Q8), C4.5(C5×D8), (C4×C8).8C10, C4.7(Q8×C10), C4.4(C5×Q16), (C4×C40).26C2, C2.11(C10×D8), C10.83(C2×D8), C4⋊Q8.11C10, C20.96(C2×Q8), C2.D8.7C10, (C2×C20).425D4, C10.42(C4⋊Q8), C10.58(C2×Q16), C2.11(C10×Q16), C42.85(C2×C10), (C2×C20).956C23, (C4×C20).369C22, (C2×C40).425C22, C22.121(D4×C10), C2.8(C5×C4⋊Q8), (C2×C4).81(C5×D4), (C5×C4⋊Q8).26C2, C4⋊C4.25(C2×C10), (C2×C8).83(C2×C10), (C5×C2.D8).16C2, (C2×C10).677(C2×D4), (C5×C4⋊C4).245C22, (C2×C4).131(C22×C10), SmallGroup(320,1001)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊2Q8
G = < a,b,c,d | a5=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 162 in 98 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C4×C8, C2.D8, C4⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C8⋊2Q8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, Q8×C10, C4×C40, C5×C2.D8, C5×C4⋊Q8, C5×C8⋊2Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, D8, Q16, C2×D4, C2×Q8, C2×C10, C4⋊Q8, C2×D8, C2×Q16, C5×D4, C5×Q8, C22×C10, C8⋊2Q8, C5×D8, C5×Q16, D4×C10, Q8×C10, C5×C4⋊Q8, C10×D8, C10×Q16, C5×C8⋊2Q8
►Regular action on 320 points
Generators in S320
(1 10 51 59 19)(2 11 52 60 20)(3 12 53 61 21)(4 13 54 62 22)(5 14 55 63 23)(6 15 56 64 24)(7 16 49 57 17)(8 9 50 58 18)(25 81 73 33 65)(26 82 74 34 66)(27 83 75 35 67)(28 84 76 36 68)(29 85 77 37 69)(30 86 78 38 70)(31 87 79 39 71)(32 88 80 40 72)(41 318 276 310 268)(42 319 277 311 269)(43 320 278 312 270)(44 313 279 305 271)(45 314 280 306 272)(46 315 273 307 265)(47 316 274 308 266)(48 317 275 309 267)(89 124 138 98 130)(90 125 139 99 131)(91 126 140 100 132)(92 127 141 101 133)(93 128 142 102 134)(94 121 143 103 135)(95 122 144 104 136)(96 123 137 97 129)(105 167 153 113 145)(106 168 154 114 146)(107 161 155 115 147)(108 162 156 116 148)(109 163 157 117 149)(110 164 158 118 150)(111 165 159 119 151)(112 166 160 120 152)(169 204 219 177 211)(170 205 220 178 212)(171 206 221 179 213)(172 207 222 180 214)(173 208 223 181 215)(174 201 224 182 216)(175 202 217 183 209)(176 203 218 184 210)(185 241 235 193 227)(186 242 236 194 228)(187 243 237 195 229)(188 244 238 196 230)(189 245 239 197 231)(190 246 240 198 232)(191 247 233 199 225)(192 248 234 200 226)(249 284 299 257 291)(250 285 300 258 292)(251 286 301 259 293)(252 287 302 260 294)(253 288 303 261 295)(254 281 304 262 296)(255 282 297 263 289)(256 283 298 264 290)
(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)
(1 107 27 131)(2 108 28 132)(3 109 29 133)(4 110 30 134)(5 111 31 135)(6 112 32 136)(7 105 25 129)(8 106 26 130)(9 168 82 89)(10 161 83 90)(11 162 84 91)(12 163 85 92)(13 164 86 93)(14 165 87 94)(15 166 88 95)(16 167 81 96)(17 145 65 97)(18 146 66 98)(19 147 67 99)(20 148 68 100)(21 149 69 101)(22 150 70 102)(23 151 71 103)(24 152 72 104)(33 137 57 113)(34 138 58 114)(35 139 59 115)(36 140 60 116)(37 141 61 117)(38 142 62 118)(39 143 63 119)(40 144 64 120)(41 201 281 244)(42 202 282 245)(43 203 283 246)(44 204 284 247)(45 205 285 248)(46 206 286 241)(47 207 287 242)(48 208 288 243)(49 153 73 123)(50 154 74 124)(51 155 75 125)(52 156 76 126)(53 157 77 127)(54 158 78 128)(55 159 79 121)(56 160 80 122)(169 249 191 271)(170 250 192 272)(171 251 185 265)(172 252 186 266)(173 253 187 267)(174 254 188 268)(175 255 189 269)(176 256 190 270)(177 257 199 279)(178 258 200 280)(179 259 193 273)(180 260 194 274)(181 261 195 275)(182 262 196 276)(183 263 197 277)(184 264 198 278)(209 289 231 311)(210 290 232 312)(211 291 225 305)(212 292 226 306)(213 293 227 307)(214 294 228 308)(215 295 229 309)(216 296 230 310)(217 297 239 319)(218 298 240 320)(219 299 233 313)(220 300 234 314)(221 301 235 315)(222 302 236 316)(223 303 237 317)(224 304 238 318)
(1 187 27 173)(2 186 28 172)(3 185 29 171)(4 192 30 170)(5 191 31 169)(6 190 32 176)(7 189 25 175)(8 188 26 174)(9 244 82 201)(10 243 83 208)(11 242 84 207)(12 241 85 206)(13 248 86 205)(14 247 87 204)(15 246 88 203)(16 245 81 202)(17 231 65 209)(18 230 66 216)(19 229 67 215)(20 228 68 214)(21 227 69 213)(22 226 70 212)(23 225 71 211)(24 232 72 210)(33 183 57 197)(34 182 58 196)(35 181 59 195)(36 180 60 194)(37 179 61 193)(38 178 62 200)(39 177 63 199)(40 184 64 198)(41 168 281 89)(42 167 282 96)(43 166 283 95)(44 165 284 94)(45 164 285 93)(46 163 286 92)(47 162 287 91)(48 161 288 90)(49 239 73 217)(50 238 74 224)(51 237 75 223)(52 236 76 222)(53 235 77 221)(54 234 78 220)(55 233 79 219)(56 240 80 218)(97 311 145 289)(98 310 146 296)(99 309 147 295)(100 308 148 294)(101 307 149 293)(102 306 150 292)(103 305 151 291)(104 312 152 290)(105 255 129 269)(106 254 130 268)(107 253 131 267)(108 252 132 266)(109 251 133 265)(110 250 134 272)(111 249 135 271)(112 256 136 270)(113 263 137 277)(114 262 138 276)(115 261 139 275)(116 260 140 274)(117 259 141 273)(118 258 142 280)(119 257 143 279)(120 264 144 278)(121 313 159 299)(122 320 160 298)(123 319 153 297)(124 318 154 304)(125 317 155 303)(126 316 156 302)(127 315 157 301)(128 314 158 300)
G:=sub<Sym(320)| (1,10,51,59,19)(2,11,52,60,20)(3,12,53,61,21)(4,13,54,62,22)(5,14,55,63,23)(6,15,56,64,24)(7,16,49,57,17)(8,9,50,58,18)(25,81,73,33,65)(26,82,74,34,66)(27,83,75,35,67)(28,84,76,36,68)(29,85,77,37,69)(30,86,78,38,70)(31,87,79,39,71)(32,88,80,40,72)(41,318,276,310,268)(42,319,277,311,269)(43,320,278,312,270)(44,313,279,305,271)(45,314,280,306,272)(46,315,273,307,265)(47,316,274,308,266)(48,317,275,309,267)(89,124,138,98,130)(90,125,139,99,131)(91,126,140,100,132)(92,127,141,101,133)(93,128,142,102,134)(94,121,143,103,135)(95,122,144,104,136)(96,123,137,97,129)(105,167,153,113,145)(106,168,154,114,146)(107,161,155,115,147)(108,162,156,116,148)(109,163,157,117,149)(110,164,158,118,150)(111,165,159,119,151)(112,166,160,120,152)(169,204,219,177,211)(170,205,220,178,212)(171,206,221,179,213)(172,207,222,180,214)(173,208,223,181,215)(174,201,224,182,216)(175,202,217,183,209)(176,203,218,184,210)(185,241,235,193,227)(186,242,236,194,228)(187,243,237,195,229)(188,244,238,196,230)(189,245,239,197,231)(190,246,240,198,232)(191,247,233,199,225)(192,248,234,200,226)(249,284,299,257,291)(250,285,300,258,292)(251,286,301,259,293)(252,287,302,260,294)(253,288,303,261,295)(254,281,304,262,296)(255,282,297,263,289)(256,283,298,264,290), (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), (1,107,27,131)(2,108,28,132)(3,109,29,133)(4,110,30,134)(5,111,31,135)(6,112,32,136)(7,105,25,129)(8,106,26,130)(9,168,82,89)(10,161,83,90)(11,162,84,91)(12,163,85,92)(13,164,86,93)(14,165,87,94)(15,166,88,95)(16,167,81,96)(17,145,65,97)(18,146,66,98)(19,147,67,99)(20,148,68,100)(21,149,69,101)(22,150,70,102)(23,151,71,103)(24,152,72,104)(33,137,57,113)(34,138,58,114)(35,139,59,115)(36,140,60,116)(37,141,61,117)(38,142,62,118)(39,143,63,119)(40,144,64,120)(41,201,281,244)(42,202,282,245)(43,203,283,246)(44,204,284,247)(45,205,285,248)(46,206,286,241)(47,207,287,242)(48,208,288,243)(49,153,73,123)(50,154,74,124)(51,155,75,125)(52,156,76,126)(53,157,77,127)(54,158,78,128)(55,159,79,121)(56,160,80,122)(169,249,191,271)(170,250,192,272)(171,251,185,265)(172,252,186,266)(173,253,187,267)(174,254,188,268)(175,255,189,269)(176,256,190,270)(177,257,199,279)(178,258,200,280)(179,259,193,273)(180,260,194,274)(181,261,195,275)(182,262,196,276)(183,263,197,277)(184,264,198,278)(209,289,231,311)(210,290,232,312)(211,291,225,305)(212,292,226,306)(213,293,227,307)(214,294,228,308)(215,295,229,309)(216,296,230,310)(217,297,239,319)(218,298,240,320)(219,299,233,313)(220,300,234,314)(221,301,235,315)(222,302,236,316)(223,303,237,317)(224,304,238,318), (1,187,27,173)(2,186,28,172)(3,185,29,171)(4,192,30,170)(5,191,31,169)(6,190,32,176)(7,189,25,175)(8,188,26,174)(9,244,82,201)(10,243,83,208)(11,242,84,207)(12,241,85,206)(13,248,86,205)(14,247,87,204)(15,246,88,203)(16,245,81,202)(17,231,65,209)(18,230,66,216)(19,229,67,215)(20,228,68,214)(21,227,69,213)(22,226,70,212)(23,225,71,211)(24,232,72,210)(33,183,57,197)(34,182,58,196)(35,181,59,195)(36,180,60,194)(37,179,61,193)(38,178,62,200)(39,177,63,199)(40,184,64,198)(41,168,281,89)(42,167,282,96)(43,166,283,95)(44,165,284,94)(45,164,285,93)(46,163,286,92)(47,162,287,91)(48,161,288,90)(49,239,73,217)(50,238,74,224)(51,237,75,223)(52,236,76,222)(53,235,77,221)(54,234,78,220)(55,233,79,219)(56,240,80,218)(97,311,145,289)(98,310,146,296)(99,309,147,295)(100,308,148,294)(101,307,149,293)(102,306,150,292)(103,305,151,291)(104,312,152,290)(105,255,129,269)(106,254,130,268)(107,253,131,267)(108,252,132,266)(109,251,133,265)(110,250,134,272)(111,249,135,271)(112,256,136,270)(113,263,137,277)(114,262,138,276)(115,261,139,275)(116,260,140,274)(117,259,141,273)(118,258,142,280)(119,257,143,279)(120,264,144,278)(121,313,159,299)(122,320,160,298)(123,319,153,297)(124,318,154,304)(125,317,155,303)(126,316,156,302)(127,315,157,301)(128,314,158,300)>;
G:=Group( (1,10,51,59,19)(2,11,52,60,20)(3,12,53,61,21)(4,13,54,62,22)(5,14,55,63,23)(6,15,56,64,24)(7,16,49,57,17)(8,9,50,58,18)(25,81,73,33,65)(26,82,74,34,66)(27,83,75,35,67)(28,84,76,36,68)(29,85,77,37,69)(30,86,78,38,70)(31,87,79,39,71)(32,88,80,40,72)(41,318,276,310,268)(42,319,277,311,269)(43,320,278,312,270)(44,313,279,305,271)(45,314,280,306,272)(46,315,273,307,265)(47,316,274,308,266)(48,317,275,309,267)(89,124,138,98,130)(90,125,139,99,131)(91,126,140,100,132)(92,127,141,101,133)(93,128,142,102,134)(94,121,143,103,135)(95,122,144,104,136)(96,123,137,97,129)(105,167,153,113,145)(106,168,154,114,146)(107,161,155,115,147)(108,162,156,116,148)(109,163,157,117,149)(110,164,158,118,150)(111,165,159,119,151)(112,166,160,120,152)(169,204,219,177,211)(170,205,220,178,212)(171,206,221,179,213)(172,207,222,180,214)(173,208,223,181,215)(174,201,224,182,216)(175,202,217,183,209)(176,203,218,184,210)(185,241,235,193,227)(186,242,236,194,228)(187,243,237,195,229)(188,244,238,196,230)(189,245,239,197,231)(190,246,240,198,232)(191,247,233,199,225)(192,248,234,200,226)(249,284,299,257,291)(250,285,300,258,292)(251,286,301,259,293)(252,287,302,260,294)(253,288,303,261,295)(254,281,304,262,296)(255,282,297,263,289)(256,283,298,264,290), (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), (1,107,27,131)(2,108,28,132)(3,109,29,133)(4,110,30,134)(5,111,31,135)(6,112,32,136)(7,105,25,129)(8,106,26,130)(9,168,82,89)(10,161,83,90)(11,162,84,91)(12,163,85,92)(13,164,86,93)(14,165,87,94)(15,166,88,95)(16,167,81,96)(17,145,65,97)(18,146,66,98)(19,147,67,99)(20,148,68,100)(21,149,69,101)(22,150,70,102)(23,151,71,103)(24,152,72,104)(33,137,57,113)(34,138,58,114)(35,139,59,115)(36,140,60,116)(37,141,61,117)(38,142,62,118)(39,143,63,119)(40,144,64,120)(41,201,281,244)(42,202,282,245)(43,203,283,246)(44,204,284,247)(45,205,285,248)(46,206,286,241)(47,207,287,242)(48,208,288,243)(49,153,73,123)(50,154,74,124)(51,155,75,125)(52,156,76,126)(53,157,77,127)(54,158,78,128)(55,159,79,121)(56,160,80,122)(169,249,191,271)(170,250,192,272)(171,251,185,265)(172,252,186,266)(173,253,187,267)(174,254,188,268)(175,255,189,269)(176,256,190,270)(177,257,199,279)(178,258,200,280)(179,259,193,273)(180,260,194,274)(181,261,195,275)(182,262,196,276)(183,263,197,277)(184,264,198,278)(209,289,231,311)(210,290,232,312)(211,291,225,305)(212,292,226,306)(213,293,227,307)(214,294,228,308)(215,295,229,309)(216,296,230,310)(217,297,239,319)(218,298,240,320)(219,299,233,313)(220,300,234,314)(221,301,235,315)(222,302,236,316)(223,303,237,317)(224,304,238,318), (1,187,27,173)(2,186,28,172)(3,185,29,171)(4,192,30,170)(5,191,31,169)(6,190,32,176)(7,189,25,175)(8,188,26,174)(9,244,82,201)(10,243,83,208)(11,242,84,207)(12,241,85,206)(13,248,86,205)(14,247,87,204)(15,246,88,203)(16,245,81,202)(17,231,65,209)(18,230,66,216)(19,229,67,215)(20,228,68,214)(21,227,69,213)(22,226,70,212)(23,225,71,211)(24,232,72,210)(33,183,57,197)(34,182,58,196)(35,181,59,195)(36,180,60,194)(37,179,61,193)(38,178,62,200)(39,177,63,199)(40,184,64,198)(41,168,281,89)(42,167,282,96)(43,166,283,95)(44,165,284,94)(45,164,285,93)(46,163,286,92)(47,162,287,91)(48,161,288,90)(49,239,73,217)(50,238,74,224)(51,237,75,223)(52,236,76,222)(53,235,77,221)(54,234,78,220)(55,233,79,219)(56,240,80,218)(97,311,145,289)(98,310,146,296)(99,309,147,295)(100,308,148,294)(101,307,149,293)(102,306,150,292)(103,305,151,291)(104,312,152,290)(105,255,129,269)(106,254,130,268)(107,253,131,267)(108,252,132,266)(109,251,133,265)(110,250,134,272)(111,249,135,271)(112,256,136,270)(113,263,137,277)(114,262,138,276)(115,261,139,275)(116,260,140,274)(117,259,141,273)(118,258,142,280)(119,257,143,279)(120,264,144,278)(121,313,159,299)(122,320,160,298)(123,319,153,297)(124,318,154,304)(125,317,155,303)(126,316,156,302)(127,315,157,301)(128,314,158,300) );
G=PermutationGroup([[(1,10,51,59,19),(2,11,52,60,20),(3,12,53,61,21),(4,13,54,62,22),(5,14,55,63,23),(6,15,56,64,24),(7,16,49,57,17),(8,9,50,58,18),(25,81,73,33,65),(26,82,74,34,66),(27,83,75,35,67),(28,84,76,36,68),(29,85,77,37,69),(30,86,78,38,70),(31,87,79,39,71),(32,88,80,40,72),(41,318,276,310,268),(42,319,277,311,269),(43,320,278,312,270),(44,313,279,305,271),(45,314,280,306,272),(46,315,273,307,265),(47,316,274,308,266),(48,317,275,309,267),(89,124,138,98,130),(90,125,139,99,131),(91,126,140,100,132),(92,127,141,101,133),(93,128,142,102,134),(94,121,143,103,135),(95,122,144,104,136),(96,123,137,97,129),(105,167,153,113,145),(106,168,154,114,146),(107,161,155,115,147),(108,162,156,116,148),(109,163,157,117,149),(110,164,158,118,150),(111,165,159,119,151),(112,166,160,120,152),(169,204,219,177,211),(170,205,220,178,212),(171,206,221,179,213),(172,207,222,180,214),(173,208,223,181,215),(174,201,224,182,216),(175,202,217,183,209),(176,203,218,184,210),(185,241,235,193,227),(186,242,236,194,228),(187,243,237,195,229),(188,244,238,196,230),(189,245,239,197,231),(190,246,240,198,232),(191,247,233,199,225),(192,248,234,200,226),(249,284,299,257,291),(250,285,300,258,292),(251,286,301,259,293),(252,287,302,260,294),(253,288,303,261,295),(254,281,304,262,296),(255,282,297,263,289),(256,283,298,264,290)], [(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)], [(1,107,27,131),(2,108,28,132),(3,109,29,133),(4,110,30,134),(5,111,31,135),(6,112,32,136),(7,105,25,129),(8,106,26,130),(9,168,82,89),(10,161,83,90),(11,162,84,91),(12,163,85,92),(13,164,86,93),(14,165,87,94),(15,166,88,95),(16,167,81,96),(17,145,65,97),(18,146,66,98),(19,147,67,99),(20,148,68,100),(21,149,69,101),(22,150,70,102),(23,151,71,103),(24,152,72,104),(33,137,57,113),(34,138,58,114),(35,139,59,115),(36,140,60,116),(37,141,61,117),(38,142,62,118),(39,143,63,119),(40,144,64,120),(41,201,281,244),(42,202,282,245),(43,203,283,246),(44,204,284,247),(45,205,285,248),(46,206,286,241),(47,207,287,242),(48,208,288,243),(49,153,73,123),(50,154,74,124),(51,155,75,125),(52,156,76,126),(53,157,77,127),(54,158,78,128),(55,159,79,121),(56,160,80,122),(169,249,191,271),(170,250,192,272),(171,251,185,265),(172,252,186,266),(173,253,187,267),(174,254,188,268),(175,255,189,269),(176,256,190,270),(177,257,199,279),(178,258,200,280),(179,259,193,273),(180,260,194,274),(181,261,195,275),(182,262,196,276),(183,263,197,277),(184,264,198,278),(209,289,231,311),(210,290,232,312),(211,291,225,305),(212,292,226,306),(213,293,227,307),(214,294,228,308),(215,295,229,309),(216,296,230,310),(217,297,239,319),(218,298,240,320),(219,299,233,313),(220,300,234,314),(221,301,235,315),(222,302,236,316),(223,303,237,317),(224,304,238,318)], [(1,187,27,173),(2,186,28,172),(3,185,29,171),(4,192,30,170),(5,191,31,169),(6,190,32,176),(7,189,25,175),(8,188,26,174),(9,244,82,201),(10,243,83,208),(11,242,84,207),(12,241,85,206),(13,248,86,205),(14,247,87,204),(15,246,88,203),(16,245,81,202),(17,231,65,209),(18,230,66,216),(19,229,67,215),(20,228,68,214),(21,227,69,213),(22,226,70,212),(23,225,71,211),(24,232,72,210),(33,183,57,197),(34,182,58,196),(35,181,59,195),(36,180,60,194),(37,179,61,193),(38,178,62,200),(39,177,63,199),(40,184,64,198),(41,168,281,89),(42,167,282,96),(43,166,283,95),(44,165,284,94),(45,164,285,93),(46,163,286,92),(47,162,287,91),(48,161,288,90),(49,239,73,217),(50,238,74,224),(51,237,75,223),(52,236,76,222),(53,235,77,221),(54,234,78,220),(55,233,79,219),(56,240,80,218),(97,311,145,289),(98,310,146,296),(99,309,147,295),(100,308,148,294),(101,307,149,293),(102,306,150,292),(103,305,151,291),(104,312,152,290),(105,255,129,269),(106,254,130,268),(107,253,131,267),(108,252,132,266),(109,251,133,265),(110,250,134,272),(111,249,135,271),(112,256,136,270),(113,263,137,277),(114,262,138,276),(115,261,139,275),(116,260,140,274),(117,259,141,273),(118,258,142,280),(119,257,143,279),(120,264,144,278),(121,313,159,299),(122,320,160,298),(123,319,153,297),(124,318,154,304),(125,317,155,303),(126,316,156,302),(127,315,157,301),(128,314,158,300)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | Q8 | D4 | D8 | Q16 | C5×Q8 | C5×D4 | C5×D8 | C5×Q16 |
| kernel | C5×C8⋊2Q8 | C4×C40 | C5×C2.D8 | C5×C4⋊Q8 | C8⋊2Q8 | C4×C8 | C2.D8 | C4⋊Q8 | C40 | C2×C20 | C20 | C20 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 4 | 2 | 4 | 4 | 16 | 8 | 16 | 16 |
Matrix representation of C5×C8⋊2Q8 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 29 | 12 | 0 | 0 |
| 29 | 29 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 29 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 2 | 6 |
| 0 | 0 | 6 | 39 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,10,0,0,0,0,10],[29,29,0,0,12,29,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0],[29,12,0,0,12,12,0,0,0,0,2,6,0,0,6,39] >;
C5×C8⋊2Q8 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_2Q_8
% in TeX
G:=Group("C5xC8:2Q8"); // GroupNames label
G:=SmallGroup(320,1001);
// by ID
G=gap.SmallGroup(320,1001);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,288,1766,1276,10085,124]);
// 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,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations