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G = C10.49(C4×D4)  order 320 = 26·5

1st non-split extension by C10 of C4×D4 acting via C4×D4/C22⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.49(C4×D4), C2.2(C20⋊Q8), C10.11(C4×Q8), (C2×C20).33Q8, Dic52(C4⋊C4), C10.D45C4, C10.10(C4⋊Q8), C2.5(C4×Dic10), C22.51(D4×D5), (C22×C4).6D10, C22.10(Q8×D5), C10.1(C4⋊D4), (C2×Dic5).15Q8, (C2×C4).20Dic10, C10.2(C22⋊Q8), C2.1(D10⋊D4), (C2×Dic5).182D4, C10.5(C42.C2), C2.C42.8D5, C2.5(Dic54D4), C22.26(C4○D20), C22.13(C2×Dic10), C23.244(C22×D5), C22.26(D42D5), (C22×C20).327C22, (C22×C10).272C23, C2.1(Dic5.Q8), C52(C23.65C23), (C22×Dic5).2C22, C10.10C42.31C2, C2.2(Dic5.14D4), C2.5(D5×C4⋊C4), C10.24(C2×C4⋊C4), (C2×C4).23(C4×D5), C22.81(C2×C4×D5), (C2×C4⋊Dic5).3C2, (C2×C10).57(C2×Q8), (C2×C4×Dic5).21C2, (C2×C20).204(C2×C4), (C2×C10).188(C2×D4), (C2×Dic5).85(C2×C4), (C2×C10.D4).4C2, (C2×C10).121(C4○D4), (C2×C10).139(C22×C4), (C5×C2.C42).15C2, SmallGroup(320,274)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.49(C4×D4)
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — C10.49(C4×D4)
C5C2×C10 — C10.49(C4×D4)
C1C23C2.C42

Generators and relations for C10.49(C4×D4)
 G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 526 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, C2.C42, C2×C42, C2×C4⋊C4 [×4], 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], C4⋊Dic5 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C10.D4 [×3], C2×C4⋊Dic5, C10.49(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, Dic10 [×2], C4×D5 [×2], C22×D5, C23.65C23, C2×Dic10, C2×C4×D5, C4○D20, D4×D5 [×2], D42D5, Q8×D5, C4×Dic10, Dic5.14D4, Dic54D4, D10⋊D4, C20⋊Q8, Dic5.Q8, D5×C4⋊C4, C10.49(C4×D4)

Smallest permutation representation of C10.49(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)
(1 217 33 250)(2 216 34 249)(3 215 35 248)(4 214 36 247)(5 213 37 246)(6 212 38 245)(7 211 39 244)(8 220 40 243)(9 219 31 242)(10 218 32 241)(11 92 293 124)(12 91 294 123)(13 100 295 122)(14 99 296 121)(15 98 297 130)(16 97 298 129)(17 96 299 128)(18 95 300 127)(19 94 291 126)(20 93 292 125)(21 229 46 236)(22 228 47 235)(23 227 48 234)(24 226 49 233)(25 225 50 232)(26 224 41 231)(27 223 42 240)(28 222 43 239)(29 221 44 238)(30 230 45 237)(51 202 83 179)(52 201 84 178)(53 210 85 177)(54 209 86 176)(55 208 87 175)(56 207 88 174)(57 206 89 173)(58 205 90 172)(59 204 81 171)(60 203 82 180)(61 191 79 184)(62 200 80 183)(63 199 71 182)(64 198 72 181)(65 197 73 190)(66 196 74 189)(67 195 75 188)(68 194 76 187)(69 193 77 186)(70 192 78 185)(101 309 119 316)(102 308 120 315)(103 307 111 314)(104 306 112 313)(105 305 113 312)(106 304 114 311)(107 303 115 320)(108 302 116 319)(109 301 117 318)(110 310 118 317)(131 274 164 267)(132 273 165 266)(133 272 166 265)(134 271 167 264)(135 280 168 263)(136 279 169 262)(137 278 170 261)(138 277 161 270)(139 276 162 269)(140 275 163 268)(141 289 159 256)(142 288 160 255)(143 287 151 254)(144 286 152 253)(145 285 153 252)(146 284 154 251)(147 283 155 260)(148 282 156 259)(149 281 157 258)(150 290 158 257)
(1 177 30 190)(2 176 21 189)(3 175 22 188)(4 174 23 187)(5 173 24 186)(6 172 25 185)(7 171 26 184)(8 180 27 183)(9 179 28 182)(10 178 29 181)(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)
(1 110 6 105)(2 101 7 106)(3 102 8 107)(4 103 9 108)(5 104 10 109)(11 233 16 238)(12 234 17 239)(13 235 18 240)(14 236 19 231)(15 237 20 232)(21 99 26 94)(22 100 27 95)(23 91 28 96)(24 92 29 97)(25 93 30 98)(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), (1,217,33,250)(2,216,34,249)(3,215,35,248)(4,214,36,247)(5,213,37,246)(6,212,38,245)(7,211,39,244)(8,220,40,243)(9,219,31,242)(10,218,32,241)(11,92,293,124)(12,91,294,123)(13,100,295,122)(14,99,296,121)(15,98,297,130)(16,97,298,129)(17,96,299,128)(18,95,300,127)(19,94,291,126)(20,93,292,125)(21,229,46,236)(22,228,47,235)(23,227,48,234)(24,226,49,233)(25,225,50,232)(26,224,41,231)(27,223,42,240)(28,222,43,239)(29,221,44,238)(30,230,45,237)(51,202,83,179)(52,201,84,178)(53,210,85,177)(54,209,86,176)(55,208,87,175)(56,207,88,174)(57,206,89,173)(58,205,90,172)(59,204,81,171)(60,203,82,180)(61,191,79,184)(62,200,80,183)(63,199,71,182)(64,198,72,181)(65,197,73,190)(66,196,74,189)(67,195,75,188)(68,194,76,187)(69,193,77,186)(70,192,78,185)(101,309,119,316)(102,308,120,315)(103,307,111,314)(104,306,112,313)(105,305,113,312)(106,304,114,311)(107,303,115,320)(108,302,116,319)(109,301,117,318)(110,310,118,317)(131,274,164,267)(132,273,165,266)(133,272,166,265)(134,271,167,264)(135,280,168,263)(136,279,169,262)(137,278,170,261)(138,277,161,270)(139,276,162,269)(140,275,163,268)(141,289,159,256)(142,288,160,255)(143,287,151,254)(144,286,152,253)(145,285,153,252)(146,284,154,251)(147,283,155,260)(148,282,156,259)(149,281,157,258)(150,290,158,257), (1,177,30,190)(2,176,21,189)(3,175,22,188)(4,174,23,187)(5,173,24,186)(6,172,25,185)(7,171,26,184)(8,180,27,183)(9,179,28,182)(10,178,29,181)(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), (1,110,6,105)(2,101,7,106)(3,102,8,107)(4,103,9,108)(5,104,10,109)(11,233,16,238)(12,234,17,239)(13,235,18,240)(14,236,19,231)(15,237,20,232)(21,99,26,94)(22,100,27,95)(23,91,28,96)(24,92,29,97)(25,93,30,98)(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), (1,217,33,250)(2,216,34,249)(3,215,35,248)(4,214,36,247)(5,213,37,246)(6,212,38,245)(7,211,39,244)(8,220,40,243)(9,219,31,242)(10,218,32,241)(11,92,293,124)(12,91,294,123)(13,100,295,122)(14,99,296,121)(15,98,297,130)(16,97,298,129)(17,96,299,128)(18,95,300,127)(19,94,291,126)(20,93,292,125)(21,229,46,236)(22,228,47,235)(23,227,48,234)(24,226,49,233)(25,225,50,232)(26,224,41,231)(27,223,42,240)(28,222,43,239)(29,221,44,238)(30,230,45,237)(51,202,83,179)(52,201,84,178)(53,210,85,177)(54,209,86,176)(55,208,87,175)(56,207,88,174)(57,206,89,173)(58,205,90,172)(59,204,81,171)(60,203,82,180)(61,191,79,184)(62,200,80,183)(63,199,71,182)(64,198,72,181)(65,197,73,190)(66,196,74,189)(67,195,75,188)(68,194,76,187)(69,193,77,186)(70,192,78,185)(101,309,119,316)(102,308,120,315)(103,307,111,314)(104,306,112,313)(105,305,113,312)(106,304,114,311)(107,303,115,320)(108,302,116,319)(109,301,117,318)(110,310,118,317)(131,274,164,267)(132,273,165,266)(133,272,166,265)(134,271,167,264)(135,280,168,263)(136,279,169,262)(137,278,170,261)(138,277,161,270)(139,276,162,269)(140,275,163,268)(141,289,159,256)(142,288,160,255)(143,287,151,254)(144,286,152,253)(145,285,153,252)(146,284,154,251)(147,283,155,260)(148,282,156,259)(149,281,157,258)(150,290,158,257), (1,177,30,190)(2,176,21,189)(3,175,22,188)(4,174,23,187)(5,173,24,186)(6,172,25,185)(7,171,26,184)(8,180,27,183)(9,179,28,182)(10,178,29,181)(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), (1,110,6,105)(2,101,7,106)(3,102,8,107)(4,103,9,108)(5,104,10,109)(11,233,16,238)(12,234,17,239)(13,235,18,240)(14,236,19,231)(15,237,20,232)(21,99,26,94)(22,100,27,95)(23,91,28,96)(24,92,29,97)(25,93,30,98)(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)], [(1,217,33,250),(2,216,34,249),(3,215,35,248),(4,214,36,247),(5,213,37,246),(6,212,38,245),(7,211,39,244),(8,220,40,243),(9,219,31,242),(10,218,32,241),(11,92,293,124),(12,91,294,123),(13,100,295,122),(14,99,296,121),(15,98,297,130),(16,97,298,129),(17,96,299,128),(18,95,300,127),(19,94,291,126),(20,93,292,125),(21,229,46,236),(22,228,47,235),(23,227,48,234),(24,226,49,233),(25,225,50,232),(26,224,41,231),(27,223,42,240),(28,222,43,239),(29,221,44,238),(30,230,45,237),(51,202,83,179),(52,201,84,178),(53,210,85,177),(54,209,86,176),(55,208,87,175),(56,207,88,174),(57,206,89,173),(58,205,90,172),(59,204,81,171),(60,203,82,180),(61,191,79,184),(62,200,80,183),(63,199,71,182),(64,198,72,181),(65,197,73,190),(66,196,74,189),(67,195,75,188),(68,194,76,187),(69,193,77,186),(70,192,78,185),(101,309,119,316),(102,308,120,315),(103,307,111,314),(104,306,112,313),(105,305,113,312),(106,304,114,311),(107,303,115,320),(108,302,116,319),(109,301,117,318),(110,310,118,317),(131,274,164,267),(132,273,165,266),(133,272,166,265),(134,271,167,264),(135,280,168,263),(136,279,169,262),(137,278,170,261),(138,277,161,270),(139,276,162,269),(140,275,163,268),(141,289,159,256),(142,288,160,255),(143,287,151,254),(144,286,152,253),(145,285,153,252),(146,284,154,251),(147,283,155,260),(148,282,156,259),(149,281,157,258),(150,290,158,257)], [(1,177,30,190),(2,176,21,189),(3,175,22,188),(4,174,23,187),(5,173,24,186),(6,172,25,185),(7,171,26,184),(8,180,27,183),(9,179,28,182),(10,178,29,181),(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)], [(1,110,6,105),(2,101,7,106),(3,102,8,107),(4,103,9,108),(5,104,10,109),(11,233,16,238),(12,234,17,239),(13,235,18,240),(14,236,19,231),(15,237,20,232),(21,99,26,94),(22,100,27,95),(23,91,28,96),(24,92,29,97),(25,93,30,98),(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···2G4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T5A5B10A···10N20A···20X
order12···2444444444···444445510···1020···20
size11···12222444410···1020202020222···24···4

68 irreducible representations

dim1111111222222222444
type+++++++--++-+--
imageC1C2C2C2C2C2C4D4Q8Q8D5C4○D4D10Dic10C4×D5C4○D20D4×D5D42D5Q8×D5
kernelC10.49(C4×D4)C10.10C42C5×C2.C42C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C10.D4C2×Dic5C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps1111318422246888422

Matrix representation of C10.49(C4×D4) in GL6(𝔽41)

060000
3470000
0040000
0004000
000010
000001
,
16150000
2250000
001000
00184000
000010
000001
,
25260000
39160000
009000
00393200
0000329
000009
,
11280000
22300000
00183900
00192300
0000112
00002230

G:=sub<GL(6,GF(41))| [0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,2,0,0,0,0,15,25,0,0,0,0,0,0,1,18,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,39,0,0,0,0,26,16,0,0,0,0,0,0,9,39,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,9,9],[11,22,0,0,0,0,28,30,0,0,0,0,0,0,18,19,0,0,0,0,39,23,0,0,0,0,0,0,11,22,0,0,0,0,2,30] >;

C10.49(C4×D4) in GAP, Magma, Sage, TeX

C_{10}._{49}(C_4\times D_4)
% in TeX

G:=Group("C10.49(C4xD4)");
// GroupNames label

G:=SmallGroup(320,274);
// by ID

G=gap.SmallGroup(320,274);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=a^5,b*a*b^-1=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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