G = C10.97(C4×D4) order 320 = 26·5
7th non-split extension by C10 of C4×D4 acting via C4×D4/C22×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.97(C4×D4), C10.32(C4×Q8), (C2×C20).250D4, C10.D4⋊10C4, C22.23(Q8×D5), (C2×Dic5).23Q8, (C22×C4).40D10, C2.2(D10⋊3Q8), C10.70(C22⋊Q8), C22.56(C4○D20), C10.18(C42.C2), C23.292(C22×D5), C10.24(C42⋊2C2), C10.55(C42⋊C2), C2.12(Dic5⋊3Q8), C22.56(D4⋊2D5), (C22×C10).342C23, (C22×C20).390C22, C2.6(Dic5.Q8), C5⋊7(C23.63C23), C22.24(Q8⋊2D5), C10.10C42.36C2, C2.4(C23.18D10), C10.74(C22.D4), (C22×Dic5).51C22, (C2×C4⋊C4).15D5, (C2×C4).41(C4×D5), C2.12(C4×C5⋊D4), (C10×C4⋊C4).27C2, (C2×C10).77(C2×Q8), (C2×C4×Dic5).36C2, C22.135(C2×C4×D5), (C2×C20).357(C2×C4), C2.5(C4⋊C4⋊D5), (C2×C10).447(C2×D4), C22.65(C2×C5⋊D4), C2.12(C4⋊C4⋊7D5), (C2×C4).168(C5⋊D4), (C2×C10).153(C4○D4), (C2×C10).219(C22×C4), (C2×Dic5).108(C2×C4), (C2×C10.D4).32C2, SmallGroup(320,605)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.97(C4×D4)
G = < a,b,c,d | a10=b4=c4=1, d2=a5, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >
Subgroups: 462 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C23, C10, C42, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.63C23, C4×Dic5, C10.D4, C5×C4⋊C4, C22×Dic5, C22×C20, C10.10C42, C2×C4×Dic5, C2×C10.D4, C10×C4⋊C4, C10.97(C4×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×D5, C5⋊D4, C22×D5, C23.63C23, C2×C4×D5, C4○D20, D4⋊2D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, Dic5⋊3Q8, Dic5.Q8, C4⋊C4⋊7D5, C4⋊C4⋊D5, C4×C5⋊D4, C23.18D10, D10⋊3Q8, C10.97(C4×D4)
►Regular action on 320 points
Generators in S320
(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 63 23 43)(2 64 24 44)(3 65 25 45)(4 66 26 46)(5 67 27 47)(6 68 28 48)(7 69 29 49)(8 70 30 50)(9 61 21 41)(10 62 22 42)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 146 101 126)(82 147 102 127)(83 148 103 128)(84 149 104 129)(85 150 105 130)(86 141 106 121)(87 142 107 122)(88 143 108 123)(89 144 109 124)(90 145 110 125)(91 156 111 136)(92 157 112 137)(93 158 113 138)(94 159 114 139)(95 160 115 140)(96 151 116 131)(97 152 117 132)(98 153 118 133)(99 154 119 134)(100 155 120 135)(161 221 181 201)(162 222 182 202)(163 223 183 203)(164 224 184 204)(165 225 185 205)(166 226 186 206)(167 227 187 207)(168 228 188 208)(169 229 189 209)(170 230 190 210)(171 231 191 211)(172 232 192 212)(173 233 193 213)(174 234 194 214)(175 235 195 215)(176 236 196 216)(177 237 197 217)(178 238 198 218)(179 239 199 219)(180 240 200 220)(241 306 261 286)(242 307 262 287)(243 308 263 288)(244 309 264 289)(245 310 265 290)(246 301 266 281)(247 302 267 282)(248 303 268 283)(249 304 269 284)(250 305 270 285)(251 316 271 296)(252 317 272 297)(253 318 273 298)(254 319 274 299)(255 320 275 300)(256 311 276 291)(257 312 277 292)(258 313 278 293)(259 314 279 294)(260 315 280 295)
(1 267 13 277)(2 266 14 276)(3 265 15 275)(4 264 16 274)(5 263 17 273)(6 262 18 272)(7 261 19 271)(8 270 20 280)(9 269 11 279)(10 268 12 278)(21 249 31 259)(22 248 32 258)(23 247 33 257)(24 246 34 256)(25 245 35 255)(26 244 36 254)(27 243 37 253)(28 242 38 252)(29 241 39 251)(30 250 40 260)(41 309 51 319)(42 308 52 318)(43 307 53 317)(44 306 54 316)(45 305 55 315)(46 304 56 314)(47 303 57 313)(48 302 58 312)(49 301 59 311)(50 310 60 320)(61 289 71 299)(62 288 72 298)(63 287 73 297)(64 286 74 296)(65 285 75 295)(66 284 76 294)(67 283 77 293)(68 282 78 292)(69 281 79 291)(70 290 80 300)(81 184 91 194)(82 183 92 193)(83 182 93 192)(84 181 94 191)(85 190 95 200)(86 189 96 199)(87 188 97 198)(88 187 98 197)(89 186 99 196)(90 185 100 195)(101 164 111 174)(102 163 112 173)(103 162 113 172)(104 161 114 171)(105 170 115 180)(106 169 116 179)(107 168 117 178)(108 167 118 177)(109 166 119 176)(110 165 120 175)(121 224 131 234)(122 223 132 233)(123 222 133 232)(124 221 134 231)(125 230 135 240)(126 229 136 239)(127 228 137 238)(128 227 138 237)(129 226 139 236)(130 225 140 235)(141 204 151 214)(142 203 152 213)(143 202 153 212)(144 201 154 211)(145 210 155 220)(146 209 156 219)(147 208 157 218)(148 207 158 217)(149 206 159 216)(150 205 160 215)
(1 88 6 83)(2 89 7 84)(3 90 8 85)(4 81 9 86)(5 82 10 87)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)(161 256 166 251)(162 257 167 252)(163 258 168 253)(164 259 169 254)(165 260 170 255)(171 246 176 241)(172 247 177 242)(173 248 178 243)(174 249 179 244)(175 250 180 245)(181 276 186 271)(182 277 187 272)(183 278 188 273)(184 279 189 274)(185 280 190 275)(191 266 196 261)(192 267 197 262)(193 268 198 263)(194 269 199 264)(195 270 200 265)(201 296 206 291)(202 297 207 292)(203 298 208 293)(204 299 209 294)(205 300 210 295)(211 286 216 281)(212 287 217 282)(213 288 218 283)(214 289 219 284)(215 290 220 285)(221 316 226 311)(222 317 227 312)(223 318 228 313)(224 319 229 314)(225 320 230 315)(231 306 236 301)(232 307 237 302)(233 308 238 303)(234 309 239 304)(235 310 240 305)
G:=sub<Sym(320)| (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), (1,63,23,43)(2,64,24,44)(3,65,25,45)(4,66,26,46)(5,67,27,47)(6,68,28,48)(7,69,29,49)(8,70,30,50)(9,61,21,41)(10,62,22,42)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,146,101,126)(82,147,102,127)(83,148,103,128)(84,149,104,129)(85,150,105,130)(86,141,106,121)(87,142,107,122)(88,143,108,123)(89,144,109,124)(90,145,110,125)(91,156,111,136)(92,157,112,137)(93,158,113,138)(94,159,114,139)(95,160,115,140)(96,151,116,131)(97,152,117,132)(98,153,118,133)(99,154,119,134)(100,155,120,135)(161,221,181,201)(162,222,182,202)(163,223,183,203)(164,224,184,204)(165,225,185,205)(166,226,186,206)(167,227,187,207)(168,228,188,208)(169,229,189,209)(170,230,190,210)(171,231,191,211)(172,232,192,212)(173,233,193,213)(174,234,194,214)(175,235,195,215)(176,236,196,216)(177,237,197,217)(178,238,198,218)(179,239,199,219)(180,240,200,220)(241,306,261,286)(242,307,262,287)(243,308,263,288)(244,309,264,289)(245,310,265,290)(246,301,266,281)(247,302,267,282)(248,303,268,283)(249,304,269,284)(250,305,270,285)(251,316,271,296)(252,317,272,297)(253,318,273,298)(254,319,274,299)(255,320,275,300)(256,311,276,291)(257,312,277,292)(258,313,278,293)(259,314,279,294)(260,315,280,295), (1,267,13,277)(2,266,14,276)(3,265,15,275)(4,264,16,274)(5,263,17,273)(6,262,18,272)(7,261,19,271)(8,270,20,280)(9,269,11,279)(10,268,12,278)(21,249,31,259)(22,248,32,258)(23,247,33,257)(24,246,34,256)(25,245,35,255)(26,244,36,254)(27,243,37,253)(28,242,38,252)(29,241,39,251)(30,250,40,260)(41,309,51,319)(42,308,52,318)(43,307,53,317)(44,306,54,316)(45,305,55,315)(46,304,56,314)(47,303,57,313)(48,302,58,312)(49,301,59,311)(50,310,60,320)(61,289,71,299)(62,288,72,298)(63,287,73,297)(64,286,74,296)(65,285,75,295)(66,284,76,294)(67,283,77,293)(68,282,78,292)(69,281,79,291)(70,290,80,300)(81,184,91,194)(82,183,92,193)(83,182,93,192)(84,181,94,191)(85,190,95,200)(86,189,96,199)(87,188,97,198)(88,187,98,197)(89,186,99,196)(90,185,100,195)(101,164,111,174)(102,163,112,173)(103,162,113,172)(104,161,114,171)(105,170,115,180)(106,169,116,179)(107,168,117,178)(108,167,118,177)(109,166,119,176)(110,165,120,175)(121,224,131,234)(122,223,132,233)(123,222,133,232)(124,221,134,231)(125,230,135,240)(126,229,136,239)(127,228,137,238)(128,227,138,237)(129,226,139,236)(130,225,140,235)(141,204,151,214)(142,203,152,213)(143,202,153,212)(144,201,154,211)(145,210,155,220)(146,209,156,219)(147,208,157,218)(148,207,158,217)(149,206,159,216)(150,205,160,215), (1,88,6,83)(2,89,7,84)(3,90,8,85)(4,81,9,86)(5,82,10,87)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155)(161,256,166,251)(162,257,167,252)(163,258,168,253)(164,259,169,254)(165,260,170,255)(171,246,176,241)(172,247,177,242)(173,248,178,243)(174,249,179,244)(175,250,180,245)(181,276,186,271)(182,277,187,272)(183,278,188,273)(184,279,189,274)(185,280,190,275)(191,266,196,261)(192,267,197,262)(193,268,198,263)(194,269,199,264)(195,270,200,265)(201,296,206,291)(202,297,207,292)(203,298,208,293)(204,299,209,294)(205,300,210,295)(211,286,216,281)(212,287,217,282)(213,288,218,283)(214,289,219,284)(215,290,220,285)(221,316,226,311)(222,317,227,312)(223,318,228,313)(224,319,229,314)(225,320,230,315)(231,306,236,301)(232,307,237,302)(233,308,238,303)(234,309,239,304)(235,310,240,305)>;
G:=Group( (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), (1,63,23,43)(2,64,24,44)(3,65,25,45)(4,66,26,46)(5,67,27,47)(6,68,28,48)(7,69,29,49)(8,70,30,50)(9,61,21,41)(10,62,22,42)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,146,101,126)(82,147,102,127)(83,148,103,128)(84,149,104,129)(85,150,105,130)(86,141,106,121)(87,142,107,122)(88,143,108,123)(89,144,109,124)(90,145,110,125)(91,156,111,136)(92,157,112,137)(93,158,113,138)(94,159,114,139)(95,160,115,140)(96,151,116,131)(97,152,117,132)(98,153,118,133)(99,154,119,134)(100,155,120,135)(161,221,181,201)(162,222,182,202)(163,223,183,203)(164,224,184,204)(165,225,185,205)(166,226,186,206)(167,227,187,207)(168,228,188,208)(169,229,189,209)(170,230,190,210)(171,231,191,211)(172,232,192,212)(173,233,193,213)(174,234,194,214)(175,235,195,215)(176,236,196,216)(177,237,197,217)(178,238,198,218)(179,239,199,219)(180,240,200,220)(241,306,261,286)(242,307,262,287)(243,308,263,288)(244,309,264,289)(245,310,265,290)(246,301,266,281)(247,302,267,282)(248,303,268,283)(249,304,269,284)(250,305,270,285)(251,316,271,296)(252,317,272,297)(253,318,273,298)(254,319,274,299)(255,320,275,300)(256,311,276,291)(257,312,277,292)(258,313,278,293)(259,314,279,294)(260,315,280,295), (1,267,13,277)(2,266,14,276)(3,265,15,275)(4,264,16,274)(5,263,17,273)(6,262,18,272)(7,261,19,271)(8,270,20,280)(9,269,11,279)(10,268,12,278)(21,249,31,259)(22,248,32,258)(23,247,33,257)(24,246,34,256)(25,245,35,255)(26,244,36,254)(27,243,37,253)(28,242,38,252)(29,241,39,251)(30,250,40,260)(41,309,51,319)(42,308,52,318)(43,307,53,317)(44,306,54,316)(45,305,55,315)(46,304,56,314)(47,303,57,313)(48,302,58,312)(49,301,59,311)(50,310,60,320)(61,289,71,299)(62,288,72,298)(63,287,73,297)(64,286,74,296)(65,285,75,295)(66,284,76,294)(67,283,77,293)(68,282,78,292)(69,281,79,291)(70,290,80,300)(81,184,91,194)(82,183,92,193)(83,182,93,192)(84,181,94,191)(85,190,95,200)(86,189,96,199)(87,188,97,198)(88,187,98,197)(89,186,99,196)(90,185,100,195)(101,164,111,174)(102,163,112,173)(103,162,113,172)(104,161,114,171)(105,170,115,180)(106,169,116,179)(107,168,117,178)(108,167,118,177)(109,166,119,176)(110,165,120,175)(121,224,131,234)(122,223,132,233)(123,222,133,232)(124,221,134,231)(125,230,135,240)(126,229,136,239)(127,228,137,238)(128,227,138,237)(129,226,139,236)(130,225,140,235)(141,204,151,214)(142,203,152,213)(143,202,153,212)(144,201,154,211)(145,210,155,220)(146,209,156,219)(147,208,157,218)(148,207,158,217)(149,206,159,216)(150,205,160,215), (1,88,6,83)(2,89,7,84)(3,90,8,85)(4,81,9,86)(5,82,10,87)(11,96,16,91)(12,97,17,92)(13,98,18,93)(14,99,19,94)(15,100,20,95)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155)(161,256,166,251)(162,257,167,252)(163,258,168,253)(164,259,169,254)(165,260,170,255)(171,246,176,241)(172,247,177,242)(173,248,178,243)(174,249,179,244)(175,250,180,245)(181,276,186,271)(182,277,187,272)(183,278,188,273)(184,279,189,274)(185,280,190,275)(191,266,196,261)(192,267,197,262)(193,268,198,263)(194,269,199,264)(195,270,200,265)(201,296,206,291)(202,297,207,292)(203,298,208,293)(204,299,209,294)(205,300,210,295)(211,286,216,281)(212,287,217,282)(213,288,218,283)(214,289,219,284)(215,290,220,285)(221,316,226,311)(222,317,227,312)(223,318,228,313)(224,319,229,314)(225,320,230,315)(231,306,236,301)(232,307,237,302)(233,308,238,303)(234,309,239,304)(235,310,240,305) );
G=PermutationGroup([[(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)], [(1,63,23,43),(2,64,24,44),(3,65,25,45),(4,66,26,46),(5,67,27,47),(6,68,28,48),(7,69,29,49),(8,70,30,50),(9,61,21,41),(10,62,22,42),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60),(81,146,101,126),(82,147,102,127),(83,148,103,128),(84,149,104,129),(85,150,105,130),(86,141,106,121),(87,142,107,122),(88,143,108,123),(89,144,109,124),(90,145,110,125),(91,156,111,136),(92,157,112,137),(93,158,113,138),(94,159,114,139),(95,160,115,140),(96,151,116,131),(97,152,117,132),(98,153,118,133),(99,154,119,134),(100,155,120,135),(161,221,181,201),(162,222,182,202),(163,223,183,203),(164,224,184,204),(165,225,185,205),(166,226,186,206),(167,227,187,207),(168,228,188,208),(169,229,189,209),(170,230,190,210),(171,231,191,211),(172,232,192,212),(173,233,193,213),(174,234,194,214),(175,235,195,215),(176,236,196,216),(177,237,197,217),(178,238,198,218),(179,239,199,219),(180,240,200,220),(241,306,261,286),(242,307,262,287),(243,308,263,288),(244,309,264,289),(245,310,265,290),(246,301,266,281),(247,302,267,282),(248,303,268,283),(249,304,269,284),(250,305,270,285),(251,316,271,296),(252,317,272,297),(253,318,273,298),(254,319,274,299),(255,320,275,300),(256,311,276,291),(257,312,277,292),(258,313,278,293),(259,314,279,294),(260,315,280,295)], [(1,267,13,277),(2,266,14,276),(3,265,15,275),(4,264,16,274),(5,263,17,273),(6,262,18,272),(7,261,19,271),(8,270,20,280),(9,269,11,279),(10,268,12,278),(21,249,31,259),(22,248,32,258),(23,247,33,257),(24,246,34,256),(25,245,35,255),(26,244,36,254),(27,243,37,253),(28,242,38,252),(29,241,39,251),(30,250,40,260),(41,309,51,319),(42,308,52,318),(43,307,53,317),(44,306,54,316),(45,305,55,315),(46,304,56,314),(47,303,57,313),(48,302,58,312),(49,301,59,311),(50,310,60,320),(61,289,71,299),(62,288,72,298),(63,287,73,297),(64,286,74,296),(65,285,75,295),(66,284,76,294),(67,283,77,293),(68,282,78,292),(69,281,79,291),(70,290,80,300),(81,184,91,194),(82,183,92,193),(83,182,93,192),(84,181,94,191),(85,190,95,200),(86,189,96,199),(87,188,97,198),(88,187,98,197),(89,186,99,196),(90,185,100,195),(101,164,111,174),(102,163,112,173),(103,162,113,172),(104,161,114,171),(105,170,115,180),(106,169,116,179),(107,168,117,178),(108,167,118,177),(109,166,119,176),(110,165,120,175),(121,224,131,234),(122,223,132,233),(123,222,133,232),(124,221,134,231),(125,230,135,240),(126,229,136,239),(127,228,137,238),(128,227,138,237),(129,226,139,236),(130,225,140,235),(141,204,151,214),(142,203,152,213),(143,202,153,212),(144,201,154,211),(145,210,155,220),(146,209,156,219),(147,208,157,218),(148,207,158,217),(149,206,159,216),(150,205,160,215)], [(1,88,6,83),(2,89,7,84),(3,90,8,85),(4,81,9,86),(5,82,10,87),(11,96,16,91),(12,97,17,92),(13,98,18,93),(14,99,19,94),(15,100,20,95),(21,106,26,101),(22,107,27,102),(23,108,28,103),(24,109,29,104),(25,110,30,105),(31,116,36,111),(32,117,37,112),(33,118,38,113),(34,119,39,114),(35,120,40,115),(41,126,46,121),(42,127,47,122),(43,128,48,123),(44,129,49,124),(45,130,50,125),(51,136,56,131),(52,137,57,132),(53,138,58,133),(54,139,59,134),(55,140,60,135),(61,146,66,141),(62,147,67,142),(63,148,68,143),(64,149,69,144),(65,150,70,145),(71,156,76,151),(72,157,77,152),(73,158,78,153),(74,159,79,154),(75,160,80,155),(161,256,166,251),(162,257,167,252),(163,258,168,253),(164,259,169,254),(165,260,170,255),(171,246,176,241),(172,247,177,242),(173,248,178,243),(174,249,179,244),(175,250,180,245),(181,276,186,271),(182,277,187,272),(183,278,188,273),(184,279,189,274),(185,280,190,275),(191,266,196,261),(192,267,197,262),(193,268,198,263),(194,269,199,264),(195,270,200,265),(201,296,206,291),(202,297,207,292),(203,298,208,293),(204,299,209,294),(205,300,210,295),(211,286,216,281),(212,287,217,282),(213,288,218,283),(214,289,219,284),(215,290,220,285),(221,316,226,311),(222,317,227,312),(223,318,228,313),(224,319,229,314),(225,320,230,315),(231,306,236,301),(232,307,237,302),(233,308,238,303),(234,309,239,304),(235,310,240,305)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | C4○D4 | D10 | C4×D5 | C5⋊D4 | C4○D20 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 |
| kernel | C10.97(C4×D4) | C10.10C42 | C2×C4×Dic5 | C2×C10.D4 | C10×C4⋊C4 | C10.D4 | C2×Dic5 | C2×C20 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 8 | 6 | 8 | 8 | 8 | 4 | 2 | 2 |
Matrix representation of C10.97(C4×D4) ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 35 | 0 | 0 |
| 0 | 0 | 15 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 26 | 26 | 0 | 0 | 0 | 0 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 33 | 0 | 0 |
| 0 | 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 27 |
| 0 | 0 | 0 | 0 | 27 | 25 |
| 15 | 26 | 0 | 0 | 0 | 0 |
| 26 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 3 |
| 0 | 0 | 0 | 0 | 3 | 21 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,20,15,0,0,0,0,35,14,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[26,26,0,0,0,0,26,15,0,0,0,0,0,0,33,13,0,0,0,0,33,8,0,0,0,0,0,0,16,27,0,0,0,0,27,25],[15,26,0,0,0,0,26,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,3,0,0,0,0,3,21] >;
C10.97(C4×D4) in GAP, Magma, Sage, TeX
C_{10}._{97}(C_4\times D_4) % in TeX
G:=Group("C10.97(C4xD4)"); // GroupNames label
G:=SmallGroup(320,605);
// by ID
G=gap.SmallGroup(320,605);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=a^5,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations