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G = C10.51(C4×D4)  order 320 = 26·5

3rd non-split extension by C10 of C4×D4 acting via C4×D4/C22⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.51(C4×D4), C10.23(C4×Q8), C10.D48C4, C22.54(D4×D5), C22.13(Q8×D5), (C2×Dic5).17Q8, C2.2(D10⋊Q8), (C2×Dic5).183D4, (C22×C4).295D10, C2.7(C42⋊D5), C10.18(C22⋊Q8), (C22×C20).5C22, C10.7(C42.C2), C2.C42.3D5, C10.7(C422C2), C2.5(Dic53Q8), C2.6(Dic54D4), C22.29(C4○D20), C23.249(C22×D5), C10.22(C42⋊C2), C2.1(D10.12D4), C22.31(D42D5), C10.10C42.3C2, (C22×C10).277C23, C2.3(Dic5.Q8), C52(C23.63C23), C10.1(C22.D4), C2.2(C23.D10), (C22×Dic5).196C22, (C2×C4).59(C4×D5), C22.84(C2×C4×D5), (C2×C10).60(C2×Q8), (C2×C4×Dic5).26C2, (C2×C20).314(C2×C4), (C2×C10).191(C2×D4), (C2×Dic5).9(C2×C4), (C2×C10).55(C4○D4), (C2×C10.D4).6C2, (C2×C10).144(C22×C4), (C5×C2.C42).23C2, SmallGroup(320,279)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.51(C4×D4)
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C10.51(C4×D4)
C5C2×C10 — C10.51(C4×D4)
C1C23C2.C42

Generators and relations for C10.51(C4×D4)
 G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 478 in 154 conjugacy classes, 67 normal (51 characteristic)
C1, C2 [×7], C4 [×12], C22 [×7], C5, C2×C4 [×2], C2×C4 [×24], C23, C10 [×7], C42 [×2], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], Dic5 [×8], C20 [×4], C2×C10 [×7], C2.C42, C2.C42 [×3], C2×C42, C2×C4⋊C4 [×2], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.63C23, C4×Dic5 [×2], C10.D4 [×4], C10.D4 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×3], C5×C2.C42, C2×C4×Dic5, C2×C10.D4 [×2], C10.51(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C22×C4, C2×D4, C2×Q8, C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×D5 [×2], C22×D5, C23.63C23, C2×C4×D5, C4○D20 [×2], D4×D5, D42D5 [×2], Q8×D5, C42⋊D5, C23.D10, Dic54D4, D10.12D4, Dic53Q8, Dic5.Q8, D10⋊Q8, C10.51(C4×D4)

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim1111112222222444
type++++++-+++--
imageC1C2C2C2C2C4D4Q8D5C4○D4D10C4×D5C4○D20D4×D5D42D5Q8×D5
kernelC10.51(C4×D4)C10.10C42C5×C2.C42C2×C4×Dic5C2×C10.D4C10.D4C2×Dic5C2×Dic5C2.C42C2×C10C22×C4C2×C4C22C22C22C22
# reps13112822286816242

Matrix representation of C10.51(C4×D4) in GL6(𝔽41)

0350000
7340000
0040000
0004000
0000400
0000040
,
17180000
34240000
00302900
00171100
0000714
00001434
,
1120000
22300000
00323900
000900
00003822
0000223
,
100000
010000
00302900
00171100
000090
000009

G:=sub<GL(6,GF(41))| [0,7,0,0,0,0,35,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[17,34,0,0,0,0,18,24,0,0,0,0,0,0,30,17,0,0,0,0,29,11,0,0,0,0,0,0,7,14,0,0,0,0,14,34],[11,22,0,0,0,0,2,30,0,0,0,0,0,0,32,0,0,0,0,0,39,9,0,0,0,0,0,0,38,22,0,0,0,0,22,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,17,0,0,0,0,29,11,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C10.51(C4×D4) in GAP, Magma, Sage, TeX

C_{10}._{51}(C_4\times D_4)
% in TeX

G:=Group("C10.51(C4xD4)");
// GroupNames label

G:=SmallGroup(320,279);
// by ID

G=gap.SmallGroup(320,279);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,64,254,219,184,12550]);
// 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,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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