direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C4⋊Q8, C20⋊7(C2×Q8), C4⋊1(Q8×C10), (C2×C20)⋊15Q8, C4.14(D4×C10), (C2×C20).431D4, C20.321(C2×D4), (C2×C42).19C10, C42.89(C2×C10), C22.64(D4×C10), (C22×Q8).8C10, C10.59(C22×Q8), C22.19(Q8×C10), (C2×C10).351C24, (C2×C20).962C23, (C4×C20).374C22, C10.187(C22×D4), C23.73(C22×C10), C22.25(C23×C10), (Q8×C10).267C22, (C22×C10).470C23, (C22×C20).445C22, (C2×C4)⋊4(C5×Q8), C2.5(Q8×C2×C10), (C2×C4×C20).42C2, C2.11(D4×C2×C10), (C2×C4).87(C5×D4), (C2×C4⋊C4).19C10, (C10×C4⋊C4).48C2, (Q8×C2×C10).18C2, C4⋊C4.65(C2×C10), (C2×C10).685(C2×D4), (C2×Q8).54(C2×C10), (C2×C10).117(C2×Q8), (C5×C4⋊C4).388C22, (C22×C4).55(C2×C10), (C2×C4).18(C22×C10), SmallGroup(320,1533)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C4⋊Q8
G = < a,b,c,d | a10=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 370 in 290 conjugacy classes, 210 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C2×C20, C2×C20, C5×Q8, C22×C10, C2×C4⋊Q8, C4×C20, C5×C4⋊C4, C22×C20, C22×C20, Q8×C10, Q8×C10, C2×C4×C20, C10×C4⋊C4, C5×C4⋊Q8, Q8×C2×C10, C10×C4⋊Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C24, C2×C10, C4⋊Q8, C22×D4, C22×Q8, C5×D4, C5×Q8, C22×C10, C2×C4⋊Q8, D4×C10, Q8×C10, C23×C10, C5×C4⋊Q8, D4×C2×C10, Q8×C2×C10, 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)
(1 176 66 154)(2 177 67 155)(3 178 68 156)(4 179 69 157)(5 180 70 158)(6 171 61 159)(7 172 62 160)(8 173 63 151)(9 174 64 152)(10 175 65 153)(11 203 319 221)(12 204 320 222)(13 205 311 223)(14 206 312 224)(15 207 313 225)(16 208 314 226)(17 209 315 227)(18 210 316 228)(19 201 317 229)(20 202 318 230)(21 199 40 211)(22 200 31 212)(23 191 32 213)(24 192 33 214)(25 193 34 215)(26 194 35 216)(27 195 36 217)(28 196 37 218)(29 197 38 219)(30 198 39 220)(41 189 59 161)(42 190 60 162)(43 181 51 163)(44 182 52 164)(45 183 53 165)(46 184 54 166)(47 185 55 167)(48 186 56 168)(49 187 57 169)(50 188 58 170)(71 133 93 111)(72 134 94 112)(73 135 95 113)(74 136 96 114)(75 137 97 115)(76 138 98 116)(77 139 99 117)(78 140 100 118)(79 131 91 119)(80 132 92 120)(81 149 109 121)(82 150 110 122)(83 141 101 123)(84 142 102 124)(85 143 103 125)(86 144 104 126)(87 145 105 127)(88 146 106 128)(89 147 107 129)(90 148 108 130)(231 271 253 293)(232 272 254 294)(233 273 255 295)(234 274 256 296)(235 275 257 297)(236 276 258 298)(237 277 259 299)(238 278 260 300)(239 279 251 291)(240 280 252 292)(241 281 269 309)(242 282 270 310)(243 283 261 301)(244 284 262 302)(245 285 263 303)(246 286 264 304)(247 287 265 305)(248 288 266 306)(249 289 267 307)(250 290 268 308)
(1 119 56 141)(2 120 57 142)(3 111 58 143)(4 112 59 144)(5 113 60 145)(6 114 51 146)(7 115 52 147)(8 116 53 148)(9 117 54 149)(10 118 55 150)(11 266 40 234)(12 267 31 235)(13 268 32 236)(14 269 33 237)(15 270 34 238)(16 261 35 239)(17 262 36 240)(18 263 37 231)(19 264 38 232)(20 265 39 233)(21 256 319 248)(22 257 320 249)(23 258 311 250)(24 259 312 241)(25 260 313 242)(26 251 314 243)(27 252 315 244)(28 253 316 245)(29 254 317 246)(30 255 318 247)(41 126 69 134)(42 127 70 135)(43 128 61 136)(44 129 62 137)(45 130 63 138)(46 121 64 139)(47 122 65 140)(48 123 66 131)(49 124 67 132)(50 125 68 133)(71 170 103 178)(72 161 104 179)(73 162 105 180)(74 163 106 171)(75 164 107 172)(76 165 108 173)(77 166 109 174)(78 167 110 175)(79 168 101 176)(80 169 102 177)(81 152 99 184)(82 153 100 185)(83 154 91 186)(84 155 92 187)(85 156 93 188)(86 157 94 189)(87 158 95 190)(88 159 96 181)(89 160 97 182)(90 151 98 183)(191 298 223 290)(192 299 224 281)(193 300 225 282)(194 291 226 283)(195 292 227 284)(196 293 228 285)(197 294 229 286)(198 295 230 287)(199 296 221 288)(200 297 222 289)(201 304 219 272)(202 305 220 273)(203 306 211 274)(204 307 212 275)(205 308 213 276)(206 309 214 277)(207 310 215 278)(208 301 216 279)(209 302 217 280)(210 303 218 271)
(1 199 56 221)(2 200 57 222)(3 191 58 223)(4 192 59 224)(5 193 60 225)(6 194 51 226)(7 195 52 227)(8 196 53 228)(9 197 54 229)(10 198 55 230)(11 154 40 186)(12 155 31 187)(13 156 32 188)(14 157 33 189)(15 158 34 190)(16 159 35 181)(17 160 36 182)(18 151 37 183)(19 152 38 184)(20 153 39 185)(21 168 319 176)(22 169 320 177)(23 170 311 178)(24 161 312 179)(25 162 313 180)(26 163 314 171)(27 164 315 172)(28 165 316 173)(29 166 317 174)(30 167 318 175)(41 206 69 214)(42 207 70 215)(43 208 61 216)(44 209 62 217)(45 210 63 218)(46 201 64 219)(47 202 65 220)(48 203 66 211)(49 204 67 212)(50 205 68 213)(71 250 103 258)(72 241 104 259)(73 242 105 260)(74 243 106 251)(75 244 107 252)(76 245 108 253)(77 246 109 254)(78 247 110 255)(79 248 101 256)(80 249 102 257)(81 232 99 264)(82 233 100 265)(83 234 91 266)(84 235 92 267)(85 236 93 268)(86 237 94 269)(87 238 95 270)(88 239 96 261)(89 240 97 262)(90 231 98 263)(111 290 143 298)(112 281 144 299)(113 282 145 300)(114 283 146 291)(115 284 147 292)(116 285 148 293)(117 286 149 294)(118 287 150 295)(119 288 141 296)(120 289 142 297)(121 272 139 304)(122 273 140 305)(123 274 131 306)(124 275 132 307)(125 276 133 308)(126 277 134 309)(127 278 135 310)(128 279 136 301)(129 280 137 302)(130 271 138 303)
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), (1,176,66,154)(2,177,67,155)(3,178,68,156)(4,179,69,157)(5,180,70,158)(6,171,61,159)(7,172,62,160)(8,173,63,151)(9,174,64,152)(10,175,65,153)(11,203,319,221)(12,204,320,222)(13,205,311,223)(14,206,312,224)(15,207,313,225)(16,208,314,226)(17,209,315,227)(18,210,316,228)(19,201,317,229)(20,202,318,230)(21,199,40,211)(22,200,31,212)(23,191,32,213)(24,192,33,214)(25,193,34,215)(26,194,35,216)(27,195,36,217)(28,196,37,218)(29,197,38,219)(30,198,39,220)(41,189,59,161)(42,190,60,162)(43,181,51,163)(44,182,52,164)(45,183,53,165)(46,184,54,166)(47,185,55,167)(48,186,56,168)(49,187,57,169)(50,188,58,170)(71,133,93,111)(72,134,94,112)(73,135,95,113)(74,136,96,114)(75,137,97,115)(76,138,98,116)(77,139,99,117)(78,140,100,118)(79,131,91,119)(80,132,92,120)(81,149,109,121)(82,150,110,122)(83,141,101,123)(84,142,102,124)(85,143,103,125)(86,144,104,126)(87,145,105,127)(88,146,106,128)(89,147,107,129)(90,148,108,130)(231,271,253,293)(232,272,254,294)(233,273,255,295)(234,274,256,296)(235,275,257,297)(236,276,258,298)(237,277,259,299)(238,278,260,300)(239,279,251,291)(240,280,252,292)(241,281,269,309)(242,282,270,310)(243,283,261,301)(244,284,262,302)(245,285,263,303)(246,286,264,304)(247,287,265,305)(248,288,266,306)(249,289,267,307)(250,290,268,308), (1,119,56,141)(2,120,57,142)(3,111,58,143)(4,112,59,144)(5,113,60,145)(6,114,51,146)(7,115,52,147)(8,116,53,148)(9,117,54,149)(10,118,55,150)(11,266,40,234)(12,267,31,235)(13,268,32,236)(14,269,33,237)(15,270,34,238)(16,261,35,239)(17,262,36,240)(18,263,37,231)(19,264,38,232)(20,265,39,233)(21,256,319,248)(22,257,320,249)(23,258,311,250)(24,259,312,241)(25,260,313,242)(26,251,314,243)(27,252,315,244)(28,253,316,245)(29,254,317,246)(30,255,318,247)(41,126,69,134)(42,127,70,135)(43,128,61,136)(44,129,62,137)(45,130,63,138)(46,121,64,139)(47,122,65,140)(48,123,66,131)(49,124,67,132)(50,125,68,133)(71,170,103,178)(72,161,104,179)(73,162,105,180)(74,163,106,171)(75,164,107,172)(76,165,108,173)(77,166,109,174)(78,167,110,175)(79,168,101,176)(80,169,102,177)(81,152,99,184)(82,153,100,185)(83,154,91,186)(84,155,92,187)(85,156,93,188)(86,157,94,189)(87,158,95,190)(88,159,96,181)(89,160,97,182)(90,151,98,183)(191,298,223,290)(192,299,224,281)(193,300,225,282)(194,291,226,283)(195,292,227,284)(196,293,228,285)(197,294,229,286)(198,295,230,287)(199,296,221,288)(200,297,222,289)(201,304,219,272)(202,305,220,273)(203,306,211,274)(204,307,212,275)(205,308,213,276)(206,309,214,277)(207,310,215,278)(208,301,216,279)(209,302,217,280)(210,303,218,271), (1,199,56,221)(2,200,57,222)(3,191,58,223)(4,192,59,224)(5,193,60,225)(6,194,51,226)(7,195,52,227)(8,196,53,228)(9,197,54,229)(10,198,55,230)(11,154,40,186)(12,155,31,187)(13,156,32,188)(14,157,33,189)(15,158,34,190)(16,159,35,181)(17,160,36,182)(18,151,37,183)(19,152,38,184)(20,153,39,185)(21,168,319,176)(22,169,320,177)(23,170,311,178)(24,161,312,179)(25,162,313,180)(26,163,314,171)(27,164,315,172)(28,165,316,173)(29,166,317,174)(30,167,318,175)(41,206,69,214)(42,207,70,215)(43,208,61,216)(44,209,62,217)(45,210,63,218)(46,201,64,219)(47,202,65,220)(48,203,66,211)(49,204,67,212)(50,205,68,213)(71,250,103,258)(72,241,104,259)(73,242,105,260)(74,243,106,251)(75,244,107,252)(76,245,108,253)(77,246,109,254)(78,247,110,255)(79,248,101,256)(80,249,102,257)(81,232,99,264)(82,233,100,265)(83,234,91,266)(84,235,92,267)(85,236,93,268)(86,237,94,269)(87,238,95,270)(88,239,96,261)(89,240,97,262)(90,231,98,263)(111,290,143,298)(112,281,144,299)(113,282,145,300)(114,283,146,291)(115,284,147,292)(116,285,148,293)(117,286,149,294)(118,287,150,295)(119,288,141,296)(120,289,142,297)(121,272,139,304)(122,273,140,305)(123,274,131,306)(124,275,132,307)(125,276,133,308)(126,277,134,309)(127,278,135,310)(128,279,136,301)(129,280,137,302)(130,271,138,303)>;
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), (1,176,66,154)(2,177,67,155)(3,178,68,156)(4,179,69,157)(5,180,70,158)(6,171,61,159)(7,172,62,160)(8,173,63,151)(9,174,64,152)(10,175,65,153)(11,203,319,221)(12,204,320,222)(13,205,311,223)(14,206,312,224)(15,207,313,225)(16,208,314,226)(17,209,315,227)(18,210,316,228)(19,201,317,229)(20,202,318,230)(21,199,40,211)(22,200,31,212)(23,191,32,213)(24,192,33,214)(25,193,34,215)(26,194,35,216)(27,195,36,217)(28,196,37,218)(29,197,38,219)(30,198,39,220)(41,189,59,161)(42,190,60,162)(43,181,51,163)(44,182,52,164)(45,183,53,165)(46,184,54,166)(47,185,55,167)(48,186,56,168)(49,187,57,169)(50,188,58,170)(71,133,93,111)(72,134,94,112)(73,135,95,113)(74,136,96,114)(75,137,97,115)(76,138,98,116)(77,139,99,117)(78,140,100,118)(79,131,91,119)(80,132,92,120)(81,149,109,121)(82,150,110,122)(83,141,101,123)(84,142,102,124)(85,143,103,125)(86,144,104,126)(87,145,105,127)(88,146,106,128)(89,147,107,129)(90,148,108,130)(231,271,253,293)(232,272,254,294)(233,273,255,295)(234,274,256,296)(235,275,257,297)(236,276,258,298)(237,277,259,299)(238,278,260,300)(239,279,251,291)(240,280,252,292)(241,281,269,309)(242,282,270,310)(243,283,261,301)(244,284,262,302)(245,285,263,303)(246,286,264,304)(247,287,265,305)(248,288,266,306)(249,289,267,307)(250,290,268,308), (1,119,56,141)(2,120,57,142)(3,111,58,143)(4,112,59,144)(5,113,60,145)(6,114,51,146)(7,115,52,147)(8,116,53,148)(9,117,54,149)(10,118,55,150)(11,266,40,234)(12,267,31,235)(13,268,32,236)(14,269,33,237)(15,270,34,238)(16,261,35,239)(17,262,36,240)(18,263,37,231)(19,264,38,232)(20,265,39,233)(21,256,319,248)(22,257,320,249)(23,258,311,250)(24,259,312,241)(25,260,313,242)(26,251,314,243)(27,252,315,244)(28,253,316,245)(29,254,317,246)(30,255,318,247)(41,126,69,134)(42,127,70,135)(43,128,61,136)(44,129,62,137)(45,130,63,138)(46,121,64,139)(47,122,65,140)(48,123,66,131)(49,124,67,132)(50,125,68,133)(71,170,103,178)(72,161,104,179)(73,162,105,180)(74,163,106,171)(75,164,107,172)(76,165,108,173)(77,166,109,174)(78,167,110,175)(79,168,101,176)(80,169,102,177)(81,152,99,184)(82,153,100,185)(83,154,91,186)(84,155,92,187)(85,156,93,188)(86,157,94,189)(87,158,95,190)(88,159,96,181)(89,160,97,182)(90,151,98,183)(191,298,223,290)(192,299,224,281)(193,300,225,282)(194,291,226,283)(195,292,227,284)(196,293,228,285)(197,294,229,286)(198,295,230,287)(199,296,221,288)(200,297,222,289)(201,304,219,272)(202,305,220,273)(203,306,211,274)(204,307,212,275)(205,308,213,276)(206,309,214,277)(207,310,215,278)(208,301,216,279)(209,302,217,280)(210,303,218,271), (1,199,56,221)(2,200,57,222)(3,191,58,223)(4,192,59,224)(5,193,60,225)(6,194,51,226)(7,195,52,227)(8,196,53,228)(9,197,54,229)(10,198,55,230)(11,154,40,186)(12,155,31,187)(13,156,32,188)(14,157,33,189)(15,158,34,190)(16,159,35,181)(17,160,36,182)(18,151,37,183)(19,152,38,184)(20,153,39,185)(21,168,319,176)(22,169,320,177)(23,170,311,178)(24,161,312,179)(25,162,313,180)(26,163,314,171)(27,164,315,172)(28,165,316,173)(29,166,317,174)(30,167,318,175)(41,206,69,214)(42,207,70,215)(43,208,61,216)(44,209,62,217)(45,210,63,218)(46,201,64,219)(47,202,65,220)(48,203,66,211)(49,204,67,212)(50,205,68,213)(71,250,103,258)(72,241,104,259)(73,242,105,260)(74,243,106,251)(75,244,107,252)(76,245,108,253)(77,246,109,254)(78,247,110,255)(79,248,101,256)(80,249,102,257)(81,232,99,264)(82,233,100,265)(83,234,91,266)(84,235,92,267)(85,236,93,268)(86,237,94,269)(87,238,95,270)(88,239,96,261)(89,240,97,262)(90,231,98,263)(111,290,143,298)(112,281,144,299)(113,282,145,300)(114,283,146,291)(115,284,147,292)(116,285,148,293)(117,286,149,294)(118,287,150,295)(119,288,141,296)(120,289,142,297)(121,272,139,304)(122,273,140,305)(123,274,131,306)(124,275,132,307)(125,276,133,308)(126,277,134,309)(127,278,135,310)(128,279,136,301)(129,280,137,302)(130,271,138,303) );
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)], [(1,176,66,154),(2,177,67,155),(3,178,68,156),(4,179,69,157),(5,180,70,158),(6,171,61,159),(7,172,62,160),(8,173,63,151),(9,174,64,152),(10,175,65,153),(11,203,319,221),(12,204,320,222),(13,205,311,223),(14,206,312,224),(15,207,313,225),(16,208,314,226),(17,209,315,227),(18,210,316,228),(19,201,317,229),(20,202,318,230),(21,199,40,211),(22,200,31,212),(23,191,32,213),(24,192,33,214),(25,193,34,215),(26,194,35,216),(27,195,36,217),(28,196,37,218),(29,197,38,219),(30,198,39,220),(41,189,59,161),(42,190,60,162),(43,181,51,163),(44,182,52,164),(45,183,53,165),(46,184,54,166),(47,185,55,167),(48,186,56,168),(49,187,57,169),(50,188,58,170),(71,133,93,111),(72,134,94,112),(73,135,95,113),(74,136,96,114),(75,137,97,115),(76,138,98,116),(77,139,99,117),(78,140,100,118),(79,131,91,119),(80,132,92,120),(81,149,109,121),(82,150,110,122),(83,141,101,123),(84,142,102,124),(85,143,103,125),(86,144,104,126),(87,145,105,127),(88,146,106,128),(89,147,107,129),(90,148,108,130),(231,271,253,293),(232,272,254,294),(233,273,255,295),(234,274,256,296),(235,275,257,297),(236,276,258,298),(237,277,259,299),(238,278,260,300),(239,279,251,291),(240,280,252,292),(241,281,269,309),(242,282,270,310),(243,283,261,301),(244,284,262,302),(245,285,263,303),(246,286,264,304),(247,287,265,305),(248,288,266,306),(249,289,267,307),(250,290,268,308)], [(1,119,56,141),(2,120,57,142),(3,111,58,143),(4,112,59,144),(5,113,60,145),(6,114,51,146),(7,115,52,147),(8,116,53,148),(9,117,54,149),(10,118,55,150),(11,266,40,234),(12,267,31,235),(13,268,32,236),(14,269,33,237),(15,270,34,238),(16,261,35,239),(17,262,36,240),(18,263,37,231),(19,264,38,232),(20,265,39,233),(21,256,319,248),(22,257,320,249),(23,258,311,250),(24,259,312,241),(25,260,313,242),(26,251,314,243),(27,252,315,244),(28,253,316,245),(29,254,317,246),(30,255,318,247),(41,126,69,134),(42,127,70,135),(43,128,61,136),(44,129,62,137),(45,130,63,138),(46,121,64,139),(47,122,65,140),(48,123,66,131),(49,124,67,132),(50,125,68,133),(71,170,103,178),(72,161,104,179),(73,162,105,180),(74,163,106,171),(75,164,107,172),(76,165,108,173),(77,166,109,174),(78,167,110,175),(79,168,101,176),(80,169,102,177),(81,152,99,184),(82,153,100,185),(83,154,91,186),(84,155,92,187),(85,156,93,188),(86,157,94,189),(87,158,95,190),(88,159,96,181),(89,160,97,182),(90,151,98,183),(191,298,223,290),(192,299,224,281),(193,300,225,282),(194,291,226,283),(195,292,227,284),(196,293,228,285),(197,294,229,286),(198,295,230,287),(199,296,221,288),(200,297,222,289),(201,304,219,272),(202,305,220,273),(203,306,211,274),(204,307,212,275),(205,308,213,276),(206,309,214,277),(207,310,215,278),(208,301,216,279),(209,302,217,280),(210,303,218,271)], [(1,199,56,221),(2,200,57,222),(3,191,58,223),(4,192,59,224),(5,193,60,225),(6,194,51,226),(7,195,52,227),(8,196,53,228),(9,197,54,229),(10,198,55,230),(11,154,40,186),(12,155,31,187),(13,156,32,188),(14,157,33,189),(15,158,34,190),(16,159,35,181),(17,160,36,182),(18,151,37,183),(19,152,38,184),(20,153,39,185),(21,168,319,176),(22,169,320,177),(23,170,311,178),(24,161,312,179),(25,162,313,180),(26,163,314,171),(27,164,315,172),(28,165,316,173),(29,166,317,174),(30,167,318,175),(41,206,69,214),(42,207,70,215),(43,208,61,216),(44,209,62,217),(45,210,63,218),(46,201,64,219),(47,202,65,220),(48,203,66,211),(49,204,67,212),(50,205,68,213),(71,250,103,258),(72,241,104,259),(73,242,105,260),(74,243,106,251),(75,244,107,252),(76,245,108,253),(77,246,109,254),(78,247,110,255),(79,248,101,256),(80,249,102,257),(81,232,99,264),(82,233,100,265),(83,234,91,266),(84,235,92,267),(85,236,93,268),(86,237,94,269),(87,238,95,270),(88,239,96,261),(89,240,97,262),(90,231,98,263),(111,290,143,298),(112,281,144,299),(113,282,145,300),(114,283,146,291),(115,284,147,292),(116,285,148,293),(117,286,149,294),(118,287,150,295),(119,288,141,296),(120,289,142,297),(121,272,139,304),(122,273,140,305),(123,274,131,306),(124,275,132,307),(125,276,133,308),(126,277,134,309),(127,278,135,310),(128,279,136,301),(129,280,137,302),(130,271,138,303)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AV | 20AW | ··· | 20CB |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | Q8 | C5×D4 | C5×Q8 |
| kernel | C10×C4⋊Q8 | C2×C4×C20 | C10×C4⋊C4 | C5×C4⋊Q8 | Q8×C2×C10 | C2×C4⋊Q8 | C2×C42 | C2×C4⋊C4 | C4⋊Q8 | C22×Q8 | C2×C20 | C2×C20 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 4 | 8 | 2 | 4 | 4 | 16 | 32 | 8 | 4 | 8 | 16 | 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 | 23 | 0 |
| 0 | 0 | 0 | 0 | 0 | 23 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 25 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 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 |
| 29 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 37 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 0 | 2 | 35 |
G:=sub<GL(6,GF(41))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,23,0,0,0,0,0,0,23],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,25,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[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],[29,12,0,0,0,0,12,12,0,0,0,0,0,0,40,0,0,0,0,0,37,1,0,0,0,0,0,0,6,2,0,0,0,0,2,35] >;
C10×C4⋊Q8 in GAP, Magma, Sage, TeX
C_{10}\times C_4\rtimes Q_8 % in TeX
G:=Group("C10xC4:Q8"); // GroupNames label
G:=SmallGroup(320,1533);
// by ID
G=gap.SmallGroup(320,1533);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446,856]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=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