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G = C42.8D10order 320 = 26·5

8th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.8D10, C10.31C4≀C2, C4⋊C4.1Dic5, (C2×C20).232D4, C42.C2.1D5, (C4×C20).236C22, C42.D5.9C2, C2.7(D42Dic5), C2.3(C20.10D4), C53(C42.2C22), C10.13(C4.10D4), C22.40(C23.D5), (C5×C4⋊C4).14C4, (C2×C20).343(C2×C4), (C2×C4).10(C2×Dic5), (C5×C42.C2).7C2, (C2×C4).166(C5⋊D4), (C2×C10).165(C22⋊C4), SmallGroup(320,101)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.8D10
C1C5C10C2×C10C2×C20C4×C20C42.D5 — C42.8D10
C5C2×C10C2×C20 — C42.8D10
C1C22C42C42.C2

Generators and relations for C42.8D10
 G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a-1b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc9 >

Subgroups: 158 in 60 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×5], C22, C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], C10, C10 [×2], C42, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C20 [×5], C2×C10, C8⋊C4 [×2], C42.C2, C52C8 [×4], C2×C20, C2×C20 [×2], C2×C20 [×2], C42.2C22, C2×C52C8 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C42.D5 [×2], C5×C42.C2, C42.8D10
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C4.10D4, C4≀C2 [×2], C2×Dic5, C5⋊D4 [×2], C42.2C22, C23.D5, C20.10D4, D42Dic5 [×2], C42.8D10

Smallest permutation representation of C42.8D10
Regular action on 320 points
Generators in S320
(1 272 306 96)(2 263 307 87)(3 274 308 98)(4 265 309 89)(5 276 310 100)(6 267 311 91)(7 278 312 82)(8 269 313 93)(9 280 314 84)(10 271 315 95)(11 262 316 86)(12 273 317 97)(13 264 318 88)(14 275 319 99)(15 266 320 90)(16 277 301 81)(17 268 302 92)(18 279 303 83)(19 270 304 94)(20 261 305 85)(21 153 127 183)(22 144 128 194)(23 155 129 185)(24 146 130 196)(25 157 131 187)(26 148 132 198)(27 159 133 189)(28 150 134 200)(29 141 135 191)(30 152 136 182)(31 143 137 193)(32 154 138 184)(33 145 139 195)(34 156 140 186)(35 147 121 197)(36 158 122 188)(37 149 123 199)(38 160 124 190)(39 151 125 181)(40 142 126 192)(41 296 253 224)(42 287 254 235)(43 298 255 226)(44 289 256 237)(45 300 257 228)(46 291 258 239)(47 282 259 230)(48 293 260 221)(49 284 241 232)(50 295 242 223)(51 286 243 234)(52 297 244 225)(53 288 245 236)(54 299 246 227)(55 290 247 238)(56 281 248 229)(57 292 249 240)(58 283 250 231)(59 294 251 222)(60 285 252 233)(61 209 178 101)(62 220 179 112)(63 211 180 103)(64 202 161 114)(65 213 162 105)(66 204 163 116)(67 215 164 107)(68 206 165 118)(69 217 166 109)(70 208 167 120)(71 219 168 111)(72 210 169 102)(73 201 170 113)(74 212 171 104)(75 203 172 115)(76 214 173 106)(77 205 174 117)(78 216 175 108)(79 207 176 119)(80 218 177 110)
(1 34 316 130)(2 131 317 35)(3 36 318 132)(4 133 319 37)(5 38 320 134)(6 135 301 39)(7 40 302 136)(8 137 303 21)(9 22 304 138)(10 139 305 23)(11 24 306 140)(12 121 307 25)(13 26 308 122)(14 123 309 27)(15 28 310 124)(16 125 311 29)(17 30 312 126)(18 127 313 31)(19 32 314 128)(20 129 315 33)(41 216 243 118)(42 119 244 217)(43 218 245 120)(44 101 246 219)(45 220 247 102)(46 103 248 201)(47 202 249 104)(48 105 250 203)(49 204 251 106)(50 107 252 205)(51 206 253 108)(52 109 254 207)(53 208 255 110)(54 111 256 209)(55 210 257 112)(56 113 258 211)(57 212 259 114)(58 115 260 213)(59 214 241 116)(60 117 242 215)(61 227 168 289)(62 290 169 228)(63 229 170 291)(64 292 171 230)(65 231 172 293)(66 294 173 232)(67 233 174 295)(68 296 175 234)(69 235 176 297)(70 298 177 236)(71 237 178 299)(72 300 179 238)(73 239 180 281)(74 282 161 240)(75 221 162 283)(76 284 163 222)(77 223 164 285)(78 286 165 224)(79 225 166 287)(80 288 167 226)(81 151 267 191)(82 192 268 152)(83 153 269 193)(84 194 270 154)(85 155 271 195)(86 196 272 156)(87 157 273 197)(88 198 274 158)(89 159 275 199)(90 200 276 160)(91 141 277 181)(92 182 278 142)(93 143 279 183)(94 184 280 144)(95 145 261 185)(96 186 262 146)(97 147 263 187)(98 188 264 148)(99 149 265 189)(100 190 266 150)
(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)
(1 219 186 237 11 209 196 227)(2 53 157 70 12 43 147 80)(3 217 188 235 13 207 198 225)(4 51 159 68 14 41 149 78)(5 215 190 233 15 205 200 223)(6 49 141 66 16 59 151 76)(7 213 192 231 17 203 182 221)(8 47 143 64 18 57 153 74)(9 211 194 229 19 201 184 239)(10 45 145 62 20 55 155 72)(21 114 93 240 31 104 83 230)(22 46 270 180 32 56 280 170)(23 112 95 238 33 102 85 228)(24 44 272 178 34 54 262 168)(25 110 97 236 35 120 87 226)(26 42 274 176 36 52 264 166)(27 108 99 234 37 118 89 224)(28 60 276 174 38 50 266 164)(29 106 81 232 39 116 91 222)(30 58 278 172 40 48 268 162)(61 140 246 86 71 130 256 96)(63 138 248 84 73 128 258 94)(65 136 250 82 75 126 260 92)(67 134 252 100 77 124 242 90)(69 132 254 98 79 122 244 88)(101 146 299 306 111 156 289 316)(103 144 281 304 113 154 291 314)(105 142 283 302 115 152 293 312)(107 160 285 320 117 150 295 310)(109 158 287 318 119 148 297 308)(121 208 263 298 131 218 273 288)(123 206 265 296 133 216 275 286)(125 204 267 294 135 214 277 284)(127 202 269 292 137 212 279 282)(129 220 271 290 139 210 261 300)(161 303 249 183 171 313 259 193)(163 301 251 181 173 311 241 191)(165 319 253 199 175 309 243 189)(167 317 255 197 177 307 245 187)(169 315 257 195 179 305 247 185)

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G5A5B8A···8H10A···10F20A···20L20M···20T
order12224444444558···810···1020···2020···20
size111122224882220···202···24···48···8

47 irreducible representations

dim1111222222444
type++++++--
imageC1C2C2C4D4D5D10Dic5C4≀C2C5⋊D4C4.10D4C20.10D4D42Dic5
kernelC42.8D10C42.D5C5×C42.C2C5×C4⋊C4C2×C20C42.C2C42C4⋊C4C10C2×C4C10C2C2
# reps1214222488148

Matrix representation of C42.8D10 in GL6(𝔽41)

9390000
40320000
0040000
0004000
0000320
0000032
,
1180000
9400000
001000
000100
0000320
000099
,
18280000
25230000
00343400
007100
0000918
00003232
,
5210000
14360000
00232500
00281800
0000400
000059

G:=sub<GL(6,GF(41))| [9,40,0,0,0,0,39,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,9,0,0,0,0,18,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[18,25,0,0,0,0,28,23,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[5,14,0,0,0,0,21,36,0,0,0,0,0,0,23,28,0,0,0,0,25,18,0,0,0,0,0,0,40,5,0,0,0,0,0,9] >;

C42.8D10 in GAP, Magma, Sage, TeX

C_4^2._8D_{10}
% in TeX

G:=Group("C4^2.8D10");
// GroupNames label

G:=SmallGroup(320,101);
// by ID

G=gap.SmallGroup(320,101);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,268,1571,570,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^-1*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b*c^9>;
// generators/relations

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