metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.76D10, C5⋊5(C8⋊Q8), C5⋊2C8⋊4Q8, 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×C5⋊2C8).133C22, SmallGroup(320,707)
Series: Derived ►Chief ►Lower central ►Upper central
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, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C8⋊Q8, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4, C5×C4⋊C4, Q8×C10, C42.D5, C10.D8, C20.Q8, C20.6Q8, C5×C4⋊Q8, C42.76D10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C8⋊Q8, Q8×D5, C2×C5⋊D4, D4.D10, C20.C23, Dic5⋊Q8, C42.76D10
►Regular action on 320 points
Generators in S320
(1 81 11 91)(2 92 12 82)(3 83 13 93)(4 94 14 84)(5 85 15 95)(6 96 16 86)(7 87 17 97)(8 98 18 88)(9 89 19 99)(10 100 20 90)(21 320 31 310)(22 311 32 301)(23 302 33 312)(24 313 34 303)(25 304 35 314)(26 315 36 305)(27 306 37 316)(28 317 38 307)(29 308 39 318)(30 319 40 309)(41 231 51 221)(42 222 52 232)(43 233 53 223)(44 224 54 234)(45 235 55 225)(46 226 56 236)(47 237 57 227)(48 228 58 238)(49 239 59 229)(50 230 60 240)(61 156 71 146)(62 147 72 157)(63 158 73 148)(64 149 74 159)(65 160 75 150)(66 151 76 141)(67 142 77 152)(68 153 78 143)(69 144 79 154)(70 155 80 145)(101 199 111 189)(102 190 112 200)(103 181 113 191)(104 192 114 182)(105 183 115 193)(106 194 116 184)(107 185 117 195)(108 196 118 186)(109 187 119 197)(110 198 120 188)(121 179 131 169)(122 170 132 180)(123 161 133 171)(124 172 134 162)(125 163 135 173)(126 174 136 164)(127 165 137 175)(128 176 138 166)(129 167 139 177)(130 178 140 168)(201 278 211 268)(202 269 212 279)(203 280 213 270)(204 271 214 261)(205 262 215 272)(206 273 216 263)(207 264 217 274)(208 275 218 265)(209 266 219 276)(210 277 220 267)(241 287 251 297)(242 298 252 288)(243 289 253 299)(244 300 254 290)(245 291 255 281)(246 282 256 292)(247 293 257 283)(248 284 258 294)(249 295 259 285)(250 286 260 296)
(1 319 264 249)(2 250 265 320)(3 301 266 251)(4 252 267 302)(5 303 268 253)(6 254 269 304)(7 305 270 255)(8 256 271 306)(9 307 272 257)(10 258 273 308)(11 309 274 259)(12 260 275 310)(13 311 276 241)(14 242 277 312)(15 313 278 243)(16 244 279 314)(17 315 280 245)(18 246 261 316)(19 317 262 247)(20 248 263 318)(21 82 296 218)(22 219 297 83)(23 84 298 220)(24 201 299 85)(25 86 300 202)(26 203 281 87)(27 88 282 204)(28 205 283 89)(29 90 284 206)(30 207 285 91)(31 92 286 208)(32 209 287 93)(33 94 288 210)(34 211 289 95)(35 96 290 212)(36 213 291 97)(37 98 292 214)(38 215 293 99)(39 100 294 216)(40 217 295 81)(41 172 158 105)(42 106 159 173)(43 174 160 107)(44 108 141 175)(45 176 142 109)(46 110 143 177)(47 178 144 111)(48 112 145 179)(49 180 146 113)(50 114 147 161)(51 162 148 115)(52 116 149 163)(53 164 150 117)(54 118 151 165)(55 166 152 119)(56 120 153 167)(57 168 154 101)(58 102 155 169)(59 170 156 103)(60 104 157 171)(61 191 239 122)(62 123 240 192)(63 193 221 124)(64 125 222 194)(65 195 223 126)(66 127 224 196)(67 197 225 128)(68 129 226 198)(69 199 227 130)(70 131 228 200)(71 181 229 132)(72 133 230 182)(73 183 231 134)(74 135 232 184)(75 185 233 136)(76 137 234 186)(77 187 235 138)(78 139 236 188)(79 189 237 140)(80 121 238 190)
(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 44 11 54)(2 117 12 107)(3 42 13 52)(4 115 14 105)(5 60 15 50)(6 113 16 103)(7 58 17 48)(8 111 18 101)(9 56 19 46)(10 109 20 119)(21 65 31 75)(22 184 32 194)(23 63 33 73)(24 182 34 192)(25 61 35 71)(26 200 36 190)(27 79 37 69)(28 198 38 188)(29 77 39 67)(30 196 40 186)(41 302 51 312)(43 320 53 310)(45 318 55 308)(47 316 57 306)(49 314 59 304)(62 85 72 95)(64 83 74 93)(66 81 76 91)(68 99 78 89)(70 97 80 87)(82 126 92 136)(84 124 94 134)(86 122 96 132)(88 140 98 130)(90 138 100 128)(102 245 112 255)(104 243 114 253)(106 241 116 251)(108 259 118 249)(110 257 120 247)(121 281 131 291)(123 299 133 289)(125 297 135 287)(127 295 137 285)(129 293 139 283)(141 274 151 264)(142 248 152 258)(143 272 153 262)(144 246 154 256)(145 270 155 280)(146 244 156 254)(147 268 157 278)(148 242 158 252)(149 266 159 276)(150 260 160 250)(161 303 171 313)(162 277 172 267)(163 301 173 311)(164 275 174 265)(165 319 175 309)(166 273 176 263)(167 317 177 307)(168 271 178 261)(169 315 179 305)(170 269 180 279)(181 202 191 212)(183 220 193 210)(185 218 195 208)(187 216 197 206)(189 214 199 204)(201 230 211 240)(203 228 213 238)(205 226 215 236)(207 224 217 234)(209 222 219 232)(221 288 231 298)(223 286 233 296)(225 284 235 294)(227 282 237 292)(229 300 239 290)
G:=sub<Sym(320)| (1,81,11,91)(2,92,12,82)(3,83,13,93)(4,94,14,84)(5,85,15,95)(6,96,16,86)(7,87,17,97)(8,98,18,88)(9,89,19,99)(10,100,20,90)(21,320,31,310)(22,311,32,301)(23,302,33,312)(24,313,34,303)(25,304,35,314)(26,315,36,305)(27,306,37,316)(28,317,38,307)(29,308,39,318)(30,319,40,309)(41,231,51,221)(42,222,52,232)(43,233,53,223)(44,224,54,234)(45,235,55,225)(46,226,56,236)(47,237,57,227)(48,228,58,238)(49,239,59,229)(50,230,60,240)(61,156,71,146)(62,147,72,157)(63,158,73,148)(64,149,74,159)(65,160,75,150)(66,151,76,141)(67,142,77,152)(68,153,78,143)(69,144,79,154)(70,155,80,145)(101,199,111,189)(102,190,112,200)(103,181,113,191)(104,192,114,182)(105,183,115,193)(106,194,116,184)(107,185,117,195)(108,196,118,186)(109,187,119,197)(110,198,120,188)(121,179,131,169)(122,170,132,180)(123,161,133,171)(124,172,134,162)(125,163,135,173)(126,174,136,164)(127,165,137,175)(128,176,138,166)(129,167,139,177)(130,178,140,168)(201,278,211,268)(202,269,212,279)(203,280,213,270)(204,271,214,261)(205,262,215,272)(206,273,216,263)(207,264,217,274)(208,275,218,265)(209,266,219,276)(210,277,220,267)(241,287,251,297)(242,298,252,288)(243,289,253,299)(244,300,254,290)(245,291,255,281)(246,282,256,292)(247,293,257,283)(248,284,258,294)(249,295,259,285)(250,286,260,296), (1,319,264,249)(2,250,265,320)(3,301,266,251)(4,252,267,302)(5,303,268,253)(6,254,269,304)(7,305,270,255)(8,256,271,306)(9,307,272,257)(10,258,273,308)(11,309,274,259)(12,260,275,310)(13,311,276,241)(14,242,277,312)(15,313,278,243)(16,244,279,314)(17,315,280,245)(18,246,261,316)(19,317,262,247)(20,248,263,318)(21,82,296,218)(22,219,297,83)(23,84,298,220)(24,201,299,85)(25,86,300,202)(26,203,281,87)(27,88,282,204)(28,205,283,89)(29,90,284,206)(30,207,285,91)(31,92,286,208)(32,209,287,93)(33,94,288,210)(34,211,289,95)(35,96,290,212)(36,213,291,97)(37,98,292,214)(38,215,293,99)(39,100,294,216)(40,217,295,81)(41,172,158,105)(42,106,159,173)(43,174,160,107)(44,108,141,175)(45,176,142,109)(46,110,143,177)(47,178,144,111)(48,112,145,179)(49,180,146,113)(50,114,147,161)(51,162,148,115)(52,116,149,163)(53,164,150,117)(54,118,151,165)(55,166,152,119)(56,120,153,167)(57,168,154,101)(58,102,155,169)(59,170,156,103)(60,104,157,171)(61,191,239,122)(62,123,240,192)(63,193,221,124)(64,125,222,194)(65,195,223,126)(66,127,224,196)(67,197,225,128)(68,129,226,198)(69,199,227,130)(70,131,228,200)(71,181,229,132)(72,133,230,182)(73,183,231,134)(74,135,232,184)(75,185,233,136)(76,137,234,186)(77,187,235,138)(78,139,236,188)(79,189,237,140)(80,121,238,190), (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), (1,44,11,54)(2,117,12,107)(3,42,13,52)(4,115,14,105)(5,60,15,50)(6,113,16,103)(7,58,17,48)(8,111,18,101)(9,56,19,46)(10,109,20,119)(21,65,31,75)(22,184,32,194)(23,63,33,73)(24,182,34,192)(25,61,35,71)(26,200,36,190)(27,79,37,69)(28,198,38,188)(29,77,39,67)(30,196,40,186)(41,302,51,312)(43,320,53,310)(45,318,55,308)(47,316,57,306)(49,314,59,304)(62,85,72,95)(64,83,74,93)(66,81,76,91)(68,99,78,89)(70,97,80,87)(82,126,92,136)(84,124,94,134)(86,122,96,132)(88,140,98,130)(90,138,100,128)(102,245,112,255)(104,243,114,253)(106,241,116,251)(108,259,118,249)(110,257,120,247)(121,281,131,291)(123,299,133,289)(125,297,135,287)(127,295,137,285)(129,293,139,283)(141,274,151,264)(142,248,152,258)(143,272,153,262)(144,246,154,256)(145,270,155,280)(146,244,156,254)(147,268,157,278)(148,242,158,252)(149,266,159,276)(150,260,160,250)(161,303,171,313)(162,277,172,267)(163,301,173,311)(164,275,174,265)(165,319,175,309)(166,273,176,263)(167,317,177,307)(168,271,178,261)(169,315,179,305)(170,269,180,279)(181,202,191,212)(183,220,193,210)(185,218,195,208)(187,216,197,206)(189,214,199,204)(201,230,211,240)(203,228,213,238)(205,226,215,236)(207,224,217,234)(209,222,219,232)(221,288,231,298)(223,286,233,296)(225,284,235,294)(227,282,237,292)(229,300,239,290)>;
G:=Group( (1,81,11,91)(2,92,12,82)(3,83,13,93)(4,94,14,84)(5,85,15,95)(6,96,16,86)(7,87,17,97)(8,98,18,88)(9,89,19,99)(10,100,20,90)(21,320,31,310)(22,311,32,301)(23,302,33,312)(24,313,34,303)(25,304,35,314)(26,315,36,305)(27,306,37,316)(28,317,38,307)(29,308,39,318)(30,319,40,309)(41,231,51,221)(42,222,52,232)(43,233,53,223)(44,224,54,234)(45,235,55,225)(46,226,56,236)(47,237,57,227)(48,228,58,238)(49,239,59,229)(50,230,60,240)(61,156,71,146)(62,147,72,157)(63,158,73,148)(64,149,74,159)(65,160,75,150)(66,151,76,141)(67,142,77,152)(68,153,78,143)(69,144,79,154)(70,155,80,145)(101,199,111,189)(102,190,112,200)(103,181,113,191)(104,192,114,182)(105,183,115,193)(106,194,116,184)(107,185,117,195)(108,196,118,186)(109,187,119,197)(110,198,120,188)(121,179,131,169)(122,170,132,180)(123,161,133,171)(124,172,134,162)(125,163,135,173)(126,174,136,164)(127,165,137,175)(128,176,138,166)(129,167,139,177)(130,178,140,168)(201,278,211,268)(202,269,212,279)(203,280,213,270)(204,271,214,261)(205,262,215,272)(206,273,216,263)(207,264,217,274)(208,275,218,265)(209,266,219,276)(210,277,220,267)(241,287,251,297)(242,298,252,288)(243,289,253,299)(244,300,254,290)(245,291,255,281)(246,282,256,292)(247,293,257,283)(248,284,258,294)(249,295,259,285)(250,286,260,296), (1,319,264,249)(2,250,265,320)(3,301,266,251)(4,252,267,302)(5,303,268,253)(6,254,269,304)(7,305,270,255)(8,256,271,306)(9,307,272,257)(10,258,273,308)(11,309,274,259)(12,260,275,310)(13,311,276,241)(14,242,277,312)(15,313,278,243)(16,244,279,314)(17,315,280,245)(18,246,261,316)(19,317,262,247)(20,248,263,318)(21,82,296,218)(22,219,297,83)(23,84,298,220)(24,201,299,85)(25,86,300,202)(26,203,281,87)(27,88,282,204)(28,205,283,89)(29,90,284,206)(30,207,285,91)(31,92,286,208)(32,209,287,93)(33,94,288,210)(34,211,289,95)(35,96,290,212)(36,213,291,97)(37,98,292,214)(38,215,293,99)(39,100,294,216)(40,217,295,81)(41,172,158,105)(42,106,159,173)(43,174,160,107)(44,108,141,175)(45,176,142,109)(46,110,143,177)(47,178,144,111)(48,112,145,179)(49,180,146,113)(50,114,147,161)(51,162,148,115)(52,116,149,163)(53,164,150,117)(54,118,151,165)(55,166,152,119)(56,120,153,167)(57,168,154,101)(58,102,155,169)(59,170,156,103)(60,104,157,171)(61,191,239,122)(62,123,240,192)(63,193,221,124)(64,125,222,194)(65,195,223,126)(66,127,224,196)(67,197,225,128)(68,129,226,198)(69,199,227,130)(70,131,228,200)(71,181,229,132)(72,133,230,182)(73,183,231,134)(74,135,232,184)(75,185,233,136)(76,137,234,186)(77,187,235,138)(78,139,236,188)(79,189,237,140)(80,121,238,190), (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), (1,44,11,54)(2,117,12,107)(3,42,13,52)(4,115,14,105)(5,60,15,50)(6,113,16,103)(7,58,17,48)(8,111,18,101)(9,56,19,46)(10,109,20,119)(21,65,31,75)(22,184,32,194)(23,63,33,73)(24,182,34,192)(25,61,35,71)(26,200,36,190)(27,79,37,69)(28,198,38,188)(29,77,39,67)(30,196,40,186)(41,302,51,312)(43,320,53,310)(45,318,55,308)(47,316,57,306)(49,314,59,304)(62,85,72,95)(64,83,74,93)(66,81,76,91)(68,99,78,89)(70,97,80,87)(82,126,92,136)(84,124,94,134)(86,122,96,132)(88,140,98,130)(90,138,100,128)(102,245,112,255)(104,243,114,253)(106,241,116,251)(108,259,118,249)(110,257,120,247)(121,281,131,291)(123,299,133,289)(125,297,135,287)(127,295,137,285)(129,293,139,283)(141,274,151,264)(142,248,152,258)(143,272,153,262)(144,246,154,256)(145,270,155,280)(146,244,156,254)(147,268,157,278)(148,242,158,252)(149,266,159,276)(150,260,160,250)(161,303,171,313)(162,277,172,267)(163,301,173,311)(164,275,174,265)(165,319,175,309)(166,273,176,263)(167,317,177,307)(168,271,178,261)(169,315,179,305)(170,269,180,279)(181,202,191,212)(183,220,193,210)(185,218,195,208)(187,216,197,206)(189,214,199,204)(201,230,211,240)(203,228,213,238)(205,226,215,236)(207,224,217,234)(209,222,219,232)(221,288,231,298)(223,286,233,296)(225,284,235,294)(227,282,237,292)(229,300,239,290) );
G=PermutationGroup([[(1,81,11,91),(2,92,12,82),(3,83,13,93),(4,94,14,84),(5,85,15,95),(6,96,16,86),(7,87,17,97),(8,98,18,88),(9,89,19,99),(10,100,20,90),(21,320,31,310),(22,311,32,301),(23,302,33,312),(24,313,34,303),(25,304,35,314),(26,315,36,305),(27,306,37,316),(28,317,38,307),(29,308,39,318),(30,319,40,309),(41,231,51,221),(42,222,52,232),(43,233,53,223),(44,224,54,234),(45,235,55,225),(46,226,56,236),(47,237,57,227),(48,228,58,238),(49,239,59,229),(50,230,60,240),(61,156,71,146),(62,147,72,157),(63,158,73,148),(64,149,74,159),(65,160,75,150),(66,151,76,141),(67,142,77,152),(68,153,78,143),(69,144,79,154),(70,155,80,145),(101,199,111,189),(102,190,112,200),(103,181,113,191),(104,192,114,182),(105,183,115,193),(106,194,116,184),(107,185,117,195),(108,196,118,186),(109,187,119,197),(110,198,120,188),(121,179,131,169),(122,170,132,180),(123,161,133,171),(124,172,134,162),(125,163,135,173),(126,174,136,164),(127,165,137,175),(128,176,138,166),(129,167,139,177),(130,178,140,168),(201,278,211,268),(202,269,212,279),(203,280,213,270),(204,271,214,261),(205,262,215,272),(206,273,216,263),(207,264,217,274),(208,275,218,265),(209,266,219,276),(210,277,220,267),(241,287,251,297),(242,298,252,288),(243,289,253,299),(244,300,254,290),(245,291,255,281),(246,282,256,292),(247,293,257,283),(248,284,258,294),(249,295,259,285),(250,286,260,296)], [(1,319,264,249),(2,250,265,320),(3,301,266,251),(4,252,267,302),(5,303,268,253),(6,254,269,304),(7,305,270,255),(8,256,271,306),(9,307,272,257),(10,258,273,308),(11,309,274,259),(12,260,275,310),(13,311,276,241),(14,242,277,312),(15,313,278,243),(16,244,279,314),(17,315,280,245),(18,246,261,316),(19,317,262,247),(20,248,263,318),(21,82,296,218),(22,219,297,83),(23,84,298,220),(24,201,299,85),(25,86,300,202),(26,203,281,87),(27,88,282,204),(28,205,283,89),(29,90,284,206),(30,207,285,91),(31,92,286,208),(32,209,287,93),(33,94,288,210),(34,211,289,95),(35,96,290,212),(36,213,291,97),(37,98,292,214),(38,215,293,99),(39,100,294,216),(40,217,295,81),(41,172,158,105),(42,106,159,173),(43,174,160,107),(44,108,141,175),(45,176,142,109),(46,110,143,177),(47,178,144,111),(48,112,145,179),(49,180,146,113),(50,114,147,161),(51,162,148,115),(52,116,149,163),(53,164,150,117),(54,118,151,165),(55,166,152,119),(56,120,153,167),(57,168,154,101),(58,102,155,169),(59,170,156,103),(60,104,157,171),(61,191,239,122),(62,123,240,192),(63,193,221,124),(64,125,222,194),(65,195,223,126),(66,127,224,196),(67,197,225,128),(68,129,226,198),(69,199,227,130),(70,131,228,200),(71,181,229,132),(72,133,230,182),(73,183,231,134),(74,135,232,184),(75,185,233,136),(76,137,234,186),(77,187,235,138),(78,139,236,188),(79,189,237,140),(80,121,238,190)], [(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)], [(1,44,11,54),(2,117,12,107),(3,42,13,52),(4,115,14,105),(5,60,15,50),(6,113,16,103),(7,58,17,48),(8,111,18,101),(9,56,19,46),(10,109,20,119),(21,65,31,75),(22,184,32,194),(23,63,33,73),(24,182,34,192),(25,61,35,71),(26,200,36,190),(27,79,37,69),(28,198,38,188),(29,77,39,67),(30,196,40,186),(41,302,51,312),(43,320,53,310),(45,318,55,308),(47,316,57,306),(49,314,59,304),(62,85,72,95),(64,83,74,93),(66,81,76,91),(68,99,78,89),(70,97,80,87),(82,126,92,136),(84,124,94,134),(86,122,96,132),(88,140,98,130),(90,138,100,128),(102,245,112,255),(104,243,114,253),(106,241,116,251),(108,259,118,249),(110,257,120,247),(121,281,131,291),(123,299,133,289),(125,297,135,287),(127,295,137,285),(129,293,139,283),(141,274,151,264),(142,248,152,258),(143,272,153,262),(144,246,154,256),(145,270,155,280),(146,244,156,254),(147,268,157,278),(148,242,158,252),(149,266,159,276),(150,260,160,250),(161,303,171,313),(162,277,172,267),(163,301,173,311),(164,275,174,265),(165,319,175,309),(166,273,176,263),(167,317,177,307),(168,271,178,261),(169,315,179,305),(170,269,180,279),(181,202,191,212),(183,220,193,210),(185,218,195,208),(187,216,197,206),(189,214,199,204),(201,230,211,240),(203,228,213,238),(205,226,215,236),(207,224,217,234),(209,222,219,232),(221,288,231,298),(223,286,233,296),(225,284,235,294),(227,282,237,292),(229,300,239,290)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | D10 | D10 | C5⋊D4 | C8⋊C22 | C8.C22 | Q8×D5 | D4.D10 | C20.C23 |
| kernel | C42.76D10 | C42.D5 | C10.D8 | C20.Q8 | C20.6Q8 | C5×C4⋊Q8 | C5⋊2C8 | C2×C20 | C4⋊Q8 | C42 | C4⋊C4 | C2×C4 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.76D10 ►in GL8(𝔽41)
| 27 | 39 | 18 | 5 | 0 | 0 | 0 | 0 |
| 2 | 15 | 36 | 7 | 0 | 0 | 0 | 0 |
| 18 | 5 | 14 | 2 | 0 | 0 | 0 | 0 |
| 36 | 7 | 39 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 39 | 12 | 16 |
| 0 | 0 | 0 | 0 | 23 | 13 | 21 | 23 |
| 0 | 0 | 0 | 0 | 38 | 15 | 5 | 21 |
| 0 | 0 | 0 | 0 | 38 | 22 | 9 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 | 40 | 4 |
| 0 | 0 | 0 | 0 | 0 | 35 | 20 | 1 |
| 0 | 0 | 35 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 29 | 12 | 1 | 37 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 5 | 0 | 29 |
| 37 | 26 | 34 | 37 | 0 | 0 | 0 | 0 |
| 2 | 4 | 36 | 7 | 0 | 0 | 0 | 0 |
| 7 | 4 | 37 | 26 | 0 | 0 | 0 | 0 |
| 5 | 34 | 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 2 | 32 | 18 |
| 0 | 0 | 0 | 0 | 28 | 29 | 0 | 18 |
| 0 | 0 | 0 | 0 | 13 | 28 | 16 | 37 |
| 0 | 0 | 0 | 0 | 0 | 14 | 14 | 25 |
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