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G = C4⋊Dic515C4order 320 = 26·5

4th semidirect product of C4⋊Dic5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic515C4, C10.75(C4×D4), (C2×C20).34Q8, C10.12(C4×Q8), C2.6(C4×Dic10), C22.55(D4×D5), (C2×C4).21Dic10, (C22×C4).10D10, C10.3(C22⋊Q8), C2.4(D208C4), (C2×Dic5).184D4, C2.1(C4.Dic10), C10.9(C42.C2), C22.31(C4○D20), (C22×C20).43C22, C22.14(C2×Dic10), C2.C42.10D5, C23.251(C22×D5), C10.17(C422C2), C10.38(C42⋊C2), C2.2(D10.12D4), C22.33(D42D5), (C22×C10).279C23, C54(C23.63C23), C22.14(Q82D5), C10.3(C22.D4), (C22×Dic5).5C22, C10.10C42.23C2, C2.3(Dic5.14D4), C2.8(C23.11D10), (C2×C4).25(C4×D5), C22.86(C2×C4×D5), (C2×C4⋊Dic5).5C2, (C2×C10).19(C2×Q8), (C2×C4×Dic5).28C2, (C2×C20).206(C2×C4), C2.2(C4⋊C4⋊D5), (C2×C10).192(C2×D4), (C2×Dic5).88(C2×C4), (C2×C10.D4).7C2, (C2×C10).180(C4○D4), (C2×C10).146(C22×C4), (C5×C2.C42).17C2, SmallGroup(320,281)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4⋊Dic515C4
C1C5C10C2×C10C22×C10C22×Dic5C2×C4⋊Dic5 — C4⋊Dic515C4
C5C2×C10 — C4⋊Dic515C4
C1C23C2.C42

Generators and relations for C4⋊Dic515C4
 G = < a,b,c,d | a4=b10=d4=1, c2=b5, ab=ba, cac-1=a-1, dad-1=a-1b5, cbc-1=b-1, bd=db, cd=dc >

Subgroups: 478 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C4 [×12], C22 [×7], C5, C2×C4 [×4], C2×C4 [×22], C23, C10 [×7], C42 [×2], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], Dic5 [×7], C20 [×5], C2×C10 [×7], C2.C42, C2.C42 [×3], C2×C42, C2×C4⋊C4 [×2], C2×Dic5 [×6], C2×Dic5 [×9], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.63C23, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×4], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×3], C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C4⋊Dic515C4
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, Dic10 [×2], C4×D5 [×2], C22×D5, C23.63C23, C2×Dic10, C2×C4×D5, C4○D20, D4×D5, D42D5 [×2], Q82D5, C4×Dic10, C23.11D10, Dic5.14D4, D10.12D4, C4.Dic10, D208C4, C4⋊C4⋊D5, C4⋊Dic515C4

Smallest permutation representation of C4⋊Dic515C4
Regular action on 320 points
Generators in S320
(1 102 22 92)(2 103 23 93)(3 104 24 94)(4 105 25 95)(5 106 26 96)(6 107 27 97)(7 108 28 98)(8 109 29 99)(9 110 30 100)(10 101 21 91)(11 239 319 248)(12 240 320 249)(13 231 311 250)(14 232 312 241)(15 233 313 242)(16 234 314 243)(17 235 315 244)(18 236 316 245)(19 237 317 246)(20 238 318 247)(31 121 41 111)(32 122 42 112)(33 123 43 113)(34 124 44 114)(35 125 45 115)(36 126 46 116)(37 127 47 117)(38 128 48 118)(39 129 49 119)(40 130 50 120)(51 141 61 131)(52 142 62 132)(53 143 63 133)(54 144 64 134)(55 145 65 135)(56 146 66 136)(57 147 67 137)(58 148 68 138)(59 149 69 139)(60 150 70 140)(71 164 81 151)(72 165 82 152)(73 166 83 153)(74 167 84 154)(75 168 85 155)(76 169 86 156)(77 170 87 157)(78 161 88 158)(79 162 89 159)(80 163 90 160)(171 251 190 270)(172 252 181 261)(173 253 182 262)(174 254 183 263)(175 255 184 264)(176 256 185 265)(177 257 186 266)(178 258 187 267)(179 259 188 268)(180 260 189 269)(191 271 210 290)(192 272 201 281)(193 273 202 282)(194 274 203 283)(195 275 204 284)(196 276 205 285)(197 277 206 286)(198 278 207 287)(199 279 208 288)(200 280 209 289)(211 291 230 310)(212 292 221 301)(213 293 222 302)(214 294 223 303)(215 295 224 304)(216 296 225 305)(217 297 226 306)(218 298 227 307)(219 299 228 308)(220 300 229 309)
(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)
(1 179 6 174)(2 178 7 173)(3 177 8 172)(4 176 9 171)(5 175 10 180)(11 170 16 165)(12 169 17 164)(13 168 18 163)(14 167 19 162)(15 166 20 161)(21 189 26 184)(22 188 27 183)(23 187 28 182)(24 186 29 181)(25 185 30 190)(31 200 36 195)(32 199 37 194)(33 198 38 193)(34 197 39 192)(35 196 40 191)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 220 56 215)(52 219 57 214)(53 218 58 213)(54 217 59 212)(55 216 60 211)(61 229 66 224)(62 228 67 223)(63 227 68 222)(64 226 69 221)(65 225 70 230)(71 240 76 235)(72 239 77 234)(73 238 78 233)(74 237 79 232)(75 236 80 231)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 260 96 255)(92 259 97 254)(93 258 98 253)(94 257 99 252)(95 256 100 251)(101 269 106 264)(102 268 107 263)(103 267 108 262)(104 266 109 261)(105 265 110 270)(111 280 116 275)(112 279 117 274)(113 278 118 273)(114 277 119 272)(115 276 120 271)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 300 136 295)(132 299 137 294)(133 298 138 293)(134 297 139 292)(135 296 140 291)(141 309 146 304)(142 308 147 303)(143 307 148 302)(144 306 149 301)(145 305 150 310)(151 320 156 315)(152 319 157 314)(153 318 158 313)(154 317 159 312)(155 316 160 311)
(1 72 32 52)(2 73 33 53)(3 74 34 54)(4 75 35 55)(5 76 36 56)(6 77 37 57)(7 78 38 58)(8 79 39 59)(9 80 40 60)(10 71 31 51)(11 274 308 254)(12 275 309 255)(13 276 310 256)(14 277 301 257)(15 278 302 258)(16 279 303 259)(17 280 304 260)(18 271 305 251)(19 272 306 252)(20 273 307 253)(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)(91 169 111 146)(92 170 112 147)(93 161 113 148)(94 162 114 149)(95 163 115 150)(96 164 116 141)(97 165 117 142)(98 166 118 143)(99 167 119 144)(100 168 120 145)(101 156 121 136)(102 157 122 137)(103 158 123 138)(104 159 124 139)(105 160 125 140)(106 151 126 131)(107 152 127 132)(108 153 128 133)(109 154 129 134)(110 155 130 135)(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)(261 317 281 297)(262 318 282 298)(263 319 283 299)(264 320 284 300)(265 311 285 291)(266 312 286 292)(267 313 287 293)(268 314 288 294)(269 315 289 295)(270 316 290 296)

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

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

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

68 conjugacy classes

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

68 irreducible representations

dim111111122222222444
type+++++++-++-+-+
imageC1C2C2C2C2C2C4D4Q8D5C4○D4D10Dic10C4×D5C4○D20D4×D5D42D5Q82D5
kernelC4⋊Dic515C4C10.10C42C5×C2.C42C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C4⋊Dic5C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps131111822286888242

Matrix representation of C4⋊Dic515C4 in GL6(𝔽41)

27340000
34140000
008200
00293300
0000309
00003211
,
100000
010000
0040000
0004000
000001
0000407
,
090000
3200000
00322900
000900
00002440
0000317
,
010000
4000000
0032000
0003200
000010
000001

G:=sub<GL(6,GF(41))| [27,34,0,0,0,0,34,14,0,0,0,0,0,0,8,29,0,0,0,0,2,33,0,0,0,0,0,0,30,32,0,0,0,0,9,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,7],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,32,0,0,0,0,0,29,9,0,0,0,0,0,0,24,3,0,0,0,0,40,17],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C4⋊Dic515C4 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_5\rtimes_{15}C_4
% in TeX

G:=Group("C4:Dic5:15C4");
// GroupNames label

G:=SmallGroup(320,281);
// by ID

G=gap.SmallGroup(320,281);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,344,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^10=d^4=1,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^5,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

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