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## 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 [×3], C2 [×4], C4 [×4], C4 [×14], C22 [×3], C22 [×4], C5, C2×C4 [×10], C2×C4 [×22], C23, C10 [×3], C10 [×4], C42 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C2×Dic5 [×12], C2×Dic5 [×6], C2×C20 [×10], C2×C20 [×4], C22×C10, C4×C4⋊C4, C4×Dic5 [×4], C4×Dic5 [×2], C10.D4 [×8], C4×C20 [×2], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C10.10C42 [×2], C2×C4×Dic5 [×2], C2×C10.D4 [×2], C2×C4×C20, C4×C10.D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, D5, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], D10 [×3], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], Dic10 [×2], C4×D5 [×6], C5⋊D4 [×2], C22×D5, C4×C4⋊C4, C10.D4 [×4], C2×Dic10, C2×C4×D5 [×3], C4○D20 [×2], C2×C5⋊D4, C4×Dic10 [×2], D5×C42, C42⋊D5, C2×C10.D4, C4×C5⋊D4 [×2], C4×C10.D4

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

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