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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), Dic53(C4⋊C4), C10.D49C4, (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(Dic53Q8), C2.5(Dic5⋊D4), C2.1(Dic5⋊Q8), C22.55(C4○D20), C10.15(C42.C2), C23.286(C22×D5), C22.51(D42D5), (C22×C20).345C22, (C22×C10).336C23, C54(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

C1C2×C10 — C10.96(C4×D4)
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — C10.96(C4×D4)
C5C2×C10 — C10.96(C4×D4)
C1C23C2×C4⋊C4

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 [×7], C4 [×14], C22 [×7], C5, C2×C4 [×4], C2×C4 [×24], C23, C10 [×7], C42 [×2], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×4], Dic5 [×4], Dic5 [×5], C20 [×5], C2×C10 [×7], C2.C42 [×2], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C2×Dic5 [×10], C2×Dic5 [×7], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.65C23, C4×Dic5 [×2], C10.D4 [×4], C10.D4 [×4], C5×C4⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×2], C2×C4×Dic5, C2×C10.D4 [×3], C10×C4⋊C4, C10.96(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], D10 [×3], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C23.65C23, C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5 [×2], C2×C5⋊D4, Dic53Q8, Dic5.Q8, D5×C4⋊C4, D10⋊Q8, C4×C5⋊D4, Dic5⋊D4, Dic5⋊Q8, C10.96(C4×D4)

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

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

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

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

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T5A5B10A···10N20A···20X
order12···2444444444···444445510···1020···20
size11···12222444410···1020202020222···24···4

68 irreducible representations

dim111111222222222444
type++++++-++++--
imageC1C2C2C2C2C4D4Q8D4D5C4○D4D10C4×D5C5⋊D4C4○D20D4×D5D42D5Q8×D5
kernelC10.96(C4×D4)C10.10C42C2×C4×Dic5C2×C10.D4C10×C4⋊C4C10.D4C2×Dic5C2×Dic5C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps121318242246888224

Matrix representation of C10.96(C4×D4) in GL6(𝔽41)

770000
34400000
007700
00344000
0000400
0000040
,
900000
090000
001000
000100
0000118
0000940
,
2290000
19190000
00122700
00252900
0000118
0000940
,
2410000
40170000
00174000
0012400
0000322
000009

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

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