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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.51(C4×D4)
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — C10.51(C4×D4)
 Lower central C5 — C2×C10 — C10.51(C4×D4)
 Upper central C1 — C23 — C2.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, C4, C22, C5, C2×C4, C2×C4, C23, C10, C42, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.63C23, C4×Dic5, C10.D4, C10.D4, C22×Dic5, C22×C20, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C10.51(C4×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×D5, C22×D5, C23.63C23, C2×C4×D5, C4○D20, D4×D5, D42D5, 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 33 171)(2 196 34 180)(3 195 35 179)(4 194 36 178)(5 193 37 177)(6 192 38 176)(7 191 39 175)(8 200 40 174)(9 199 31 173)(10 198 32 172)(11 207 48 190)(12 206 49 189)(13 205 50 188)(14 204 41 187)(15 203 42 186)(16 202 43 185)(17 201 44 184)(18 210 45 183)(19 209 46 182)(20 208 47 181)(21 170 309 141)(22 169 310 150)(23 168 301 149)(24 167 302 148)(25 166 303 147)(26 165 304 146)(27 164 305 145)(28 163 306 144)(29 162 307 143)(30 161 308 142)(51 240 80 214)(52 239 71 213)(53 238 72 212)(54 237 73 211)(55 236 74 220)(56 235 75 219)(57 234 76 218)(58 233 77 217)(59 232 78 216)(60 231 79 215)(61 246 89 229)(62 245 90 228)(63 244 81 227)(64 243 82 226)(65 242 83 225)(66 241 84 224)(67 250 85 223)(68 249 86 222)(69 248 87 221)(70 247 88 230)(91 280 120 254)(92 279 111 253)(93 278 112 252)(94 277 113 251)(95 276 114 260)(96 275 115 259)(97 274 116 258)(98 273 117 257)(99 272 118 256)(100 271 119 255)(101 286 129 269)(102 285 130 268)(103 284 121 267)(104 283 122 266)(105 282 123 265)(106 281 124 264)(107 290 125 263)(108 289 126 262)(109 288 127 261)(110 287 128 270)(131 320 160 294)(132 319 151 293)(133 318 152 292)(134 317 153 291)(135 316 154 300)(136 315 155 299)(137 314 156 298)(138 313 157 297)(139 312 158 296)(140 311 159 295)
(1 25 13 312)(2 24 14 311)(3 23 15 320)(4 22 16 319)(5 21 17 318)(6 30 18 317)(7 29 19 316)(8 28 20 315)(9 27 11 314)(10 26 12 313)(31 305 48 298)(32 304 49 297)(33 303 50 296)(34 302 41 295)(35 301 42 294)(36 310 43 293)(37 309 44 292)(38 308 45 291)(39 307 46 300)(40 306 47 299)(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)(91 240 109 248)(92 239 110 247)(93 238 101 246)(94 237 102 245)(95 236 103 244)(96 235 104 243)(97 234 105 242)(98 233 106 241)(99 232 107 250)(100 231 108 249)(111 213 128 230)(112 212 129 229)(113 211 130 228)(114 220 121 227)(115 219 122 226)(116 218 123 225)(117 217 124 224)(118 216 125 223)(119 215 126 222)(120 214 127 221)(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)(11 123 16 128)(12 124 17 129)(13 125 18 130)(14 126 19 121)(15 127 20 122)(21 229 26 224)(22 230 27 225)(23 221 28 226)(24 222 29 227)(25 223 30 228)(31 97 36 92)(32 98 37 93)(33 99 38 94)(34 100 39 95)(35 91 40 96)(41 108 46 103)(42 109 47 104)(43 110 48 105)(44 101 49 106)(45 102 50 107)(51 155 56 160)(52 156 57 151)(53 157 58 152)(54 158 59 153)(55 159 60 154)(61 165 66 170)(62 166 67 161)(63 167 68 162)(64 168 69 163)(65 169 70 164)(71 137 76 132)(72 138 77 133)(73 139 78 134)(74 140 79 135)(75 131 80 136)(81 148 86 143)(82 149 87 144)(83 150 88 145)(84 141 89 146)(85 142 90 147)(171 272 176 277)(172 273 177 278)(173 274 178 279)(174 275 179 280)(175 276 180 271)(181 283 186 288)(182 284 187 289)(183 285 188 290)(184 286 189 281)(185 287 190 282)(191 260 196 255)(192 251 197 256)(193 252 198 257)(194 253 199 258)(195 254 200 259)(201 269 206 264)(202 270 207 265)(203 261 208 266)(204 262 209 267)(205 263 210 268)(211 312 216 317)(212 313 217 318)(213 314 218 319)(214 315 219 320)(215 316 220 311)(231 300 236 295)(232 291 237 296)(233 292 238 297)(234 293 239 298)(235 294 240 299)(241 309 246 304)(242 310 247 305)(243 301 248 306)(244 302 249 307)(245 303 250 308)

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

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

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

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 4 4 4 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 C4 D4 Q8 D5 C4○D4 D10 C4×D5 C4○D20 D4×D5 D4⋊2D5 Q8×D5 kernel C10.51(C4×D4) C10.10C42 C5×C2.C42 C2×C4×Dic5 C2×C10.D4 C10.D4 C2×Dic5 C2×Dic5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 3 1 1 2 8 2 2 2 8 6 8 16 2 4 2

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

 0 35 0 0 0 0 7 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 18 0 0 0 0 34 24 0 0 0 0 0 0 30 29 0 0 0 0 17 11 0 0 0 0 0 0 7 14 0 0 0 0 14 34
,
 11 2 0 0 0 0 22 30 0 0 0 0 0 0 32 39 0 0 0 0 0 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 29 0 0 0 0 17 11 0 0 0 0 0 0 9 0 0 0 0 0 0 9

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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