direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C4.Q8, (C2×C8)⋊7C20, C8⋊8(C2×C20), C40⋊44(C2×C4), (C2×C40)⋊27C4, C4.1(Q8×C10), C20.93(C4⋊C4), (C2×C20).76Q8, C20.90(C2×Q8), (C2×C20).417D4, C2.3(C10×SD16), C23.56(C5×D4), C4.24(C22×C20), (C22×C8).13C10, (C22×C40).31C2, (C2×C10).46SD16, C10.83(C2×SD16), C22.47(D4×C10), (C2×C20).898C23, C20.241(C22×C4), (C2×C40).435C22, (C22×C10).217D4, C22.12(C5×SD16), (C22×C20).587C22, C4.13(C5×C4⋊C4), C2.11(C10×C4⋊C4), C10.90(C2×C4⋊C4), (C2×C4).72(C5×D4), (C10×C4⋊C4).42C2, (C2×C4⋊C4).13C10, (C2×C4).18(C5×Q8), C4⋊C4.41(C2×C10), (C2×C8).92(C2×C10), (C2×C4).74(C2×C20), C22.20(C5×C4⋊C4), (C2×C10).91(C4⋊C4), (C2×C20).508(C2×C4), (C2×C10).623(C2×D4), (C5×C4⋊C4).362C22, (C2×C4).73(C22×C10), (C22×C4).116(C2×C10), SmallGroup(320,926)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C4.Q8
G = < a,b,c,d | a10=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 194 in 130 conjugacy classes, 98 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C20, C20, C20, C2×C10, C2×C10, C4.Q8, C2×C4⋊C4, C22×C8, C40, C2×C20, C2×C20, C2×C20, C22×C10, C2×C4.Q8, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, C5×C4.Q8, C10×C4⋊C4, C22×C40, C10×C4.Q8
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C4.Q8, C2×C4⋊C4, C2×SD16, C2×C20, C5×D4, C5×Q8, C22×C10, C2×C4.Q8, C5×C4⋊C4, C5×SD16, C22×C20, D4×C10, Q8×C10, C5×C4.Q8, C10×C4⋊C4, C10×SD16, C10×C4.Q8
►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 83 149 129)(2 84 150 130)(3 85 141 121)(4 86 142 122)(5 87 143 123)(6 88 144 124)(7 89 145 125)(8 90 146 126)(9 81 147 127)(10 82 148 128)(11 60 45 28)(12 51 46 29)(13 52 47 30)(14 53 48 21)(15 54 49 22)(16 55 50 23)(17 56 41 24)(18 57 42 25)(19 58 43 26)(20 59 44 27)(31 319 71 64)(32 320 72 65)(33 311 73 66)(34 312 74 67)(35 313 75 68)(36 314 76 69)(37 315 77 70)(38 316 78 61)(39 317 79 62)(40 318 80 63)(91 138 109 116)(92 139 110 117)(93 140 101 118)(94 131 102 119)(95 132 103 120)(96 133 104 111)(97 134 105 112)(98 135 106 113)(99 136 107 114)(100 137 108 115)(151 208 189 162)(152 209 190 163)(153 210 181 164)(154 201 182 165)(155 202 183 166)(156 203 184 167)(157 204 185 168)(158 205 186 169)(159 206 187 170)(160 207 188 161)(171 218 225 196)(172 219 226 197)(173 220 227 198)(174 211 228 199)(175 212 229 200)(176 213 230 191)(177 214 221 192)(178 215 222 193)(179 216 223 194)(180 217 224 195)(231 242 269 288)(232 243 270 289)(233 244 261 290)(234 245 262 281)(235 246 263 282)(236 247 264 283)(237 248 265 284)(238 249 266 285)(239 250 267 286)(240 241 268 287)(251 276 305 298)(252 277 306 299)(253 278 307 300)(254 279 308 291)(255 280 309 292)(256 271 310 293)(257 272 301 294)(258 273 302 295)(259 274 303 296)(260 275 304 297)
(1 204 104 218 149 168 96 196)(2 205 105 219 150 169 97 197)(3 206 106 220 141 170 98 198)(4 207 107 211 142 161 99 199)(5 208 108 212 143 162 100 200)(6 209 109 213 144 163 91 191)(7 210 110 214 145 164 92 192)(8 201 101 215 146 165 93 193)(9 202 102 216 147 166 94 194)(10 203 103 217 148 167 95 195)(11 258 31 234 45 302 71 262)(12 259 32 235 46 303 72 263)(13 260 33 236 47 304 73 264)(14 251 34 237 48 305 74 265)(15 252 35 238 49 306 75 266)(16 253 36 239 50 307 76 267)(17 254 37 240 41 308 77 268)(18 255 38 231 42 309 78 269)(19 256 39 232 43 310 79 270)(20 257 40 233 44 301 80 261)(21 298 67 284 53 276 312 248)(22 299 68 285 54 277 313 249)(23 300 69 286 55 278 314 250)(24 291 70 287 56 279 315 241)(25 292 61 288 57 280 316 242)(26 293 62 289 58 271 317 243)(27 294 63 290 59 272 318 244)(28 295 64 281 60 273 319 245)(29 296 65 282 51 274 320 246)(30 297 66 283 52 275 311 247)(81 183 119 223 127 155 131 179)(82 184 120 224 128 156 132 180)(83 185 111 225 129 157 133 171)(84 186 112 226 130 158 134 172)(85 187 113 227 121 159 135 173)(86 188 114 228 122 160 136 174)(87 189 115 229 123 151 137 175)(88 190 116 230 124 152 138 176)(89 181 117 221 125 153 139 177)(90 182 118 222 126 154 140 178)
(1 237 133 276)(2 238 134 277)(3 239 135 278)(4 240 136 279)(5 231 137 280)(6 232 138 271)(7 233 139 272)(8 234 140 273)(9 235 131 274)(10 236 132 275)(11 154 64 193)(12 155 65 194)(13 156 66 195)(14 157 67 196)(15 158 68 197)(16 159 69 198)(17 160 70 199)(18 151 61 200)(19 152 62 191)(20 153 63 192)(21 204 74 171)(22 205 75 172)(23 206 76 173)(24 207 77 174)(25 208 78 175)(26 209 79 176)(27 210 80 177)(28 201 71 178)(29 202 72 179)(30 203 73 180)(31 222 60 165)(32 223 51 166)(33 224 52 167)(34 225 53 168)(35 226 54 169)(36 227 55 170)(37 228 56 161)(38 229 57 162)(39 230 58 163)(40 221 59 164)(41 188 315 211)(42 189 316 212)(43 190 317 213)(44 181 318 214)(45 182 319 215)(46 183 320 216)(47 184 311 217)(48 185 312 218)(49 186 313 219)(50 187 314 220)(81 282 102 259)(82 283 103 260)(83 284 104 251)(84 285 105 252)(85 286 106 253)(86 287 107 254)(87 288 108 255)(88 289 109 256)(89 290 110 257)(90 281 101 258)(91 310 124 243)(92 301 125 244)(93 302 126 245)(94 303 127 246)(95 304 128 247)(96 305 129 248)(97 306 130 249)(98 307 121 250)(99 308 122 241)(100 309 123 242)(111 298 149 265)(112 299 150 266)(113 300 141 267)(114 291 142 268)(115 292 143 269)(116 293 144 270)(117 294 145 261)(118 295 146 262)(119 296 147 263)(120 297 148 264)
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), (1,83,149,129)(2,84,150,130)(3,85,141,121)(4,86,142,122)(5,87,143,123)(6,88,144,124)(7,89,145,125)(8,90,146,126)(9,81,147,127)(10,82,148,128)(11,60,45,28)(12,51,46,29)(13,52,47,30)(14,53,48,21)(15,54,49,22)(16,55,50,23)(17,56,41,24)(18,57,42,25)(19,58,43,26)(20,59,44,27)(31,319,71,64)(32,320,72,65)(33,311,73,66)(34,312,74,67)(35,313,75,68)(36,314,76,69)(37,315,77,70)(38,316,78,61)(39,317,79,62)(40,318,80,63)(91,138,109,116)(92,139,110,117)(93,140,101,118)(94,131,102,119)(95,132,103,120)(96,133,104,111)(97,134,105,112)(98,135,106,113)(99,136,107,114)(100,137,108,115)(151,208,189,162)(152,209,190,163)(153,210,181,164)(154,201,182,165)(155,202,183,166)(156,203,184,167)(157,204,185,168)(158,205,186,169)(159,206,187,170)(160,207,188,161)(171,218,225,196)(172,219,226,197)(173,220,227,198)(174,211,228,199)(175,212,229,200)(176,213,230,191)(177,214,221,192)(178,215,222,193)(179,216,223,194)(180,217,224,195)(231,242,269,288)(232,243,270,289)(233,244,261,290)(234,245,262,281)(235,246,263,282)(236,247,264,283)(237,248,265,284)(238,249,266,285)(239,250,267,286)(240,241,268,287)(251,276,305,298)(252,277,306,299)(253,278,307,300)(254,279,308,291)(255,280,309,292)(256,271,310,293)(257,272,301,294)(258,273,302,295)(259,274,303,296)(260,275,304,297), (1,204,104,218,149,168,96,196)(2,205,105,219,150,169,97,197)(3,206,106,220,141,170,98,198)(4,207,107,211,142,161,99,199)(5,208,108,212,143,162,100,200)(6,209,109,213,144,163,91,191)(7,210,110,214,145,164,92,192)(8,201,101,215,146,165,93,193)(9,202,102,216,147,166,94,194)(10,203,103,217,148,167,95,195)(11,258,31,234,45,302,71,262)(12,259,32,235,46,303,72,263)(13,260,33,236,47,304,73,264)(14,251,34,237,48,305,74,265)(15,252,35,238,49,306,75,266)(16,253,36,239,50,307,76,267)(17,254,37,240,41,308,77,268)(18,255,38,231,42,309,78,269)(19,256,39,232,43,310,79,270)(20,257,40,233,44,301,80,261)(21,298,67,284,53,276,312,248)(22,299,68,285,54,277,313,249)(23,300,69,286,55,278,314,250)(24,291,70,287,56,279,315,241)(25,292,61,288,57,280,316,242)(26,293,62,289,58,271,317,243)(27,294,63,290,59,272,318,244)(28,295,64,281,60,273,319,245)(29,296,65,282,51,274,320,246)(30,297,66,283,52,275,311,247)(81,183,119,223,127,155,131,179)(82,184,120,224,128,156,132,180)(83,185,111,225,129,157,133,171)(84,186,112,226,130,158,134,172)(85,187,113,227,121,159,135,173)(86,188,114,228,122,160,136,174)(87,189,115,229,123,151,137,175)(88,190,116,230,124,152,138,176)(89,181,117,221,125,153,139,177)(90,182,118,222,126,154,140,178), (1,237,133,276)(2,238,134,277)(3,239,135,278)(4,240,136,279)(5,231,137,280)(6,232,138,271)(7,233,139,272)(8,234,140,273)(9,235,131,274)(10,236,132,275)(11,154,64,193)(12,155,65,194)(13,156,66,195)(14,157,67,196)(15,158,68,197)(16,159,69,198)(17,160,70,199)(18,151,61,200)(19,152,62,191)(20,153,63,192)(21,204,74,171)(22,205,75,172)(23,206,76,173)(24,207,77,174)(25,208,78,175)(26,209,79,176)(27,210,80,177)(28,201,71,178)(29,202,72,179)(30,203,73,180)(31,222,60,165)(32,223,51,166)(33,224,52,167)(34,225,53,168)(35,226,54,169)(36,227,55,170)(37,228,56,161)(38,229,57,162)(39,230,58,163)(40,221,59,164)(41,188,315,211)(42,189,316,212)(43,190,317,213)(44,181,318,214)(45,182,319,215)(46,183,320,216)(47,184,311,217)(48,185,312,218)(49,186,313,219)(50,187,314,220)(81,282,102,259)(82,283,103,260)(83,284,104,251)(84,285,105,252)(85,286,106,253)(86,287,107,254)(87,288,108,255)(88,289,109,256)(89,290,110,257)(90,281,101,258)(91,310,124,243)(92,301,125,244)(93,302,126,245)(94,303,127,246)(95,304,128,247)(96,305,129,248)(97,306,130,249)(98,307,121,250)(99,308,122,241)(100,309,123,242)(111,298,149,265)(112,299,150,266)(113,300,141,267)(114,291,142,268)(115,292,143,269)(116,293,144,270)(117,294,145,261)(118,295,146,262)(119,296,147,263)(120,297,148,264)>;
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), (1,83,149,129)(2,84,150,130)(3,85,141,121)(4,86,142,122)(5,87,143,123)(6,88,144,124)(7,89,145,125)(8,90,146,126)(9,81,147,127)(10,82,148,128)(11,60,45,28)(12,51,46,29)(13,52,47,30)(14,53,48,21)(15,54,49,22)(16,55,50,23)(17,56,41,24)(18,57,42,25)(19,58,43,26)(20,59,44,27)(31,319,71,64)(32,320,72,65)(33,311,73,66)(34,312,74,67)(35,313,75,68)(36,314,76,69)(37,315,77,70)(38,316,78,61)(39,317,79,62)(40,318,80,63)(91,138,109,116)(92,139,110,117)(93,140,101,118)(94,131,102,119)(95,132,103,120)(96,133,104,111)(97,134,105,112)(98,135,106,113)(99,136,107,114)(100,137,108,115)(151,208,189,162)(152,209,190,163)(153,210,181,164)(154,201,182,165)(155,202,183,166)(156,203,184,167)(157,204,185,168)(158,205,186,169)(159,206,187,170)(160,207,188,161)(171,218,225,196)(172,219,226,197)(173,220,227,198)(174,211,228,199)(175,212,229,200)(176,213,230,191)(177,214,221,192)(178,215,222,193)(179,216,223,194)(180,217,224,195)(231,242,269,288)(232,243,270,289)(233,244,261,290)(234,245,262,281)(235,246,263,282)(236,247,264,283)(237,248,265,284)(238,249,266,285)(239,250,267,286)(240,241,268,287)(251,276,305,298)(252,277,306,299)(253,278,307,300)(254,279,308,291)(255,280,309,292)(256,271,310,293)(257,272,301,294)(258,273,302,295)(259,274,303,296)(260,275,304,297), (1,204,104,218,149,168,96,196)(2,205,105,219,150,169,97,197)(3,206,106,220,141,170,98,198)(4,207,107,211,142,161,99,199)(5,208,108,212,143,162,100,200)(6,209,109,213,144,163,91,191)(7,210,110,214,145,164,92,192)(8,201,101,215,146,165,93,193)(9,202,102,216,147,166,94,194)(10,203,103,217,148,167,95,195)(11,258,31,234,45,302,71,262)(12,259,32,235,46,303,72,263)(13,260,33,236,47,304,73,264)(14,251,34,237,48,305,74,265)(15,252,35,238,49,306,75,266)(16,253,36,239,50,307,76,267)(17,254,37,240,41,308,77,268)(18,255,38,231,42,309,78,269)(19,256,39,232,43,310,79,270)(20,257,40,233,44,301,80,261)(21,298,67,284,53,276,312,248)(22,299,68,285,54,277,313,249)(23,300,69,286,55,278,314,250)(24,291,70,287,56,279,315,241)(25,292,61,288,57,280,316,242)(26,293,62,289,58,271,317,243)(27,294,63,290,59,272,318,244)(28,295,64,281,60,273,319,245)(29,296,65,282,51,274,320,246)(30,297,66,283,52,275,311,247)(81,183,119,223,127,155,131,179)(82,184,120,224,128,156,132,180)(83,185,111,225,129,157,133,171)(84,186,112,226,130,158,134,172)(85,187,113,227,121,159,135,173)(86,188,114,228,122,160,136,174)(87,189,115,229,123,151,137,175)(88,190,116,230,124,152,138,176)(89,181,117,221,125,153,139,177)(90,182,118,222,126,154,140,178), (1,237,133,276)(2,238,134,277)(3,239,135,278)(4,240,136,279)(5,231,137,280)(6,232,138,271)(7,233,139,272)(8,234,140,273)(9,235,131,274)(10,236,132,275)(11,154,64,193)(12,155,65,194)(13,156,66,195)(14,157,67,196)(15,158,68,197)(16,159,69,198)(17,160,70,199)(18,151,61,200)(19,152,62,191)(20,153,63,192)(21,204,74,171)(22,205,75,172)(23,206,76,173)(24,207,77,174)(25,208,78,175)(26,209,79,176)(27,210,80,177)(28,201,71,178)(29,202,72,179)(30,203,73,180)(31,222,60,165)(32,223,51,166)(33,224,52,167)(34,225,53,168)(35,226,54,169)(36,227,55,170)(37,228,56,161)(38,229,57,162)(39,230,58,163)(40,221,59,164)(41,188,315,211)(42,189,316,212)(43,190,317,213)(44,181,318,214)(45,182,319,215)(46,183,320,216)(47,184,311,217)(48,185,312,218)(49,186,313,219)(50,187,314,220)(81,282,102,259)(82,283,103,260)(83,284,104,251)(84,285,105,252)(85,286,106,253)(86,287,107,254)(87,288,108,255)(88,289,109,256)(89,290,110,257)(90,281,101,258)(91,310,124,243)(92,301,125,244)(93,302,126,245)(94,303,127,246)(95,304,128,247)(96,305,129,248)(97,306,130,249)(98,307,121,250)(99,308,122,241)(100,309,123,242)(111,298,149,265)(112,299,150,266)(113,300,141,267)(114,291,142,268)(115,292,143,269)(116,293,144,270)(117,294,145,261)(118,295,146,262)(119,296,147,263)(120,297,148,264) );
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)], [(1,83,149,129),(2,84,150,130),(3,85,141,121),(4,86,142,122),(5,87,143,123),(6,88,144,124),(7,89,145,125),(8,90,146,126),(9,81,147,127),(10,82,148,128),(11,60,45,28),(12,51,46,29),(13,52,47,30),(14,53,48,21),(15,54,49,22),(16,55,50,23),(17,56,41,24),(18,57,42,25),(19,58,43,26),(20,59,44,27),(31,319,71,64),(32,320,72,65),(33,311,73,66),(34,312,74,67),(35,313,75,68),(36,314,76,69),(37,315,77,70),(38,316,78,61),(39,317,79,62),(40,318,80,63),(91,138,109,116),(92,139,110,117),(93,140,101,118),(94,131,102,119),(95,132,103,120),(96,133,104,111),(97,134,105,112),(98,135,106,113),(99,136,107,114),(100,137,108,115),(151,208,189,162),(152,209,190,163),(153,210,181,164),(154,201,182,165),(155,202,183,166),(156,203,184,167),(157,204,185,168),(158,205,186,169),(159,206,187,170),(160,207,188,161),(171,218,225,196),(172,219,226,197),(173,220,227,198),(174,211,228,199),(175,212,229,200),(176,213,230,191),(177,214,221,192),(178,215,222,193),(179,216,223,194),(180,217,224,195),(231,242,269,288),(232,243,270,289),(233,244,261,290),(234,245,262,281),(235,246,263,282),(236,247,264,283),(237,248,265,284),(238,249,266,285),(239,250,267,286),(240,241,268,287),(251,276,305,298),(252,277,306,299),(253,278,307,300),(254,279,308,291),(255,280,309,292),(256,271,310,293),(257,272,301,294),(258,273,302,295),(259,274,303,296),(260,275,304,297)], [(1,204,104,218,149,168,96,196),(2,205,105,219,150,169,97,197),(3,206,106,220,141,170,98,198),(4,207,107,211,142,161,99,199),(5,208,108,212,143,162,100,200),(6,209,109,213,144,163,91,191),(7,210,110,214,145,164,92,192),(8,201,101,215,146,165,93,193),(9,202,102,216,147,166,94,194),(10,203,103,217,148,167,95,195),(11,258,31,234,45,302,71,262),(12,259,32,235,46,303,72,263),(13,260,33,236,47,304,73,264),(14,251,34,237,48,305,74,265),(15,252,35,238,49,306,75,266),(16,253,36,239,50,307,76,267),(17,254,37,240,41,308,77,268),(18,255,38,231,42,309,78,269),(19,256,39,232,43,310,79,270),(20,257,40,233,44,301,80,261),(21,298,67,284,53,276,312,248),(22,299,68,285,54,277,313,249),(23,300,69,286,55,278,314,250),(24,291,70,287,56,279,315,241),(25,292,61,288,57,280,316,242),(26,293,62,289,58,271,317,243),(27,294,63,290,59,272,318,244),(28,295,64,281,60,273,319,245),(29,296,65,282,51,274,320,246),(30,297,66,283,52,275,311,247),(81,183,119,223,127,155,131,179),(82,184,120,224,128,156,132,180),(83,185,111,225,129,157,133,171),(84,186,112,226,130,158,134,172),(85,187,113,227,121,159,135,173),(86,188,114,228,122,160,136,174),(87,189,115,229,123,151,137,175),(88,190,116,230,124,152,138,176),(89,181,117,221,125,153,139,177),(90,182,118,222,126,154,140,178)], [(1,237,133,276),(2,238,134,277),(3,239,135,278),(4,240,136,279),(5,231,137,280),(6,232,138,271),(7,233,139,272),(8,234,140,273),(9,235,131,274),(10,236,132,275),(11,154,64,193),(12,155,65,194),(13,156,66,195),(14,157,67,196),(15,158,68,197),(16,159,69,198),(17,160,70,199),(18,151,61,200),(19,152,62,191),(20,153,63,192),(21,204,74,171),(22,205,75,172),(23,206,76,173),(24,207,77,174),(25,208,78,175),(26,209,79,176),(27,210,80,177),(28,201,71,178),(29,202,72,179),(30,203,73,180),(31,222,60,165),(32,223,51,166),(33,224,52,167),(34,225,53,168),(35,226,54,169),(36,227,55,170),(37,228,56,161),(38,229,57,162),(39,230,58,163),(40,221,59,164),(41,188,315,211),(42,189,316,212),(43,190,317,213),(44,181,318,214),(45,182,319,215),(46,183,320,216),(47,184,311,217),(48,185,312,218),(49,186,313,219),(50,187,314,220),(81,282,102,259),(82,283,103,260),(83,284,104,251),(84,285,105,252),(85,286,106,253),(86,287,107,254),(87,288,108,255),(88,289,109,256),(89,290,110,257),(90,281,101,258),(91,310,124,243),(92,301,125,244),(93,302,126,245),(94,303,127,246),(95,304,128,247),(96,305,129,248),(97,306,130,249),(98,307,121,250),(99,308,122,241),(100,309,123,242),(111,298,149,265),(112,299,150,266),(113,300,141,267),(114,291,142,268),(115,292,143,269),(116,293,144,270),(117,294,145,261),(118,295,146,262),(119,296,147,263),(120,297,148,264)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AV | 40A | ··· | 40AF |
| order | 1 | 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 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D4 | Q8 | D4 | SD16 | C5×D4 | C5×Q8 | C5×D4 | C5×SD16 |
| kernel | C10×C4.Q8 | C5×C4.Q8 | C10×C4⋊C4 | C22×C40 | C2×C40 | C2×C4.Q8 | C4.Q8 | C2×C4⋊C4 | C22×C8 | C2×C8 | C2×C20 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 16 | 8 | 4 | 32 | 1 | 2 | 1 | 8 | 4 | 8 | 4 | 32 |
Matrix representation of C10×C4.Q8 ►in GL6(𝔽41)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 15 |
| 0 | 0 | 0 | 0 | 26 | 26 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 29 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 15 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 34 |
| 0 | 0 | 0 | 0 | 34 | 27 |
G:=sub<GL(6,GF(41))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,26,26,0,0,0,0,15,26],[12,29,0,0,0,0,29,29,0,0,0,0,0,0,26,15,0,0,0,0,15,15,0,0,0,0,0,0,14,34,0,0,0,0,34,27] >;
C10×C4.Q8 in GAP, Magma, Sage, TeX
C_{10}\times C_4.Q_8 % in TeX
G:=Group("C10xC4.Q8"); // GroupNames label
G:=SmallGroup(320,926);
// by ID
G=gap.SmallGroup(320,926);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=b^-1*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^3>;
// generators/relations