Copied to
clipboard

## G = C4×C10.D4order 320 = 26·5

### Direct product of C4 and C10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×C10.D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — C4×C10.D4
 Lower central C5 — C10 — C4×C10.D4
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C4×C10.D4
G = < a,b,c,d | a4=b10=c4=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 510 in 194 conjugacy classes, 107 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×C4⋊C4, C4×Dic5, C4×Dic5, C10.D4, C4×C20, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C2×C4×Dic5, C2×C10.D4, C2×C4×C20, C4×C10.D4
Quotients:

Smallest permutation representation of C4×C10.D4
Regular action on 320 points
Generators in S320
(1 117 18 135)(2 118 19 136)(3 119 20 137)(4 120 11 138)(5 111 12 139)(6 112 13 140)(7 113 14 131)(8 114 15 132)(9 115 16 133)(10 116 17 134)(21 104 59 82)(22 105 60 83)(23 106 51 84)(24 107 52 85)(25 108 53 86)(26 109 54 87)(27 110 55 88)(28 101 56 89)(29 102 57 90)(30 103 58 81)(31 224 317 201)(32 225 318 202)(33 226 319 203)(34 227 320 204)(35 228 311 205)(36 229 312 206)(37 230 313 207)(38 221 314 208)(39 222 315 209)(40 223 316 210)(41 177 63 160)(42 178 64 151)(43 179 65 152)(44 180 66 153)(45 171 67 154)(46 172 68 155)(47 173 69 156)(48 174 70 157)(49 175 61 158)(50 176 62 159)(71 144 99 122)(72 145 100 123)(73 146 91 124)(74 147 92 125)(75 148 93 126)(76 149 94 127)(77 150 95 128)(78 141 96 129)(79 142 97 130)(80 143 98 121)(161 294 184 277)(162 295 185 278)(163 296 186 279)(164 297 187 280)(165 298 188 271)(166 299 189 272)(167 300 190 273)(168 291 181 274)(169 292 182 275)(170 293 183 276)(191 268 218 245)(192 269 219 246)(193 270 220 247)(194 261 211 248)(195 262 212 249)(196 263 213 250)(197 264 214 241)(198 265 215 242)(199 266 216 243)(200 267 217 244)(231 308 258 285)(232 309 259 286)(233 310 260 287)(234 301 251 288)(235 302 252 289)(236 303 253 290)(237 304 254 281)(238 305 255 282)(239 306 256 283)(240 307 257 284)
(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)
(1 193 55 167)(2 192 56 166)(3 191 57 165)(4 200 58 164)(5 199 59 163)(6 198 60 162)(7 197 51 161)(8 196 52 170)(9 195 53 169)(10 194 54 168)(11 217 30 187)(12 216 21 186)(13 215 22 185)(14 214 23 184)(15 213 24 183)(16 212 25 182)(17 211 26 181)(18 220 27 190)(19 219 28 189)(20 218 29 188)(31 153 304 124)(32 152 305 123)(33 151 306 122)(34 160 307 121)(35 159 308 130)(36 158 309 129)(37 157 310 128)(38 156 301 127)(39 155 302 126)(40 154 303 125)(41 257 80 227)(42 256 71 226)(43 255 72 225)(44 254 73 224)(45 253 74 223)(46 252 75 222)(47 251 76 221)(48 260 77 230)(49 259 78 229)(50 258 79 228)(61 232 96 206)(62 231 97 205)(63 240 98 204)(64 239 99 203)(65 238 100 202)(66 237 91 201)(67 236 92 210)(68 235 93 209)(69 234 94 208)(70 233 95 207)(81 297 120 267)(82 296 111 266)(83 295 112 265)(84 294 113 264)(85 293 114 263)(86 292 115 262)(87 291 116 261)(88 300 117 270)(89 299 118 269)(90 298 119 268)(101 272 136 246)(102 271 137 245)(103 280 138 244)(104 279 139 243)(105 278 140 242)(106 277 131 241)(107 276 132 250)(108 275 133 249)(109 274 134 248)(110 273 135 247)(141 312 175 286)(142 311 176 285)(143 320 177 284)(144 319 178 283)(145 318 179 282)(146 317 180 281)(147 316 171 290)(148 315 172 289)(149 314 173 288)(150 313 174 287)
(1 282 6 287)(2 281 7 286)(3 290 8 285)(4 289 9 284)(5 288 10 283)(11 302 16 307)(12 301 17 306)(13 310 18 305)(14 309 19 304)(15 308 20 303)(21 38 26 33)(22 37 27 32)(23 36 28 31)(24 35 29 40)(25 34 30 39)(41 244 46 249)(42 243 47 248)(43 242 48 247)(44 241 49 246)(45 250 50 245)(51 312 56 317)(52 311 57 316)(53 320 58 315)(54 319 59 314)(55 318 60 313)(61 269 66 264)(62 268 67 263)(63 267 68 262)(64 266 69 261)(65 265 70 270)(71 279 76 274)(72 278 77 273)(73 277 78 272)(74 276 79 271)(75 275 80 280)(81 209 86 204)(82 208 87 203)(83 207 88 202)(84 206 89 201)(85 205 90 210)(91 294 96 299)(92 293 97 298)(93 292 98 297)(94 291 99 296)(95 300 100 295)(101 224 106 229)(102 223 107 228)(103 222 108 227)(104 221 109 226)(105 230 110 225)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)(121 187 126 182)(122 186 127 181)(123 185 128 190)(124 184 129 189)(125 183 130 188)(131 259 136 254)(132 258 137 253)(133 257 138 252)(134 256 139 251)(135 255 140 260)(141 166 146 161)(142 165 147 170)(143 164 148 169)(144 163 149 168)(145 162 150 167)(151 216 156 211)(152 215 157 220)(153 214 158 219)(154 213 159 218)(155 212 160 217)(171 196 176 191)(172 195 177 200)(173 194 178 199)(174 193 179 198)(175 192 180 197)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D5 C4○D4 D10 Dic10 C4×D5 C5⋊D4 C4○D20 kernel C4×C10.D4 C10.10C42 C2×C4×Dic5 C2×C10.D4 C2×C4×C20 C4×Dic5 C10.D4 C2×C20 C2×C20 C2×C42 C2×C10 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 2 2 1 8 16 2 2 2 4 6 8 24 8 16

Matrix representation of C4×C10.D4 in GL4(𝔽41) generated by

 32 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 40 0 0 0 0 40 34 0 0 7 7
,
 1 0 0 0 0 40 0 0 0 0 1 24 0 0 17 40
,
 1 0 0 0 0 9 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [32,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,7,0,0,34,7],[1,0,0,0,0,40,0,0,0,0,1,17,0,0,24,40],[1,0,0,0,0,9,0,0,0,0,0,1,0,0,1,0] >;

C4×C10.D4 in GAP, Magma, Sage, TeX

C_4\times C_{10}.D_4
% in TeX

G:=Group("C4xC10.D4");
// GroupNames label

G:=SmallGroup(320,558);
// by ID

G=gap.SmallGroup(320,558);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,758,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^10=c^4=1,d^2=b^5,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=c^-1>;
// generators/relations

׿
×
𝔽