direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C4⋊Q8, C4⋊1(Q8×C10), C20⋊7(C2×Q8), (C2×C20)⋊15Q8, C4.14(D4×C10), C20.321(C2×D4), (C2×C20).431D4, (C2×C42).19C10, C42.89(C2×C10), C22.64(D4×C10), (C22×Q8).8C10, C10.59(C22×Q8), C22.19(Q8×C10), (C2×C20).962C23, (C2×C10).351C24, (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), (C10×C4⋊C4).48C2, (C2×C4⋊C4).19C10, (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
Subgroups: 370 in 290 conjugacy classes, 210 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×8], C22, C22 [×6], C5, C2×C4 [×26], C2×C4 [×8], Q8 [×16], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×6], C2×Q8 [×8], C2×Q8 [×8], C20 [×12], C20 [×8], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C22×Q8 [×2], C2×C20 [×26], C2×C20 [×8], C5×Q8 [×16], C22×C10, C2×C4⋊Q8, C4×C20 [×4], C5×C4⋊C4 [×16], C22×C20, C22×C20 [×6], Q8×C10 [×8], Q8×C10 [×8], C2×C4×C20, C10×C4⋊C4 [×4], C5×C4⋊Q8 [×8], Q8×C2×C10 [×2], C10×C4⋊Q8
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], Q8 [×8], C23 [×15], C10 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C2×C10 [×35], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C5×D4 [×4], C5×Q8 [×8], C22×C10 [×15], C2×C4⋊Q8, D4×C10 [×6], Q8×C10 [×12], C23×C10, C5×C4⋊Q8 [×4], D4×C2×C10, Q8×C2×C10 [×2], C10×C4⋊Q8
Generators and relations
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 >
►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 48 154)(2 177 49 155)(3 178 50 156)(4 179 41 157)(5 180 42 158)(6 171 43 159)(7 172 44 160)(8 173 45 151)(9 174 46 152)(10 175 47 153)(11 200 31 212)(12 191 32 213)(13 192 33 214)(14 193 34 215)(15 194 35 216)(16 195 36 217)(17 196 37 218)(18 197 38 219)(19 198 39 220)(20 199 40 211)(21 206 312 230)(22 207 313 221)(23 208 314 222)(24 209 315 223)(25 210 316 224)(26 201 317 225)(27 202 318 226)(28 203 319 227)(29 204 320 228)(30 205 311 229)(51 163 70 187)(52 164 61 188)(53 165 62 189)(54 166 63 190)(55 167 64 181)(56 168 65 182)(57 169 66 183)(58 170 67 184)(59 161 68 185)(60 162 69 186)(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)(81 145 105 121)(82 146 106 122)(83 147 107 123)(84 148 108 124)(85 149 109 125)(86 150 110 126)(87 141 101 127)(88 142 102 128)(89 143 103 129)(90 144 104 130)(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)(241 281 265 305)(242 282 266 306)(243 283 267 307)(244 284 268 308)(245 285 269 309)(246 286 270 310)(247 287 261 301)(248 288 262 302)(249 289 263 303)(250 290 264 304)
(1 119 56 147)(2 120 57 148)(3 111 58 149)(4 112 59 150)(5 113 60 141)(6 114 51 142)(7 115 52 143)(8 116 53 144)(9 117 54 145)(10 118 55 146)(11 257 320 249)(12 258 311 250)(13 259 312 241)(14 260 313 242)(15 251 314 243)(16 252 315 244)(17 253 316 245)(18 254 317 246)(19 255 318 247)(20 256 319 248)(21 265 33 237)(22 266 34 238)(23 267 35 239)(24 268 36 240)(25 269 37 231)(26 270 38 232)(27 261 39 233)(28 262 40 234)(29 263 31 235)(30 264 32 236)(41 134 68 126)(42 135 69 127)(43 136 70 128)(44 137 61 129)(45 138 62 130)(46 139 63 121)(47 140 64 122)(48 131 65 123)(49 132 66 124)(50 133 67 125)(71 170 109 178)(72 161 110 179)(73 162 101 180)(74 163 102 171)(75 164 103 172)(76 165 104 173)(77 166 105 174)(78 167 106 175)(79 168 107 176)(80 169 108 177)(81 152 99 190)(82 153 100 181)(83 154 91 182)(84 155 92 183)(85 156 93 184)(86 157 94 185)(87 158 95 186)(88 159 96 187)(89 160 97 188)(90 151 98 189)(191 298 229 290)(192 299 230 281)(193 300 221 282)(194 291 222 283)(195 292 223 284)(196 293 224 285)(197 294 225 286)(198 295 226 287)(199 296 227 288)(200 297 228 289)(201 310 219 272)(202 301 220 273)(203 302 211 274)(204 303 212 275)(205 304 213 276)(206 305 214 277)(207 306 215 278)(208 307 216 279)(209 308 217 280)(210 309 218 271)
(1 199 56 227)(2 200 57 228)(3 191 58 229)(4 192 59 230)(5 193 60 221)(6 194 51 222)(7 195 52 223)(8 196 53 224)(9 197 54 225)(10 198 55 226)(11 169 320 177)(12 170 311 178)(13 161 312 179)(14 162 313 180)(15 163 314 171)(16 164 315 172)(17 165 316 173)(18 166 317 174)(19 167 318 175)(20 168 319 176)(21 157 33 185)(22 158 34 186)(23 159 35 187)(24 160 36 188)(25 151 37 189)(26 152 38 190)(27 153 39 181)(28 154 40 182)(29 155 31 183)(30 156 32 184)(41 214 68 206)(42 215 69 207)(43 216 70 208)(44 217 61 209)(45 218 62 210)(46 219 63 201)(47 220 64 202)(48 211 65 203)(49 212 66 204)(50 213 67 205)(71 250 109 258)(72 241 110 259)(73 242 101 260)(74 243 102 251)(75 244 103 252)(76 245 104 253)(77 246 105 254)(78 247 106 255)(79 248 107 256)(80 249 108 257)(81 232 99 270)(82 233 100 261)(83 234 91 262)(84 235 92 263)(85 236 93 264)(86 237 94 265)(87 238 95 266)(88 239 96 267)(89 240 97 268)(90 231 98 269)(111 290 149 298)(112 281 150 299)(113 282 141 300)(114 283 142 291)(115 284 143 292)(116 285 144 293)(117 286 145 294)(118 287 146 295)(119 288 147 296)(120 289 148 297)(121 272 139 310)(122 273 140 301)(123 274 131 302)(124 275 132 303)(125 276 133 304)(126 277 134 305)(127 278 135 306)(128 279 136 307)(129 280 137 308)(130 271 138 309)
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), (1,176,48,154)(2,177,49,155)(3,178,50,156)(4,179,41,157)(5,180,42,158)(6,171,43,159)(7,172,44,160)(8,173,45,151)(9,174,46,152)(10,175,47,153)(11,200,31,212)(12,191,32,213)(13,192,33,214)(14,193,34,215)(15,194,35,216)(16,195,36,217)(17,196,37,218)(18,197,38,219)(19,198,39,220)(20,199,40,211)(21,206,312,230)(22,207,313,221)(23,208,314,222)(24,209,315,223)(25,210,316,224)(26,201,317,225)(27,202,318,226)(28,203,319,227)(29,204,320,228)(30,205,311,229)(51,163,70,187)(52,164,61,188)(53,165,62,189)(54,166,63,190)(55,167,64,181)(56,168,65,182)(57,169,66,183)(58,170,67,184)(59,161,68,185)(60,162,69,186)(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)(81,145,105,121)(82,146,106,122)(83,147,107,123)(84,148,108,124)(85,149,109,125)(86,150,110,126)(87,141,101,127)(88,142,102,128)(89,143,103,129)(90,144,104,130)(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)(241,281,265,305)(242,282,266,306)(243,283,267,307)(244,284,268,308)(245,285,269,309)(246,286,270,310)(247,287,261,301)(248,288,262,302)(249,289,263,303)(250,290,264,304), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,257,320,249)(12,258,311,250)(13,259,312,241)(14,260,313,242)(15,251,314,243)(16,252,315,244)(17,253,316,245)(18,254,317,246)(19,255,318,247)(20,256,319,248)(21,265,33,237)(22,266,34,238)(23,267,35,239)(24,268,36,240)(25,269,37,231)(26,270,38,232)(27,261,39,233)(28,262,40,234)(29,263,31,235)(30,264,32,236)(41,134,68,126)(42,135,69,127)(43,136,70,128)(44,137,61,129)(45,138,62,130)(46,139,63,121)(47,140,64,122)(48,131,65,123)(49,132,66,124)(50,133,67,125)(71,170,109,178)(72,161,110,179)(73,162,101,180)(74,163,102,171)(75,164,103,172)(76,165,104,173)(77,166,105,174)(78,167,106,175)(79,168,107,176)(80,169,108,177)(81,152,99,190)(82,153,100,181)(83,154,91,182)(84,155,92,183)(85,156,93,184)(86,157,94,185)(87,158,95,186)(88,159,96,187)(89,160,97,188)(90,151,98,189)(191,298,229,290)(192,299,230,281)(193,300,221,282)(194,291,222,283)(195,292,223,284)(196,293,224,285)(197,294,225,286)(198,295,226,287)(199,296,227,288)(200,297,228,289)(201,310,219,272)(202,301,220,273)(203,302,211,274)(204,303,212,275)(205,304,213,276)(206,305,214,277)(207,306,215,278)(208,307,216,279)(209,308,217,280)(210,309,218,271), (1,199,56,227)(2,200,57,228)(3,191,58,229)(4,192,59,230)(5,193,60,221)(6,194,51,222)(7,195,52,223)(8,196,53,224)(9,197,54,225)(10,198,55,226)(11,169,320,177)(12,170,311,178)(13,161,312,179)(14,162,313,180)(15,163,314,171)(16,164,315,172)(17,165,316,173)(18,166,317,174)(19,167,318,175)(20,168,319,176)(21,157,33,185)(22,158,34,186)(23,159,35,187)(24,160,36,188)(25,151,37,189)(26,152,38,190)(27,153,39,181)(28,154,40,182)(29,155,31,183)(30,156,32,184)(41,214,68,206)(42,215,69,207)(43,216,70,208)(44,217,61,209)(45,218,62,210)(46,219,63,201)(47,220,64,202)(48,211,65,203)(49,212,66,204)(50,213,67,205)(71,250,109,258)(72,241,110,259)(73,242,101,260)(74,243,102,251)(75,244,103,252)(76,245,104,253)(77,246,105,254)(78,247,106,255)(79,248,107,256)(80,249,108,257)(81,232,99,270)(82,233,100,261)(83,234,91,262)(84,235,92,263)(85,236,93,264)(86,237,94,265)(87,238,95,266)(88,239,96,267)(89,240,97,268)(90,231,98,269)(111,290,149,298)(112,281,150,299)(113,282,141,300)(114,283,142,291)(115,284,143,292)(116,285,144,293)(117,286,145,294)(118,287,146,295)(119,288,147,296)(120,289,148,297)(121,272,139,310)(122,273,140,301)(123,274,131,302)(124,275,132,303)(125,276,133,304)(126,277,134,305)(127,278,135,306)(128,279,136,307)(129,280,137,308)(130,271,138,309)>;
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), (1,176,48,154)(2,177,49,155)(3,178,50,156)(4,179,41,157)(5,180,42,158)(6,171,43,159)(7,172,44,160)(8,173,45,151)(9,174,46,152)(10,175,47,153)(11,200,31,212)(12,191,32,213)(13,192,33,214)(14,193,34,215)(15,194,35,216)(16,195,36,217)(17,196,37,218)(18,197,38,219)(19,198,39,220)(20,199,40,211)(21,206,312,230)(22,207,313,221)(23,208,314,222)(24,209,315,223)(25,210,316,224)(26,201,317,225)(27,202,318,226)(28,203,319,227)(29,204,320,228)(30,205,311,229)(51,163,70,187)(52,164,61,188)(53,165,62,189)(54,166,63,190)(55,167,64,181)(56,168,65,182)(57,169,66,183)(58,170,67,184)(59,161,68,185)(60,162,69,186)(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)(81,145,105,121)(82,146,106,122)(83,147,107,123)(84,148,108,124)(85,149,109,125)(86,150,110,126)(87,141,101,127)(88,142,102,128)(89,143,103,129)(90,144,104,130)(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)(241,281,265,305)(242,282,266,306)(243,283,267,307)(244,284,268,308)(245,285,269,309)(246,286,270,310)(247,287,261,301)(248,288,262,302)(249,289,263,303)(250,290,264,304), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,257,320,249)(12,258,311,250)(13,259,312,241)(14,260,313,242)(15,251,314,243)(16,252,315,244)(17,253,316,245)(18,254,317,246)(19,255,318,247)(20,256,319,248)(21,265,33,237)(22,266,34,238)(23,267,35,239)(24,268,36,240)(25,269,37,231)(26,270,38,232)(27,261,39,233)(28,262,40,234)(29,263,31,235)(30,264,32,236)(41,134,68,126)(42,135,69,127)(43,136,70,128)(44,137,61,129)(45,138,62,130)(46,139,63,121)(47,140,64,122)(48,131,65,123)(49,132,66,124)(50,133,67,125)(71,170,109,178)(72,161,110,179)(73,162,101,180)(74,163,102,171)(75,164,103,172)(76,165,104,173)(77,166,105,174)(78,167,106,175)(79,168,107,176)(80,169,108,177)(81,152,99,190)(82,153,100,181)(83,154,91,182)(84,155,92,183)(85,156,93,184)(86,157,94,185)(87,158,95,186)(88,159,96,187)(89,160,97,188)(90,151,98,189)(191,298,229,290)(192,299,230,281)(193,300,221,282)(194,291,222,283)(195,292,223,284)(196,293,224,285)(197,294,225,286)(198,295,226,287)(199,296,227,288)(200,297,228,289)(201,310,219,272)(202,301,220,273)(203,302,211,274)(204,303,212,275)(205,304,213,276)(206,305,214,277)(207,306,215,278)(208,307,216,279)(209,308,217,280)(210,309,218,271), (1,199,56,227)(2,200,57,228)(3,191,58,229)(4,192,59,230)(5,193,60,221)(6,194,51,222)(7,195,52,223)(8,196,53,224)(9,197,54,225)(10,198,55,226)(11,169,320,177)(12,170,311,178)(13,161,312,179)(14,162,313,180)(15,163,314,171)(16,164,315,172)(17,165,316,173)(18,166,317,174)(19,167,318,175)(20,168,319,176)(21,157,33,185)(22,158,34,186)(23,159,35,187)(24,160,36,188)(25,151,37,189)(26,152,38,190)(27,153,39,181)(28,154,40,182)(29,155,31,183)(30,156,32,184)(41,214,68,206)(42,215,69,207)(43,216,70,208)(44,217,61,209)(45,218,62,210)(46,219,63,201)(47,220,64,202)(48,211,65,203)(49,212,66,204)(50,213,67,205)(71,250,109,258)(72,241,110,259)(73,242,101,260)(74,243,102,251)(75,244,103,252)(76,245,104,253)(77,246,105,254)(78,247,106,255)(79,248,107,256)(80,249,108,257)(81,232,99,270)(82,233,100,261)(83,234,91,262)(84,235,92,263)(85,236,93,264)(86,237,94,265)(87,238,95,266)(88,239,96,267)(89,240,97,268)(90,231,98,269)(111,290,149,298)(112,281,150,299)(113,282,141,300)(114,283,142,291)(115,284,143,292)(116,285,144,293)(117,286,145,294)(118,287,146,295)(119,288,147,296)(120,289,148,297)(121,272,139,310)(122,273,140,301)(123,274,131,302)(124,275,132,303)(125,276,133,304)(126,277,134,305)(127,278,135,306)(128,279,136,307)(129,280,137,308)(130,271,138,309) );
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)], [(1,176,48,154),(2,177,49,155),(3,178,50,156),(4,179,41,157),(5,180,42,158),(6,171,43,159),(7,172,44,160),(8,173,45,151),(9,174,46,152),(10,175,47,153),(11,200,31,212),(12,191,32,213),(13,192,33,214),(14,193,34,215),(15,194,35,216),(16,195,36,217),(17,196,37,218),(18,197,38,219),(19,198,39,220),(20,199,40,211),(21,206,312,230),(22,207,313,221),(23,208,314,222),(24,209,315,223),(25,210,316,224),(26,201,317,225),(27,202,318,226),(28,203,319,227),(29,204,320,228),(30,205,311,229),(51,163,70,187),(52,164,61,188),(53,165,62,189),(54,166,63,190),(55,167,64,181),(56,168,65,182),(57,169,66,183),(58,170,67,184),(59,161,68,185),(60,162,69,186),(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),(81,145,105,121),(82,146,106,122),(83,147,107,123),(84,148,108,124),(85,149,109,125),(86,150,110,126),(87,141,101,127),(88,142,102,128),(89,143,103,129),(90,144,104,130),(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),(241,281,265,305),(242,282,266,306),(243,283,267,307),(244,284,268,308),(245,285,269,309),(246,286,270,310),(247,287,261,301),(248,288,262,302),(249,289,263,303),(250,290,264,304)], [(1,119,56,147),(2,120,57,148),(3,111,58,149),(4,112,59,150),(5,113,60,141),(6,114,51,142),(7,115,52,143),(8,116,53,144),(9,117,54,145),(10,118,55,146),(11,257,320,249),(12,258,311,250),(13,259,312,241),(14,260,313,242),(15,251,314,243),(16,252,315,244),(17,253,316,245),(18,254,317,246),(19,255,318,247),(20,256,319,248),(21,265,33,237),(22,266,34,238),(23,267,35,239),(24,268,36,240),(25,269,37,231),(26,270,38,232),(27,261,39,233),(28,262,40,234),(29,263,31,235),(30,264,32,236),(41,134,68,126),(42,135,69,127),(43,136,70,128),(44,137,61,129),(45,138,62,130),(46,139,63,121),(47,140,64,122),(48,131,65,123),(49,132,66,124),(50,133,67,125),(71,170,109,178),(72,161,110,179),(73,162,101,180),(74,163,102,171),(75,164,103,172),(76,165,104,173),(77,166,105,174),(78,167,106,175),(79,168,107,176),(80,169,108,177),(81,152,99,190),(82,153,100,181),(83,154,91,182),(84,155,92,183),(85,156,93,184),(86,157,94,185),(87,158,95,186),(88,159,96,187),(89,160,97,188),(90,151,98,189),(191,298,229,290),(192,299,230,281),(193,300,221,282),(194,291,222,283),(195,292,223,284),(196,293,224,285),(197,294,225,286),(198,295,226,287),(199,296,227,288),(200,297,228,289),(201,310,219,272),(202,301,220,273),(203,302,211,274),(204,303,212,275),(205,304,213,276),(206,305,214,277),(207,306,215,278),(208,307,216,279),(209,308,217,280),(210,309,218,271)], [(1,199,56,227),(2,200,57,228),(3,191,58,229),(4,192,59,230),(5,193,60,221),(6,194,51,222),(7,195,52,223),(8,196,53,224),(9,197,54,225),(10,198,55,226),(11,169,320,177),(12,170,311,178),(13,161,312,179),(14,162,313,180),(15,163,314,171),(16,164,315,172),(17,165,316,173),(18,166,317,174),(19,167,318,175),(20,168,319,176),(21,157,33,185),(22,158,34,186),(23,159,35,187),(24,160,36,188),(25,151,37,189),(26,152,38,190),(27,153,39,181),(28,154,40,182),(29,155,31,183),(30,156,32,184),(41,214,68,206),(42,215,69,207),(43,216,70,208),(44,217,61,209),(45,218,62,210),(46,219,63,201),(47,220,64,202),(48,211,65,203),(49,212,66,204),(50,213,67,205),(71,250,109,258),(72,241,110,259),(73,242,101,260),(74,243,102,251),(75,244,103,252),(76,245,104,253),(77,246,105,254),(78,247,106,255),(79,248,107,256),(80,249,108,257),(81,232,99,270),(82,233,100,261),(83,234,91,262),(84,235,92,263),(85,236,93,264),(86,237,94,265),(87,238,95,266),(88,239,96,267),(89,240,97,268),(90,231,98,269),(111,290,149,298),(112,281,150,299),(113,282,141,300),(114,283,142,291),(115,284,143,292),(116,285,144,293),(117,286,145,294),(118,287,146,295),(119,288,147,296),(120,289,148,297),(121,272,139,310),(122,273,140,301),(123,274,131,302),(124,275,132,303),(125,276,133,304),(126,277,134,305),(127,278,135,306),(128,279,136,307),(129,280,137,308),(130,271,138,309)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
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ℤ
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