G = C10.96(C4×D4) order 320 = 26·5
6th non-split extension by C10 of C4×D4 acting via C4×D4/C22×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.96(C4×D4), C10.28(C4×Q8), Dic5⋊3(C4⋊C4), C10.D4⋊9C4, (C2×C20).249D4, C10.24(C4⋊Q8), C22.18(Q8×D5), (C2×Dic5).20Q8, C22.104(D4×D5), (C22×C4).37D10, C10.87(C4⋊D4), C2.5(D10⋊Q8), (C2×Dic5).231D4, C10.43(C22⋊Q8), C2.9(Dic5⋊3Q8), C2.5(Dic5⋊D4), C2.1(Dic5⋊Q8), C22.55(C4○D20), C10.15(C42.C2), C23.286(C22×D5), C22.51(D4⋊2D5), (C22×C20).345C22, (C22×C10).336C23, C5⋊4(C23.65C23), C2.5(Dic5.Q8), C10.10C42.14C2, (C22×Dic5).48C22, C2.19(D5×C4⋊C4), C10.41(C2×C4⋊C4), (C2×C4⋊C4).10D5, (C2×C4).40(C4×D5), C2.11(C4×C5⋊D4), (C10×C4⋊C4).26C2, (C2×C10).72(C2×Q8), (C2×C4×Dic5).35C2, C22.130(C2×C4×D5), (C2×C20).356(C2×C4), (C2×C10).445(C2×D4), (C2×C4).99(C5⋊D4), C22.60(C2×C5⋊D4), (C2×C10).149(C4○D4), (C2×C10).213(C22×C4), (C2×Dic5).106(C2×C4), (C2×C10.D4).29C2, SmallGroup(320,599)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.96(C4×D4)
G = < a,b,c,d | a10=b4=c4=1, d2=a5, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 510 in 170 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C23, C10, C42, C4⋊C4, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.65C23, C4×Dic5, C10.D4, C10.D4, C5×C4⋊C4, C22×Dic5, C22×C20, C10.10C42, C2×C4×Dic5, C2×C10.D4, C10×C4⋊C4, C10.96(C4×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C4×D5, C5⋊D4, C22×D5, C23.65C23, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, C2×C5⋊D4, Dic5⋊3Q8, Dic5.Q8, D5×C4⋊C4, D10⋊Q8, C4×C5⋊D4, Dic5⋊D4, Dic5⋊Q8, C10.96(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 78 38 58)(2 79 39 59)(3 80 40 60)(4 71 31 51)(5 72 32 52)(6 73 33 53)(7 74 34 54)(8 75 35 55)(9 76 36 56)(10 77 37 57)(11 281 306 261)(12 282 307 262)(13 283 308 263)(14 284 309 264)(15 285 310 265)(16 286 301 266)(17 287 302 267)(18 288 303 268)(19 289 304 269)(20 290 305 270)(21 81 41 61)(22 82 42 62)(23 83 43 63)(24 84 44 64)(25 85 45 65)(26 86 46 66)(27 87 47 67)(28 88 48 68)(29 89 49 69)(30 90 50 70)(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)(101 167 121 146)(102 168 122 147)(103 169 123 148)(104 170 124 149)(105 161 125 150)(106 162 126 141)(107 163 127 142)(108 164 128 143)(109 165 129 144)(110 166 130 145)(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)(181 241 201 221)(182 242 202 222)(183 243 203 223)(184 244 204 224)(185 245 205 225)(186 246 206 226)(187 247 207 227)(188 248 208 228)(189 249 209 229)(190 250 210 230)(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)
(1 185 30 172)(2 184 21 171)(3 183 22 180)(4 182 23 179)(5 181 24 178)(6 190 25 177)(7 189 26 176)(8 188 27 175)(9 187 28 174)(10 186 29 173)(11 170 313 157)(12 169 314 156)(13 168 315 155)(14 167 316 154)(15 166 317 153)(16 165 318 152)(17 164 319 151)(18 163 320 160)(19 162 311 159)(20 161 312 158)(31 202 43 199)(32 201 44 198)(33 210 45 197)(34 209 46 196)(35 208 47 195)(36 207 48 194)(37 206 49 193)(38 205 50 192)(39 204 41 191)(40 203 42 200)(51 222 63 219)(52 221 64 218)(53 230 65 217)(54 229 66 216)(55 228 67 215)(56 227 68 214)(57 226 69 213)(58 225 70 212)(59 224 61 211)(60 223 62 220)(71 242 83 239)(72 241 84 238)(73 250 85 237)(74 249 86 236)(75 248 87 235)(76 247 88 234)(77 246 89 233)(78 245 90 232)(79 244 81 231)(80 243 82 240)(91 262 103 259)(92 261 104 258)(93 270 105 257)(94 269 106 256)(95 268 107 255)(96 267 108 254)(97 266 109 253)(98 265 110 252)(99 264 101 251)(100 263 102 260)(111 282 123 279)(112 281 124 278)(113 290 125 277)(114 289 126 276)(115 288 127 275)(116 287 128 274)(117 286 129 273)(118 285 130 272)(119 284 121 271)(120 283 122 280)(131 302 143 299)(132 301 144 298)(133 310 145 297)(134 309 146 296)(135 308 147 295)(136 307 148 294)(137 306 149 293)(138 305 150 292)(139 304 141 291)(140 303 142 300)
(1 93 6 98)(2 94 7 99)(3 95 8 100)(4 96 9 91)(5 97 10 92)(11 233 16 238)(12 234 17 239)(13 235 18 240)(14 236 19 231)(15 237 20 232)(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)(81 167 86 162)(82 168 87 163)(83 169 88 164)(84 170 89 165)(85 161 90 166)(171 269 176 264)(172 270 177 265)(173 261 178 266)(174 262 179 267)(175 263 180 268)(181 253 186 258)(182 254 187 259)(183 255 188 260)(184 256 189 251)(185 257 190 252)(191 289 196 284)(192 290 197 285)(193 281 198 286)(194 282 199 287)(195 283 200 288)(201 273 206 278)(202 274 207 279)(203 275 208 280)(204 276 209 271)(205 277 210 272)(211 309 216 304)(212 310 217 305)(213 301 218 306)(214 302 219 307)(215 303 220 308)(221 293 226 298)(222 294 227 299)(223 295 228 300)(224 296 229 291)(225 297 230 292)(241 313 246 318)(242 314 247 319)(243 315 248 320)(244 316 249 311)(245 317 250 312)
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), (1,78,38,58)(2,79,39,59)(3,80,40,60)(4,71,31,51)(5,72,32,52)(6,73,33,53)(7,74,34,54)(8,75,35,55)(9,76,36,56)(10,77,37,57)(11,281,306,261)(12,282,307,262)(13,283,308,263)(14,284,309,264)(15,285,310,265)(16,286,301,266)(17,287,302,267)(18,288,303,268)(19,289,304,269)(20,290,305,270)(21,81,41,61)(22,82,42,62)(23,83,43,63)(24,84,44,64)(25,85,45,65)(26,86,46,66)(27,87,47,67)(28,88,48,68)(29,89,49,69)(30,90,50,70)(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)(101,167,121,146)(102,168,122,147)(103,169,123,148)(104,170,124,149)(105,161,125,150)(106,162,126,141)(107,163,127,142)(108,164,128,143)(109,165,129,144)(110,166,130,145)(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)(181,241,201,221)(182,242,202,222)(183,243,203,223)(184,244,204,224)(185,245,205,225)(186,246,206,226)(187,247,207,227)(188,248,208,228)(189,249,209,229)(190,250,210,230)(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), (1,185,30,172)(2,184,21,171)(3,183,22,180)(4,182,23,179)(5,181,24,178)(6,190,25,177)(7,189,26,176)(8,188,27,175)(9,187,28,174)(10,186,29,173)(11,170,313,157)(12,169,314,156)(13,168,315,155)(14,167,316,154)(15,166,317,153)(16,165,318,152)(17,164,319,151)(18,163,320,160)(19,162,311,159)(20,161,312,158)(31,202,43,199)(32,201,44,198)(33,210,45,197)(34,209,46,196)(35,208,47,195)(36,207,48,194)(37,206,49,193)(38,205,50,192)(39,204,41,191)(40,203,42,200)(51,222,63,219)(52,221,64,218)(53,230,65,217)(54,229,66,216)(55,228,67,215)(56,227,68,214)(57,226,69,213)(58,225,70,212)(59,224,61,211)(60,223,62,220)(71,242,83,239)(72,241,84,238)(73,250,85,237)(74,249,86,236)(75,248,87,235)(76,247,88,234)(77,246,89,233)(78,245,90,232)(79,244,81,231)(80,243,82,240)(91,262,103,259)(92,261,104,258)(93,270,105,257)(94,269,106,256)(95,268,107,255)(96,267,108,254)(97,266,109,253)(98,265,110,252)(99,264,101,251)(100,263,102,260)(111,282,123,279)(112,281,124,278)(113,290,125,277)(114,289,126,276)(115,288,127,275)(116,287,128,274)(117,286,129,273)(118,285,130,272)(119,284,121,271)(120,283,122,280)(131,302,143,299)(132,301,144,298)(133,310,145,297)(134,309,146,296)(135,308,147,295)(136,307,148,294)(137,306,149,293)(138,305,150,292)(139,304,141,291)(140,303,142,300), (1,93,6,98)(2,94,7,99)(3,95,8,100)(4,96,9,91)(5,97,10,92)(11,233,16,238)(12,234,17,239)(13,235,18,240)(14,236,19,231)(15,237,20,232)(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)(81,167,86,162)(82,168,87,163)(83,169,88,164)(84,170,89,165)(85,161,90,166)(171,269,176,264)(172,270,177,265)(173,261,178,266)(174,262,179,267)(175,263,180,268)(181,253,186,258)(182,254,187,259)(183,255,188,260)(184,256,189,251)(185,257,190,252)(191,289,196,284)(192,290,197,285)(193,281,198,286)(194,282,199,287)(195,283,200,288)(201,273,206,278)(202,274,207,279)(203,275,208,280)(204,276,209,271)(205,277,210,272)(211,309,216,304)(212,310,217,305)(213,301,218,306)(214,302,219,307)(215,303,220,308)(221,293,226,298)(222,294,227,299)(223,295,228,300)(224,296,229,291)(225,297,230,292)(241,313,246,318)(242,314,247,319)(243,315,248,320)(244,316,249,311)(245,317,250,312)>;
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), (1,78,38,58)(2,79,39,59)(3,80,40,60)(4,71,31,51)(5,72,32,52)(6,73,33,53)(7,74,34,54)(8,75,35,55)(9,76,36,56)(10,77,37,57)(11,281,306,261)(12,282,307,262)(13,283,308,263)(14,284,309,264)(15,285,310,265)(16,286,301,266)(17,287,302,267)(18,288,303,268)(19,289,304,269)(20,290,305,270)(21,81,41,61)(22,82,42,62)(23,83,43,63)(24,84,44,64)(25,85,45,65)(26,86,46,66)(27,87,47,67)(28,88,48,68)(29,89,49,69)(30,90,50,70)(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)(101,167,121,146)(102,168,122,147)(103,169,123,148)(104,170,124,149)(105,161,125,150)(106,162,126,141)(107,163,127,142)(108,164,128,143)(109,165,129,144)(110,166,130,145)(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)(181,241,201,221)(182,242,202,222)(183,243,203,223)(184,244,204,224)(185,245,205,225)(186,246,206,226)(187,247,207,227)(188,248,208,228)(189,249,209,229)(190,250,210,230)(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), (1,185,30,172)(2,184,21,171)(3,183,22,180)(4,182,23,179)(5,181,24,178)(6,190,25,177)(7,189,26,176)(8,188,27,175)(9,187,28,174)(10,186,29,173)(11,170,313,157)(12,169,314,156)(13,168,315,155)(14,167,316,154)(15,166,317,153)(16,165,318,152)(17,164,319,151)(18,163,320,160)(19,162,311,159)(20,161,312,158)(31,202,43,199)(32,201,44,198)(33,210,45,197)(34,209,46,196)(35,208,47,195)(36,207,48,194)(37,206,49,193)(38,205,50,192)(39,204,41,191)(40,203,42,200)(51,222,63,219)(52,221,64,218)(53,230,65,217)(54,229,66,216)(55,228,67,215)(56,227,68,214)(57,226,69,213)(58,225,70,212)(59,224,61,211)(60,223,62,220)(71,242,83,239)(72,241,84,238)(73,250,85,237)(74,249,86,236)(75,248,87,235)(76,247,88,234)(77,246,89,233)(78,245,90,232)(79,244,81,231)(80,243,82,240)(91,262,103,259)(92,261,104,258)(93,270,105,257)(94,269,106,256)(95,268,107,255)(96,267,108,254)(97,266,109,253)(98,265,110,252)(99,264,101,251)(100,263,102,260)(111,282,123,279)(112,281,124,278)(113,290,125,277)(114,289,126,276)(115,288,127,275)(116,287,128,274)(117,286,129,273)(118,285,130,272)(119,284,121,271)(120,283,122,280)(131,302,143,299)(132,301,144,298)(133,310,145,297)(134,309,146,296)(135,308,147,295)(136,307,148,294)(137,306,149,293)(138,305,150,292)(139,304,141,291)(140,303,142,300), (1,93,6,98)(2,94,7,99)(3,95,8,100)(4,96,9,91)(5,97,10,92)(11,233,16,238)(12,234,17,239)(13,235,18,240)(14,236,19,231)(15,237,20,232)(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)(81,167,86,162)(82,168,87,163)(83,169,88,164)(84,170,89,165)(85,161,90,166)(171,269,176,264)(172,270,177,265)(173,261,178,266)(174,262,179,267)(175,263,180,268)(181,253,186,258)(182,254,187,259)(183,255,188,260)(184,256,189,251)(185,257,190,252)(191,289,196,284)(192,290,197,285)(193,281,198,286)(194,282,199,287)(195,283,200,288)(201,273,206,278)(202,274,207,279)(203,275,208,280)(204,276,209,271)(205,277,210,272)(211,309,216,304)(212,310,217,305)(213,301,218,306)(214,302,219,307)(215,303,220,308)(221,293,226,298)(222,294,227,299)(223,295,228,300)(224,296,229,291)(225,297,230,292)(241,313,246,318)(242,314,247,319)(243,315,248,320)(244,316,249,311)(245,317,250,312) );
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)], [(1,78,38,58),(2,79,39,59),(3,80,40,60),(4,71,31,51),(5,72,32,52),(6,73,33,53),(7,74,34,54),(8,75,35,55),(9,76,36,56),(10,77,37,57),(11,281,306,261),(12,282,307,262),(13,283,308,263),(14,284,309,264),(15,285,310,265),(16,286,301,266),(17,287,302,267),(18,288,303,268),(19,289,304,269),(20,290,305,270),(21,81,41,61),(22,82,42,62),(23,83,43,63),(24,84,44,64),(25,85,45,65),(26,86,46,66),(27,87,47,67),(28,88,48,68),(29,89,49,69),(30,90,50,70),(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),(101,167,121,146),(102,168,122,147),(103,169,123,148),(104,170,124,149),(105,161,125,150),(106,162,126,141),(107,163,127,142),(108,164,128,143),(109,165,129,144),(110,166,130,145),(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),(181,241,201,221),(182,242,202,222),(183,243,203,223),(184,244,204,224),(185,245,205,225),(186,246,206,226),(187,247,207,227),(188,248,208,228),(189,249,209,229),(190,250,210,230),(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)], [(1,185,30,172),(2,184,21,171),(3,183,22,180),(4,182,23,179),(5,181,24,178),(6,190,25,177),(7,189,26,176),(8,188,27,175),(9,187,28,174),(10,186,29,173),(11,170,313,157),(12,169,314,156),(13,168,315,155),(14,167,316,154),(15,166,317,153),(16,165,318,152),(17,164,319,151),(18,163,320,160),(19,162,311,159),(20,161,312,158),(31,202,43,199),(32,201,44,198),(33,210,45,197),(34,209,46,196),(35,208,47,195),(36,207,48,194),(37,206,49,193),(38,205,50,192),(39,204,41,191),(40,203,42,200),(51,222,63,219),(52,221,64,218),(53,230,65,217),(54,229,66,216),(55,228,67,215),(56,227,68,214),(57,226,69,213),(58,225,70,212),(59,224,61,211),(60,223,62,220),(71,242,83,239),(72,241,84,238),(73,250,85,237),(74,249,86,236),(75,248,87,235),(76,247,88,234),(77,246,89,233),(78,245,90,232),(79,244,81,231),(80,243,82,240),(91,262,103,259),(92,261,104,258),(93,270,105,257),(94,269,106,256),(95,268,107,255),(96,267,108,254),(97,266,109,253),(98,265,110,252),(99,264,101,251),(100,263,102,260),(111,282,123,279),(112,281,124,278),(113,290,125,277),(114,289,126,276),(115,288,127,275),(116,287,128,274),(117,286,129,273),(118,285,130,272),(119,284,121,271),(120,283,122,280),(131,302,143,299),(132,301,144,298),(133,310,145,297),(134,309,146,296),(135,308,147,295),(136,307,148,294),(137,306,149,293),(138,305,150,292),(139,304,141,291),(140,303,142,300)], [(1,93,6,98),(2,94,7,99),(3,95,8,100),(4,96,9,91),(5,97,10,92),(11,233,16,238),(12,234,17,239),(13,235,18,240),(14,236,19,231),(15,237,20,232),(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),(81,167,86,162),(82,168,87,163),(83,169,88,164),(84,170,89,165),(85,161,90,166),(171,269,176,264),(172,270,177,265),(173,261,178,266),(174,262,179,267),(175,263,180,268),(181,253,186,258),(182,254,187,259),(183,255,188,260),(184,256,189,251),(185,257,190,252),(191,289,196,284),(192,290,197,285),(193,281,198,286),(194,282,199,287),(195,283,200,288),(201,273,206,278),(202,274,207,279),(203,275,208,280),(204,276,209,271),(205,277,210,272),(211,309,216,304),(212,310,217,305),(213,301,218,306),(214,302,219,307),(215,303,220,308),(221,293,226,298),(222,294,227,299),(223,295,228,300),(224,296,229,291),(225,297,230,292),(241,313,246,318),(242,314,247,319),(243,315,248,320),(244,316,249,311),(245,317,250,312)]])
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 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D5 | C4○D4 | D10 | C4×D5 | C5⋊D4 | C4○D20 | D4×D5 | D4⋊2D5 | Q8×D5 |
| kernel | C10.96(C4×D4) | C10.10C42 | C2×C4×Dic5 | C2×C10.D4 | C10×C4⋊C4 | C10.D4 | C2×Dic5 | C2×Dic5 | C2×C20 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 8 | 2 | 4 | 2 | 2 | 4 | 6 | 8 | 8 | 8 | 2 | 2 | 4 |
Matrix representation of C10.96(C4×D4) ►in GL6(𝔽41)
| 7 | 7 | 0 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 22 | 9 | 0 | 0 | 0 | 0 |
| 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 27 | 0 | 0 |
| 0 | 0 | 25 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 24 | 1 | 0 | 0 | 0 | 0 |
| 40 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 2 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [7,34,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,9,0,0,0,0,18,40],[22,19,0,0,0,0,9,19,0,0,0,0,0,0,12,25,0,0,0,0,27,29,0,0,0,0,0,0,1,9,0,0,0,0,18,40],[24,40,0,0,0,0,1,17,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,32,0,0,0,0,0,2,9] >;
C10.96(C4×D4) in GAP, Magma, Sage, TeX
C_{10}._{96}(C_4\times D_4) % in TeX
G:=Group("C10.96(C4xD4)"); // GroupNames label
G:=SmallGroup(320,599);
// by ID
G=gap.SmallGroup(320,599);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,120,219,58,12550]);
// 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,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations