Copied to
clipboard

G = C42.76D10order 320 = 26·5

76th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.76D10, C55(C8⋊Q8), C52C84Q8, C4⋊Q8.6D5, C4.36(Q8×D5), C4⋊C4.81D10, C20.37(C2×Q8), (C2×C20).293D4, C10.32(C4⋊Q8), C20.6Q8.8C2, C10.97(C8⋊C22), (C2×C20).398C23, (C4×C20).127C22, C10.D8.16C2, C20.Q8.17C2, C42.D5.7C2, C10.93(C8.C22), C4⋊Dic5.158C22, C2.12(Dic5⋊Q8), C2.18(D4.D10), C2.14(C20.C23), (C5×C4⋊Q8).6C2, (C2×C10).529(C2×D4), (C2×C4).71(C5⋊D4), (C5×C4⋊C4).128C22, (C2×C4).495(C22×D5), C22.201(C2×C5⋊D4), (C2×C52C8).133C22, SmallGroup(320,707)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.76D10
C1C5C10C2×C10C2×C20C2×C52C8C42.D5 — C42.76D10
C5C10C2×C20 — C42.76D10
C1C22C42C4⋊Q8

Generators and relations for C42.76D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc9 >

Subgroups: 270 in 90 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C22, C5, C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42, C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×4], C2×C10, C8⋊C4, C4.Q8 [×2], C2.D8 [×2], C42.C2, C4⋊Q8, C52C8 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C8⋊Q8, C2×C52C8 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, Q8×C10, C42.D5, C10.D8 [×2], C20.Q8 [×2], C20.6Q8, C5×C4⋊Q8, C42.76D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C8⋊C22, C8.C22, C5⋊D4 [×2], C22×D5, C8⋊Q8, Q8×D5 [×2], C2×C5⋊D4, D4.D10, C20.C23, Dic5⋊Q8, C42.76D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12224444444455888810···1020···2020···20
size1111224488404022202020202···24···48···8

44 irreducible representations

dim11111122222244444
type++++++-+++++--
imageC1C2C2C2C2C2Q8D4D5D10D10C5⋊D4C8⋊C22C8.C22Q8×D5D4.D10C20.C23
kernelC42.76D10C42.D5C10.D8C20.Q8C20.6Q8C5×C4⋊Q8C52C8C2×C20C4⋊Q8C42C4⋊C4C2×C4C10C10C4C2C2
# reps11221142224811444

Matrix representation of C42.76D10 in GL8(𝔽41)

27391850000
2153670000
1851420000
36739260000
00007391216
000023132123
00003815521
00003822916
,
10000000
01000000
00100000
00010000
00000100
000040000
00001229404
0000035201
,
0035350000
006400000
66000000
351000000
00000010
00002912137
00001000
0000365029
,
372634370000
243670000
7437260000
534240000
00001223218
00002829018
000013281637
00000141425

G:=sub<GL(8,GF(41))| [27,2,18,36,0,0,0,0,39,15,5,7,0,0,0,0,18,36,14,39,0,0,0,0,5,7,2,26,0,0,0,0,0,0,0,0,7,23,38,38,0,0,0,0,39,13,15,22,0,0,0,0,12,21,5,9,0,0,0,0,16,23,21,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,12,0,0,0,0,0,1,0,29,35,0,0,0,0,0,0,40,20,0,0,0,0,0,0,4,1],[0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,0,0,0,29,1,36,0,0,0,0,0,12,0,5,0,0,0,0,1,1,0,0,0,0,0,0,0,37,0,29],[37,2,7,5,0,0,0,0,26,4,4,34,0,0,0,0,34,36,37,2,0,0,0,0,37,7,26,4,0,0,0,0,0,0,0,0,12,28,13,0,0,0,0,0,2,29,28,14,0,0,0,0,32,0,16,14,0,0,0,0,18,18,37,25] >;

C42.76D10 in GAP, Magma, Sage, TeX

C_4^2._{76}D_{10}
% in TeX

G:=Group("C4^2.76D10");
// GroupNames label

G:=SmallGroup(320,707);
// by ID

G=gap.SmallGroup(320,707);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,64,422,135,58,438,102,12550]);
// Polycyclic

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

׿
×
𝔽