direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×Q8⋊C4, Q8⋊3(C2×C20), (C2×Q8)⋊5C20, (Q8×C10)⋊25C4, C4.52(D4×C10), C2.1(C10×Q16), C20.459(C2×D4), (C2×C20).415D4, C4.2(C22×C20), (C22×C8).5C10, C10.48(C2×Q16), (C2×C10).20Q16, C2.2(C10×SD16), C23.55(C5×D4), C22.5(C5×Q16), (C22×C40).11C2, (C2×C10).45SD16, C10.82(C2×SD16), C22.42(D4×C10), (C22×Q8).4C10, (C2×C40).357C22, C20.206(C22×C4), (C2×C20).891C23, (C22×C10).216D4, C22.11(C5×SD16), C20.128(C22⋊C4), (Q8×C10).250C22, (C22×C20).582C22, (C5×Q8)⋊30(C2×C4), (C2×C4).69(C5×D4), (C10×C4⋊C4).41C2, (C2×C4⋊C4).12C10, (Q8×C2×C10).14C2, C4⋊C4.37(C2×C10), (C2×C8).60(C2×C10), (C2×C4).48(C2×C20), C4.13(C5×C22⋊C4), (C2×C20).442(C2×C4), (C2×C10).618(C2×D4), C2.18(C10×C22⋊C4), (C2×Q8).35(C2×C10), (C5×C4⋊C4).358C22, C10.147(C2×C22⋊C4), (C2×C4).66(C22×C10), C22.34(C5×C22⋊C4), (C22×C4).111(C2×C10), (C2×C10).203(C22⋊C4), SmallGroup(320,916)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×Q8⋊C4
G = < a,b,c,d | a10=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 242 in 162 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C20, C20, C20, C2×C10, C2×C10, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C2×Q8⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C22×C20, C22×C20, Q8×C10, Q8×C10, C5×Q8⋊C4, C10×C4⋊C4, C22×C40, Q8×C2×C10, C10×Q8⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C20, C2×C10, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C2×C20, C5×D4, C22×C10, C2×Q8⋊C4, C5×C22⋊C4, C5×SD16, C5×Q16, C22×C20, D4×C10, C5×Q8⋊C4, C10×C22⋊C4, C10×SD16, C10×Q16, C10×Q8⋊C4
►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)
(1 132 86 96)(2 133 87 97)(3 134 88 98)(4 135 89 99)(5 136 90 100)(6 137 81 91)(7 138 82 92)(8 139 83 93)(9 140 84 94)(10 131 85 95)(11 77 65 23)(12 78 66 24)(13 79 67 25)(14 80 68 26)(15 71 69 27)(16 72 70 28)(17 73 61 29)(18 74 62 30)(19 75 63 21)(20 76 64 22)(31 49 311 52)(32 50 312 53)(33 41 313 54)(34 42 314 55)(35 43 315 56)(36 44 316 57)(37 45 317 58)(38 46 318 59)(39 47 319 60)(40 48 320 51)(101 142 116 122)(102 143 117 123)(103 144 118 124)(104 145 119 125)(105 146 120 126)(106 147 111 127)(107 148 112 128)(108 149 113 129)(109 150 114 130)(110 141 115 121)(151 211 206 175)(152 212 207 176)(153 213 208 177)(154 214 209 178)(155 215 210 179)(156 216 201 180)(157 217 202 171)(158 218 203 172)(159 219 204 173)(160 220 205 174)(161 197 223 182)(162 198 224 183)(163 199 225 184)(164 200 226 185)(165 191 227 186)(166 192 228 187)(167 193 229 188)(168 194 230 189)(169 195 221 190)(170 196 222 181)(231 291 286 255)(232 292 287 256)(233 293 288 257)(234 294 289 258)(235 295 290 259)(236 296 281 260)(237 297 282 251)(238 298 283 252)(239 299 284 253)(240 300 285 254)(241 277 303 262)(242 278 304 263)(243 279 305 264)(244 280 306 265)(245 271 307 266)(246 272 308 267)(247 273 309 268)(248 274 310 269)(249 275 301 270)(250 276 302 261)
(1 202 86 157)(2 203 87 158)(3 204 88 159)(4 205 89 160)(5 206 90 151)(6 207 81 152)(7 208 82 153)(8 209 83 154)(9 210 84 155)(10 201 85 156)(11 280 65 265)(12 271 66 266)(13 272 67 267)(14 273 68 268)(15 274 69 269)(16 275 70 270)(17 276 61 261)(18 277 62 262)(19 278 63 263)(20 279 64 264)(21 304 75 242)(22 305 76 243)(23 306 77 244)(24 307 78 245)(25 308 79 246)(26 309 80 247)(27 310 71 248)(28 301 72 249)(29 302 73 250)(30 303 74 241)(31 260 311 296)(32 251 312 297)(33 252 313 298)(34 253 314 299)(35 254 315 300)(36 255 316 291)(37 256 317 292)(38 257 318 293)(39 258 319 294)(40 259 320 295)(41 283 54 238)(42 284 55 239)(43 285 56 240)(44 286 57 231)(45 287 58 232)(46 288 59 233)(47 289 60 234)(48 290 51 235)(49 281 52 236)(50 282 53 237)(91 176 137 212)(92 177 138 213)(93 178 139 214)(94 179 140 215)(95 180 131 216)(96 171 132 217)(97 172 133 218)(98 173 134 219)(99 174 135 220)(100 175 136 211)(101 186 116 191)(102 187 117 192)(103 188 118 193)(104 189 119 194)(105 190 120 195)(106 181 111 196)(107 182 112 197)(108 183 113 198)(109 184 114 199)(110 185 115 200)(121 164 141 226)(122 165 142 227)(123 166 143 228)(124 167 144 229)(125 168 145 230)(126 169 146 221)(127 170 147 222)(128 161 148 223)(129 162 149 224)(130 163 150 225)
(1 237 141 244)(2 238 142 245)(3 239 143 246)(4 240 144 247)(5 231 145 248)(6 232 146 249)(7 233 147 250)(8 234 148 241)(9 235 149 242)(10 236 150 243)(11 202 312 226)(12 203 313 227)(13 204 314 228)(14 205 315 229)(15 206 316 230)(16 207 317 221)(17 208 318 222)(18 209 319 223)(19 210 320 224)(20 201 311 225)(21 179 48 183)(22 180 49 184)(23 171 50 185)(24 172 41 186)(25 173 42 187)(26 174 43 188)(27 175 44 189)(28 176 45 190)(29 177 46 181)(30 178 47 182)(31 163 64 156)(32 164 65 157)(33 165 66 158)(34 166 67 159)(35 167 68 160)(36 168 69 151)(37 169 70 152)(38 170 61 153)(39 161 62 154)(40 162 63 155)(51 198 75 215)(52 199 76 216)(53 200 77 217)(54 191 78 218)(55 192 79 219)(56 193 80 220)(57 194 71 211)(58 195 72 212)(59 196 73 213)(60 197 74 214)(81 287 126 301)(82 288 127 302)(83 289 128 303)(84 290 129 304)(85 281 130 305)(86 282 121 306)(87 283 122 307)(88 284 123 308)(89 285 124 309)(90 286 125 310)(91 292 105 275)(92 293 106 276)(93 294 107 277)(94 295 108 278)(95 296 109 279)(96 297 110 280)(97 298 101 271)(98 299 102 272)(99 300 103 273)(100 291 104 274)(111 261 138 257)(112 262 139 258)(113 263 140 259)(114 264 131 260)(115 265 132 251)(116 266 133 252)(117 267 134 253)(118 268 135 254)(119 269 136 255)(120 270 137 256)
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), (1,132,86,96)(2,133,87,97)(3,134,88,98)(4,135,89,99)(5,136,90,100)(6,137,81,91)(7,138,82,92)(8,139,83,93)(9,140,84,94)(10,131,85,95)(11,77,65,23)(12,78,66,24)(13,79,67,25)(14,80,68,26)(15,71,69,27)(16,72,70,28)(17,73,61,29)(18,74,62,30)(19,75,63,21)(20,76,64,22)(31,49,311,52)(32,50,312,53)(33,41,313,54)(34,42,314,55)(35,43,315,56)(36,44,316,57)(37,45,317,58)(38,46,318,59)(39,47,319,60)(40,48,320,51)(101,142,116,122)(102,143,117,123)(103,144,118,124)(104,145,119,125)(105,146,120,126)(106,147,111,127)(107,148,112,128)(108,149,113,129)(109,150,114,130)(110,141,115,121)(151,211,206,175)(152,212,207,176)(153,213,208,177)(154,214,209,178)(155,215,210,179)(156,216,201,180)(157,217,202,171)(158,218,203,172)(159,219,204,173)(160,220,205,174)(161,197,223,182)(162,198,224,183)(163,199,225,184)(164,200,226,185)(165,191,227,186)(166,192,228,187)(167,193,229,188)(168,194,230,189)(169,195,221,190)(170,196,222,181)(231,291,286,255)(232,292,287,256)(233,293,288,257)(234,294,289,258)(235,295,290,259)(236,296,281,260)(237,297,282,251)(238,298,283,252)(239,299,284,253)(240,300,285,254)(241,277,303,262)(242,278,304,263)(243,279,305,264)(244,280,306,265)(245,271,307,266)(246,272,308,267)(247,273,309,268)(248,274,310,269)(249,275,301,270)(250,276,302,261), (1,202,86,157)(2,203,87,158)(3,204,88,159)(4,205,89,160)(5,206,90,151)(6,207,81,152)(7,208,82,153)(8,209,83,154)(9,210,84,155)(10,201,85,156)(11,280,65,265)(12,271,66,266)(13,272,67,267)(14,273,68,268)(15,274,69,269)(16,275,70,270)(17,276,61,261)(18,277,62,262)(19,278,63,263)(20,279,64,264)(21,304,75,242)(22,305,76,243)(23,306,77,244)(24,307,78,245)(25,308,79,246)(26,309,80,247)(27,310,71,248)(28,301,72,249)(29,302,73,250)(30,303,74,241)(31,260,311,296)(32,251,312,297)(33,252,313,298)(34,253,314,299)(35,254,315,300)(36,255,316,291)(37,256,317,292)(38,257,318,293)(39,258,319,294)(40,259,320,295)(41,283,54,238)(42,284,55,239)(43,285,56,240)(44,286,57,231)(45,287,58,232)(46,288,59,233)(47,289,60,234)(48,290,51,235)(49,281,52,236)(50,282,53,237)(91,176,137,212)(92,177,138,213)(93,178,139,214)(94,179,140,215)(95,180,131,216)(96,171,132,217)(97,172,133,218)(98,173,134,219)(99,174,135,220)(100,175,136,211)(101,186,116,191)(102,187,117,192)(103,188,118,193)(104,189,119,194)(105,190,120,195)(106,181,111,196)(107,182,112,197)(108,183,113,198)(109,184,114,199)(110,185,115,200)(121,164,141,226)(122,165,142,227)(123,166,143,228)(124,167,144,229)(125,168,145,230)(126,169,146,221)(127,170,147,222)(128,161,148,223)(129,162,149,224)(130,163,150,225), (1,237,141,244)(2,238,142,245)(3,239,143,246)(4,240,144,247)(5,231,145,248)(6,232,146,249)(7,233,147,250)(8,234,148,241)(9,235,149,242)(10,236,150,243)(11,202,312,226)(12,203,313,227)(13,204,314,228)(14,205,315,229)(15,206,316,230)(16,207,317,221)(17,208,318,222)(18,209,319,223)(19,210,320,224)(20,201,311,225)(21,179,48,183)(22,180,49,184)(23,171,50,185)(24,172,41,186)(25,173,42,187)(26,174,43,188)(27,175,44,189)(28,176,45,190)(29,177,46,181)(30,178,47,182)(31,163,64,156)(32,164,65,157)(33,165,66,158)(34,166,67,159)(35,167,68,160)(36,168,69,151)(37,169,70,152)(38,170,61,153)(39,161,62,154)(40,162,63,155)(51,198,75,215)(52,199,76,216)(53,200,77,217)(54,191,78,218)(55,192,79,219)(56,193,80,220)(57,194,71,211)(58,195,72,212)(59,196,73,213)(60,197,74,214)(81,287,126,301)(82,288,127,302)(83,289,128,303)(84,290,129,304)(85,281,130,305)(86,282,121,306)(87,283,122,307)(88,284,123,308)(89,285,124,309)(90,286,125,310)(91,292,105,275)(92,293,106,276)(93,294,107,277)(94,295,108,278)(95,296,109,279)(96,297,110,280)(97,298,101,271)(98,299,102,272)(99,300,103,273)(100,291,104,274)(111,261,138,257)(112,262,139,258)(113,263,140,259)(114,264,131,260)(115,265,132,251)(116,266,133,252)(117,267,134,253)(118,268,135,254)(119,269,136,255)(120,270,137,256)>;
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), (1,132,86,96)(2,133,87,97)(3,134,88,98)(4,135,89,99)(5,136,90,100)(6,137,81,91)(7,138,82,92)(8,139,83,93)(9,140,84,94)(10,131,85,95)(11,77,65,23)(12,78,66,24)(13,79,67,25)(14,80,68,26)(15,71,69,27)(16,72,70,28)(17,73,61,29)(18,74,62,30)(19,75,63,21)(20,76,64,22)(31,49,311,52)(32,50,312,53)(33,41,313,54)(34,42,314,55)(35,43,315,56)(36,44,316,57)(37,45,317,58)(38,46,318,59)(39,47,319,60)(40,48,320,51)(101,142,116,122)(102,143,117,123)(103,144,118,124)(104,145,119,125)(105,146,120,126)(106,147,111,127)(107,148,112,128)(108,149,113,129)(109,150,114,130)(110,141,115,121)(151,211,206,175)(152,212,207,176)(153,213,208,177)(154,214,209,178)(155,215,210,179)(156,216,201,180)(157,217,202,171)(158,218,203,172)(159,219,204,173)(160,220,205,174)(161,197,223,182)(162,198,224,183)(163,199,225,184)(164,200,226,185)(165,191,227,186)(166,192,228,187)(167,193,229,188)(168,194,230,189)(169,195,221,190)(170,196,222,181)(231,291,286,255)(232,292,287,256)(233,293,288,257)(234,294,289,258)(235,295,290,259)(236,296,281,260)(237,297,282,251)(238,298,283,252)(239,299,284,253)(240,300,285,254)(241,277,303,262)(242,278,304,263)(243,279,305,264)(244,280,306,265)(245,271,307,266)(246,272,308,267)(247,273,309,268)(248,274,310,269)(249,275,301,270)(250,276,302,261), (1,202,86,157)(2,203,87,158)(3,204,88,159)(4,205,89,160)(5,206,90,151)(6,207,81,152)(7,208,82,153)(8,209,83,154)(9,210,84,155)(10,201,85,156)(11,280,65,265)(12,271,66,266)(13,272,67,267)(14,273,68,268)(15,274,69,269)(16,275,70,270)(17,276,61,261)(18,277,62,262)(19,278,63,263)(20,279,64,264)(21,304,75,242)(22,305,76,243)(23,306,77,244)(24,307,78,245)(25,308,79,246)(26,309,80,247)(27,310,71,248)(28,301,72,249)(29,302,73,250)(30,303,74,241)(31,260,311,296)(32,251,312,297)(33,252,313,298)(34,253,314,299)(35,254,315,300)(36,255,316,291)(37,256,317,292)(38,257,318,293)(39,258,319,294)(40,259,320,295)(41,283,54,238)(42,284,55,239)(43,285,56,240)(44,286,57,231)(45,287,58,232)(46,288,59,233)(47,289,60,234)(48,290,51,235)(49,281,52,236)(50,282,53,237)(91,176,137,212)(92,177,138,213)(93,178,139,214)(94,179,140,215)(95,180,131,216)(96,171,132,217)(97,172,133,218)(98,173,134,219)(99,174,135,220)(100,175,136,211)(101,186,116,191)(102,187,117,192)(103,188,118,193)(104,189,119,194)(105,190,120,195)(106,181,111,196)(107,182,112,197)(108,183,113,198)(109,184,114,199)(110,185,115,200)(121,164,141,226)(122,165,142,227)(123,166,143,228)(124,167,144,229)(125,168,145,230)(126,169,146,221)(127,170,147,222)(128,161,148,223)(129,162,149,224)(130,163,150,225), (1,237,141,244)(2,238,142,245)(3,239,143,246)(4,240,144,247)(5,231,145,248)(6,232,146,249)(7,233,147,250)(8,234,148,241)(9,235,149,242)(10,236,150,243)(11,202,312,226)(12,203,313,227)(13,204,314,228)(14,205,315,229)(15,206,316,230)(16,207,317,221)(17,208,318,222)(18,209,319,223)(19,210,320,224)(20,201,311,225)(21,179,48,183)(22,180,49,184)(23,171,50,185)(24,172,41,186)(25,173,42,187)(26,174,43,188)(27,175,44,189)(28,176,45,190)(29,177,46,181)(30,178,47,182)(31,163,64,156)(32,164,65,157)(33,165,66,158)(34,166,67,159)(35,167,68,160)(36,168,69,151)(37,169,70,152)(38,170,61,153)(39,161,62,154)(40,162,63,155)(51,198,75,215)(52,199,76,216)(53,200,77,217)(54,191,78,218)(55,192,79,219)(56,193,80,220)(57,194,71,211)(58,195,72,212)(59,196,73,213)(60,197,74,214)(81,287,126,301)(82,288,127,302)(83,289,128,303)(84,290,129,304)(85,281,130,305)(86,282,121,306)(87,283,122,307)(88,284,123,308)(89,285,124,309)(90,286,125,310)(91,292,105,275)(92,293,106,276)(93,294,107,277)(94,295,108,278)(95,296,109,279)(96,297,110,280)(97,298,101,271)(98,299,102,272)(99,300,103,273)(100,291,104,274)(111,261,138,257)(112,262,139,258)(113,263,140,259)(114,264,131,260)(115,265,132,251)(116,266,133,252)(117,267,134,253)(118,268,135,254)(119,269,136,255)(120,270,137,256) );
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)], [(1,132,86,96),(2,133,87,97),(3,134,88,98),(4,135,89,99),(5,136,90,100),(6,137,81,91),(7,138,82,92),(8,139,83,93),(9,140,84,94),(10,131,85,95),(11,77,65,23),(12,78,66,24),(13,79,67,25),(14,80,68,26),(15,71,69,27),(16,72,70,28),(17,73,61,29),(18,74,62,30),(19,75,63,21),(20,76,64,22),(31,49,311,52),(32,50,312,53),(33,41,313,54),(34,42,314,55),(35,43,315,56),(36,44,316,57),(37,45,317,58),(38,46,318,59),(39,47,319,60),(40,48,320,51),(101,142,116,122),(102,143,117,123),(103,144,118,124),(104,145,119,125),(105,146,120,126),(106,147,111,127),(107,148,112,128),(108,149,113,129),(109,150,114,130),(110,141,115,121),(151,211,206,175),(152,212,207,176),(153,213,208,177),(154,214,209,178),(155,215,210,179),(156,216,201,180),(157,217,202,171),(158,218,203,172),(159,219,204,173),(160,220,205,174),(161,197,223,182),(162,198,224,183),(163,199,225,184),(164,200,226,185),(165,191,227,186),(166,192,228,187),(167,193,229,188),(168,194,230,189),(169,195,221,190),(170,196,222,181),(231,291,286,255),(232,292,287,256),(233,293,288,257),(234,294,289,258),(235,295,290,259),(236,296,281,260),(237,297,282,251),(238,298,283,252),(239,299,284,253),(240,300,285,254),(241,277,303,262),(242,278,304,263),(243,279,305,264),(244,280,306,265),(245,271,307,266),(246,272,308,267),(247,273,309,268),(248,274,310,269),(249,275,301,270),(250,276,302,261)], [(1,202,86,157),(2,203,87,158),(3,204,88,159),(4,205,89,160),(5,206,90,151),(6,207,81,152),(7,208,82,153),(8,209,83,154),(9,210,84,155),(10,201,85,156),(11,280,65,265),(12,271,66,266),(13,272,67,267),(14,273,68,268),(15,274,69,269),(16,275,70,270),(17,276,61,261),(18,277,62,262),(19,278,63,263),(20,279,64,264),(21,304,75,242),(22,305,76,243),(23,306,77,244),(24,307,78,245),(25,308,79,246),(26,309,80,247),(27,310,71,248),(28,301,72,249),(29,302,73,250),(30,303,74,241),(31,260,311,296),(32,251,312,297),(33,252,313,298),(34,253,314,299),(35,254,315,300),(36,255,316,291),(37,256,317,292),(38,257,318,293),(39,258,319,294),(40,259,320,295),(41,283,54,238),(42,284,55,239),(43,285,56,240),(44,286,57,231),(45,287,58,232),(46,288,59,233),(47,289,60,234),(48,290,51,235),(49,281,52,236),(50,282,53,237),(91,176,137,212),(92,177,138,213),(93,178,139,214),(94,179,140,215),(95,180,131,216),(96,171,132,217),(97,172,133,218),(98,173,134,219),(99,174,135,220),(100,175,136,211),(101,186,116,191),(102,187,117,192),(103,188,118,193),(104,189,119,194),(105,190,120,195),(106,181,111,196),(107,182,112,197),(108,183,113,198),(109,184,114,199),(110,185,115,200),(121,164,141,226),(122,165,142,227),(123,166,143,228),(124,167,144,229),(125,168,145,230),(126,169,146,221),(127,170,147,222),(128,161,148,223),(129,162,149,224),(130,163,150,225)], [(1,237,141,244),(2,238,142,245),(3,239,143,246),(4,240,144,247),(5,231,145,248),(6,232,146,249),(7,233,147,250),(8,234,148,241),(9,235,149,242),(10,236,150,243),(11,202,312,226),(12,203,313,227),(13,204,314,228),(14,205,315,229),(15,206,316,230),(16,207,317,221),(17,208,318,222),(18,209,319,223),(19,210,320,224),(20,201,311,225),(21,179,48,183),(22,180,49,184),(23,171,50,185),(24,172,41,186),(25,173,42,187),(26,174,43,188),(27,175,44,189),(28,176,45,190),(29,177,46,181),(30,178,47,182),(31,163,64,156),(32,164,65,157),(33,165,66,158),(34,166,67,159),(35,167,68,160),(36,168,69,151),(37,169,70,152),(38,170,61,153),(39,161,62,154),(40,162,63,155),(51,198,75,215),(52,199,76,216),(53,200,77,217),(54,191,78,218),(55,192,79,219),(56,193,80,220),(57,194,71,211),(58,195,72,212),(59,196,73,213),(60,197,74,214),(81,287,126,301),(82,288,127,302),(83,289,128,303),(84,290,129,304),(85,281,130,305),(86,282,121,306),(87,283,122,307),(88,284,123,308),(89,285,124,309),(90,286,125,310),(91,292,105,275),(92,293,106,276),(93,294,107,277),(94,295,108,278),(95,296,109,279),(96,297,110,280),(97,298,101,271),(98,299,102,272),(99,300,103,273),(100,291,104,274),(111,261,138,257),(112,262,139,258),(113,263,140,259),(114,264,131,260),(115,265,132,251),(116,266,133,252),(117,267,134,253),(118,268,135,254),(119,269,136,255),(120,270,137,256)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | SD16 | Q16 | C5×D4 | C5×D4 | C5×SD16 | C5×Q16 |
| kernel | C10×Q8⋊C4 | C5×Q8⋊C4 | C10×C4⋊C4 | C22×C40 | Q8×C2×C10 | Q8×C10 | C2×Q8⋊C4 | Q8⋊C4 | C2×C4⋊C4 | C22×C8 | C22×Q8 | C2×Q8 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 3 | 1 | 4 | 4 | 12 | 4 | 16 | 16 |
Matrix representation of C10×Q8⋊C4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 |
| 0 | 0 | 23 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 14 |
| 0 | 0 | 0 | 27 | 20 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 17 | 1 | 0 | 0 |
| 0 | 40 | 24 | 0 | 0 |
| 0 | 0 | 0 | 31 | 26 |
| 0 | 0 | 0 | 23 | 10 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,21,27,0,0,0,14,20],[9,0,0,0,0,0,17,40,0,0,0,1,24,0,0,0,0,0,31,23,0,0,0,26,10] >;
C10×Q8⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times Q_8\rtimes C_4 % in TeX
G:=Group("C10xQ8:C4"); // GroupNames label
G:=SmallGroup(320,916);
// by ID
G=gap.SmallGroup(320,916);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations