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G = Q8×C3×C15order 360 = 23·32·5

Direct product of C3×C15 and Q8

direct product, metacyclic, nilpotent (class 2), monomial

Aliases: Q8×C3×C15, C60.13C6, C12.5C30, C10.7C62, C4.(C3×C30), C6.9(C2×C30), C20.3(C3×C6), C2.2(C6×C30), (C3×C60).11C2, (C3×C12).5C10, C30.32(C2×C6), (C3×C30).58C22, (C3×C6).17(C2×C10), SmallGroup(360,117)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C3×C15
 Chief series C1 — C2 — C10 — C30 — C3×C30 — C3×C60 — Q8×C3×C15
 Lower central C1 — C2 — Q8×C3×C15
 Upper central C1 — C3×C30 — Q8×C3×C15

Generators and relations for Q8×C3×C15
G = < a,b,c,d | a3=b15=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 72, all normal (12 characteristic)
C1, C2, C3, C4, C5, C6, Q8, C32, C10, C12, C15, C3×C6, C20, C3×Q8, C30, C3×C12, C5×Q8, C3×C15, C60, Q8×C32, C3×C30, Q8×C15, C3×C60, Q8×C3×C15
Quotients: C1, C2, C3, C22, C5, C6, Q8, C32, C10, C2×C6, C15, C3×C6, C2×C10, C3×Q8, C30, C62, C5×Q8, C3×C15, C2×C30, Q8×C32, C3×C30, Q8×C15, C6×C30, Q8×C3×C15

Smallest permutation representation of Q8×C3×C15
Regular action on 360 points
Generators in S360
(1 25 253)(2 26 254)(3 27 255)(4 28 241)(5 29 242)(6 30 243)(7 16 244)(8 17 245)(9 18 246)(10 19 247)(11 20 248)(12 21 249)(13 22 250)(14 23 251)(15 24 252)(31 222 237)(32 223 238)(33 224 239)(34 225 240)(35 211 226)(36 212 227)(37 213 228)(38 214 229)(39 215 230)(40 216 231)(41 217 232)(42 218 233)(43 219 234)(44 220 235)(45 221 236)(46 357 195)(47 358 181)(48 359 182)(49 360 183)(50 346 184)(51 347 185)(52 348 186)(53 349 187)(54 350 188)(55 351 189)(56 352 190)(57 353 191)(58 354 192)(59 355 193)(60 356 194)(61 297 138)(62 298 139)(63 299 140)(64 300 141)(65 286 142)(66 287 143)(67 288 144)(68 289 145)(69 290 146)(70 291 147)(71 292 148)(72 293 149)(73 294 150)(74 295 136)(75 296 137)(76 197 158)(77 198 159)(78 199 160)(79 200 161)(80 201 162)(81 202 163)(82 203 164)(83 204 165)(84 205 151)(85 206 152)(86 207 153)(87 208 154)(88 209 155)(89 210 156)(90 196 157)(91 337 272)(92 338 273)(93 339 274)(94 340 275)(95 341 276)(96 342 277)(97 343 278)(98 344 279)(99 345 280)(100 331 281)(101 332 282)(102 333 283)(103 334 284)(104 335 285)(105 336 271)(106 322 304)(107 323 305)(108 324 306)(109 325 307)(110 326 308)(111 327 309)(112 328 310)(113 329 311)(114 330 312)(115 316 313)(116 317 314)(117 318 315)(118 319 301)(119 320 302)(120 321 303)(121 257 169)(122 258 170)(123 259 171)(124 260 172)(125 261 173)(126 262 174)(127 263 175)(128 264 176)(129 265 177)(130 266 178)(131 267 179)(132 268 180)(133 269 166)(134 270 167)(135 256 168)
(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)(316 317 318 319 320 321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345)(346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)
(1 225 347 319)(2 211 348 320)(3 212 349 321)(4 213 350 322)(5 214 351 323)(6 215 352 324)(7 216 353 325)(8 217 354 326)(9 218 355 327)(10 219 356 328)(11 220 357 329)(12 221 358 330)(13 222 359 316)(14 223 360 317)(15 224 346 318)(16 231 191 307)(17 232 192 308)(18 233 193 309)(19 234 194 310)(20 235 195 311)(21 236 181 312)(22 237 182 313)(23 238 183 314)(24 239 184 315)(25 240 185 301)(26 226 186 302)(27 227 187 303)(28 228 188 304)(29 229 189 305)(30 230 190 306)(31 48 115 250)(32 49 116 251)(33 50 117 252)(34 51 118 253)(35 52 119 254)(36 53 120 255)(37 54 106 241)(38 55 107 242)(39 56 108 243)(40 57 109 244)(41 58 110 245)(42 59 111 246)(43 60 112 247)(44 46 113 248)(45 47 114 249)(61 151 121 101)(62 152 122 102)(63 153 123 103)(64 154 124 104)(65 155 125 105)(66 156 126 91)(67 157 127 92)(68 158 128 93)(69 159 129 94)(70 160 130 95)(71 161 131 96)(72 162 132 97)(73 163 133 98)(74 164 134 99)(75 165 135 100)(76 264 339 289)(77 265 340 290)(78 266 341 291)(79 267 342 292)(80 268 343 293)(81 269 344 294)(82 270 345 295)(83 256 331 296)(84 257 332 297)(85 258 333 298)(86 259 334 299)(87 260 335 300)(88 261 336 286)(89 262 337 287)(90 263 338 288)(136 203 167 280)(137 204 168 281)(138 205 169 282)(139 206 170 283)(140 207 171 284)(141 208 172 285)(142 209 173 271)(143 210 174 272)(144 196 175 273)(145 197 176 274)(146 198 177 275)(147 199 178 276)(148 200 179 277)(149 201 180 278)(150 202 166 279)
(1 132 347 72)(2 133 348 73)(3 134 349 74)(4 135 350 75)(5 121 351 61)(6 122 352 62)(7 123 353 63)(8 124 354 64)(9 125 355 65)(10 126 356 66)(11 127 357 67)(12 128 358 68)(13 129 359 69)(14 130 360 70)(15 131 346 71)(16 259 191 299)(17 260 192 300)(18 261 193 286)(19 262 194 287)(20 263 195 288)(21 264 181 289)(22 265 182 290)(23 266 183 291)(24 267 184 292)(25 268 185 293)(26 269 186 294)(27 270 187 295)(28 256 188 296)(29 257 189 297)(30 258 190 298)(31 198 115 275)(32 199 116 276)(33 200 117 277)(34 201 118 278)(35 202 119 279)(36 203 120 280)(37 204 106 281)(38 205 107 282)(39 206 108 283)(40 207 109 284)(41 208 110 285)(42 209 111 271)(43 210 112 272)(44 196 113 273)(45 197 114 274)(46 144 248 175)(47 145 249 176)(48 146 250 177)(49 147 251 178)(50 148 252 179)(51 149 253 180)(52 150 254 166)(53 136 255 167)(54 137 241 168)(55 138 242 169)(56 139 243 170)(57 140 244 171)(58 141 245 172)(59 142 246 173)(60 143 247 174)(76 312 339 236)(77 313 340 237)(78 314 341 238)(79 315 342 239)(80 301 343 240)(81 302 344 226)(82 303 345 227)(83 304 331 228)(84 305 332 229)(85 306 333 230)(86 307 334 231)(87 308 335 232)(88 309 336 233)(89 310 337 234)(90 311 338 235)(91 219 156 328)(92 220 157 329)(93 221 158 330)(94 222 159 316)(95 223 160 317)(96 224 161 318)(97 225 162 319)(98 211 163 320)(99 212 164 321)(100 213 165 322)(101 214 151 323)(102 215 152 324)(103 216 153 325)(104 217 154 326)(105 218 155 327)

G:=sub<Sym(360)| (1,25,253)(2,26,254)(3,27,255)(4,28,241)(5,29,242)(6,30,243)(7,16,244)(8,17,245)(9,18,246)(10,19,247)(11,20,248)(12,21,249)(13,22,250)(14,23,251)(15,24,252)(31,222,237)(32,223,238)(33,224,239)(34,225,240)(35,211,226)(36,212,227)(37,213,228)(38,214,229)(39,215,230)(40,216,231)(41,217,232)(42,218,233)(43,219,234)(44,220,235)(45,221,236)(46,357,195)(47,358,181)(48,359,182)(49,360,183)(50,346,184)(51,347,185)(52,348,186)(53,349,187)(54,350,188)(55,351,189)(56,352,190)(57,353,191)(58,354,192)(59,355,193)(60,356,194)(61,297,138)(62,298,139)(63,299,140)(64,300,141)(65,286,142)(66,287,143)(67,288,144)(68,289,145)(69,290,146)(70,291,147)(71,292,148)(72,293,149)(73,294,150)(74,295,136)(75,296,137)(76,197,158)(77,198,159)(78,199,160)(79,200,161)(80,201,162)(81,202,163)(82,203,164)(83,204,165)(84,205,151)(85,206,152)(86,207,153)(87,208,154)(88,209,155)(89,210,156)(90,196,157)(91,337,272)(92,338,273)(93,339,274)(94,340,275)(95,341,276)(96,342,277)(97,343,278)(98,344,279)(99,345,280)(100,331,281)(101,332,282)(102,333,283)(103,334,284)(104,335,285)(105,336,271)(106,322,304)(107,323,305)(108,324,306)(109,325,307)(110,326,308)(111,327,309)(112,328,310)(113,329,311)(114,330,312)(115,316,313)(116,317,314)(117,318,315)(118,319,301)(119,320,302)(120,321,303)(121,257,169)(122,258,170)(123,259,171)(124,260,172)(125,261,173)(126,262,174)(127,263,175)(128,264,176)(129,265,177)(130,266,178)(131,267,179)(132,268,180)(133,269,166)(134,270,167)(135,256,168), (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)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345)(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,225,347,319)(2,211,348,320)(3,212,349,321)(4,213,350,322)(5,214,351,323)(6,215,352,324)(7,216,353,325)(8,217,354,326)(9,218,355,327)(10,219,356,328)(11,220,357,329)(12,221,358,330)(13,222,359,316)(14,223,360,317)(15,224,346,318)(16,231,191,307)(17,232,192,308)(18,233,193,309)(19,234,194,310)(20,235,195,311)(21,236,181,312)(22,237,182,313)(23,238,183,314)(24,239,184,315)(25,240,185,301)(26,226,186,302)(27,227,187,303)(28,228,188,304)(29,229,189,305)(30,230,190,306)(31,48,115,250)(32,49,116,251)(33,50,117,252)(34,51,118,253)(35,52,119,254)(36,53,120,255)(37,54,106,241)(38,55,107,242)(39,56,108,243)(40,57,109,244)(41,58,110,245)(42,59,111,246)(43,60,112,247)(44,46,113,248)(45,47,114,249)(61,151,121,101)(62,152,122,102)(63,153,123,103)(64,154,124,104)(65,155,125,105)(66,156,126,91)(67,157,127,92)(68,158,128,93)(69,159,129,94)(70,160,130,95)(71,161,131,96)(72,162,132,97)(73,163,133,98)(74,164,134,99)(75,165,135,100)(76,264,339,289)(77,265,340,290)(78,266,341,291)(79,267,342,292)(80,268,343,293)(81,269,344,294)(82,270,345,295)(83,256,331,296)(84,257,332,297)(85,258,333,298)(86,259,334,299)(87,260,335,300)(88,261,336,286)(89,262,337,287)(90,263,338,288)(136,203,167,280)(137,204,168,281)(138,205,169,282)(139,206,170,283)(140,207,171,284)(141,208,172,285)(142,209,173,271)(143,210,174,272)(144,196,175,273)(145,197,176,274)(146,198,177,275)(147,199,178,276)(148,200,179,277)(149,201,180,278)(150,202,166,279), (1,132,347,72)(2,133,348,73)(3,134,349,74)(4,135,350,75)(5,121,351,61)(6,122,352,62)(7,123,353,63)(8,124,354,64)(9,125,355,65)(10,126,356,66)(11,127,357,67)(12,128,358,68)(13,129,359,69)(14,130,360,70)(15,131,346,71)(16,259,191,299)(17,260,192,300)(18,261,193,286)(19,262,194,287)(20,263,195,288)(21,264,181,289)(22,265,182,290)(23,266,183,291)(24,267,184,292)(25,268,185,293)(26,269,186,294)(27,270,187,295)(28,256,188,296)(29,257,189,297)(30,258,190,298)(31,198,115,275)(32,199,116,276)(33,200,117,277)(34,201,118,278)(35,202,119,279)(36,203,120,280)(37,204,106,281)(38,205,107,282)(39,206,108,283)(40,207,109,284)(41,208,110,285)(42,209,111,271)(43,210,112,272)(44,196,113,273)(45,197,114,274)(46,144,248,175)(47,145,249,176)(48,146,250,177)(49,147,251,178)(50,148,252,179)(51,149,253,180)(52,150,254,166)(53,136,255,167)(54,137,241,168)(55,138,242,169)(56,139,243,170)(57,140,244,171)(58,141,245,172)(59,142,246,173)(60,143,247,174)(76,312,339,236)(77,313,340,237)(78,314,341,238)(79,315,342,239)(80,301,343,240)(81,302,344,226)(82,303,345,227)(83,304,331,228)(84,305,332,229)(85,306,333,230)(86,307,334,231)(87,308,335,232)(88,309,336,233)(89,310,337,234)(90,311,338,235)(91,219,156,328)(92,220,157,329)(93,221,158,330)(94,222,159,316)(95,223,160,317)(96,224,161,318)(97,225,162,319)(98,211,163,320)(99,212,164,321)(100,213,165,322)(101,214,151,323)(102,215,152,324)(103,216,153,325)(104,217,154,326)(105,218,155,327)>;

G:=Group( (1,25,253)(2,26,254)(3,27,255)(4,28,241)(5,29,242)(6,30,243)(7,16,244)(8,17,245)(9,18,246)(10,19,247)(11,20,248)(12,21,249)(13,22,250)(14,23,251)(15,24,252)(31,222,237)(32,223,238)(33,224,239)(34,225,240)(35,211,226)(36,212,227)(37,213,228)(38,214,229)(39,215,230)(40,216,231)(41,217,232)(42,218,233)(43,219,234)(44,220,235)(45,221,236)(46,357,195)(47,358,181)(48,359,182)(49,360,183)(50,346,184)(51,347,185)(52,348,186)(53,349,187)(54,350,188)(55,351,189)(56,352,190)(57,353,191)(58,354,192)(59,355,193)(60,356,194)(61,297,138)(62,298,139)(63,299,140)(64,300,141)(65,286,142)(66,287,143)(67,288,144)(68,289,145)(69,290,146)(70,291,147)(71,292,148)(72,293,149)(73,294,150)(74,295,136)(75,296,137)(76,197,158)(77,198,159)(78,199,160)(79,200,161)(80,201,162)(81,202,163)(82,203,164)(83,204,165)(84,205,151)(85,206,152)(86,207,153)(87,208,154)(88,209,155)(89,210,156)(90,196,157)(91,337,272)(92,338,273)(93,339,274)(94,340,275)(95,341,276)(96,342,277)(97,343,278)(98,344,279)(99,345,280)(100,331,281)(101,332,282)(102,333,283)(103,334,284)(104,335,285)(105,336,271)(106,322,304)(107,323,305)(108,324,306)(109,325,307)(110,326,308)(111,327,309)(112,328,310)(113,329,311)(114,330,312)(115,316,313)(116,317,314)(117,318,315)(118,319,301)(119,320,302)(120,321,303)(121,257,169)(122,258,170)(123,259,171)(124,260,172)(125,261,173)(126,262,174)(127,263,175)(128,264,176)(129,265,177)(130,266,178)(131,267,179)(132,268,180)(133,269,166)(134,270,167)(135,256,168), (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)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345)(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,225,347,319)(2,211,348,320)(3,212,349,321)(4,213,350,322)(5,214,351,323)(6,215,352,324)(7,216,353,325)(8,217,354,326)(9,218,355,327)(10,219,356,328)(11,220,357,329)(12,221,358,330)(13,222,359,316)(14,223,360,317)(15,224,346,318)(16,231,191,307)(17,232,192,308)(18,233,193,309)(19,234,194,310)(20,235,195,311)(21,236,181,312)(22,237,182,313)(23,238,183,314)(24,239,184,315)(25,240,185,301)(26,226,186,302)(27,227,187,303)(28,228,188,304)(29,229,189,305)(30,230,190,306)(31,48,115,250)(32,49,116,251)(33,50,117,252)(34,51,118,253)(35,52,119,254)(36,53,120,255)(37,54,106,241)(38,55,107,242)(39,56,108,243)(40,57,109,244)(41,58,110,245)(42,59,111,246)(43,60,112,247)(44,46,113,248)(45,47,114,249)(61,151,121,101)(62,152,122,102)(63,153,123,103)(64,154,124,104)(65,155,125,105)(66,156,126,91)(67,157,127,92)(68,158,128,93)(69,159,129,94)(70,160,130,95)(71,161,131,96)(72,162,132,97)(73,163,133,98)(74,164,134,99)(75,165,135,100)(76,264,339,289)(77,265,340,290)(78,266,341,291)(79,267,342,292)(80,268,343,293)(81,269,344,294)(82,270,345,295)(83,256,331,296)(84,257,332,297)(85,258,333,298)(86,259,334,299)(87,260,335,300)(88,261,336,286)(89,262,337,287)(90,263,338,288)(136,203,167,280)(137,204,168,281)(138,205,169,282)(139,206,170,283)(140,207,171,284)(141,208,172,285)(142,209,173,271)(143,210,174,272)(144,196,175,273)(145,197,176,274)(146,198,177,275)(147,199,178,276)(148,200,179,277)(149,201,180,278)(150,202,166,279), (1,132,347,72)(2,133,348,73)(3,134,349,74)(4,135,350,75)(5,121,351,61)(6,122,352,62)(7,123,353,63)(8,124,354,64)(9,125,355,65)(10,126,356,66)(11,127,357,67)(12,128,358,68)(13,129,359,69)(14,130,360,70)(15,131,346,71)(16,259,191,299)(17,260,192,300)(18,261,193,286)(19,262,194,287)(20,263,195,288)(21,264,181,289)(22,265,182,290)(23,266,183,291)(24,267,184,292)(25,268,185,293)(26,269,186,294)(27,270,187,295)(28,256,188,296)(29,257,189,297)(30,258,190,298)(31,198,115,275)(32,199,116,276)(33,200,117,277)(34,201,118,278)(35,202,119,279)(36,203,120,280)(37,204,106,281)(38,205,107,282)(39,206,108,283)(40,207,109,284)(41,208,110,285)(42,209,111,271)(43,210,112,272)(44,196,113,273)(45,197,114,274)(46,144,248,175)(47,145,249,176)(48,146,250,177)(49,147,251,178)(50,148,252,179)(51,149,253,180)(52,150,254,166)(53,136,255,167)(54,137,241,168)(55,138,242,169)(56,139,243,170)(57,140,244,171)(58,141,245,172)(59,142,246,173)(60,143,247,174)(76,312,339,236)(77,313,340,237)(78,314,341,238)(79,315,342,239)(80,301,343,240)(81,302,344,226)(82,303,345,227)(83,304,331,228)(84,305,332,229)(85,306,333,230)(86,307,334,231)(87,308,335,232)(88,309,336,233)(89,310,337,234)(90,311,338,235)(91,219,156,328)(92,220,157,329)(93,221,158,330)(94,222,159,316)(95,223,160,317)(96,224,161,318)(97,225,162,319)(98,211,163,320)(99,212,164,321)(100,213,165,322)(101,214,151,323)(102,215,152,324)(103,216,153,325)(104,217,154,326)(105,218,155,327) );

G=PermutationGroup([[(1,25,253),(2,26,254),(3,27,255),(4,28,241),(5,29,242),(6,30,243),(7,16,244),(8,17,245),(9,18,246),(10,19,247),(11,20,248),(12,21,249),(13,22,250),(14,23,251),(15,24,252),(31,222,237),(32,223,238),(33,224,239),(34,225,240),(35,211,226),(36,212,227),(37,213,228),(38,214,229),(39,215,230),(40,216,231),(41,217,232),(42,218,233),(43,219,234),(44,220,235),(45,221,236),(46,357,195),(47,358,181),(48,359,182),(49,360,183),(50,346,184),(51,347,185),(52,348,186),(53,349,187),(54,350,188),(55,351,189),(56,352,190),(57,353,191),(58,354,192),(59,355,193),(60,356,194),(61,297,138),(62,298,139),(63,299,140),(64,300,141),(65,286,142),(66,287,143),(67,288,144),(68,289,145),(69,290,146),(70,291,147),(71,292,148),(72,293,149),(73,294,150),(74,295,136),(75,296,137),(76,197,158),(77,198,159),(78,199,160),(79,200,161),(80,201,162),(81,202,163),(82,203,164),(83,204,165),(84,205,151),(85,206,152),(86,207,153),(87,208,154),(88,209,155),(89,210,156),(90,196,157),(91,337,272),(92,338,273),(93,339,274),(94,340,275),(95,341,276),(96,342,277),(97,343,278),(98,344,279),(99,345,280),(100,331,281),(101,332,282),(102,333,283),(103,334,284),(104,335,285),(105,336,271),(106,322,304),(107,323,305),(108,324,306),(109,325,307),(110,326,308),(111,327,309),(112,328,310),(113,329,311),(114,330,312),(115,316,313),(116,317,314),(117,318,315),(118,319,301),(119,320,302),(120,321,303),(121,257,169),(122,258,170),(123,259,171),(124,260,172),(125,261,173),(126,262,174),(127,263,175),(128,264,176),(129,265,177),(130,266,178),(131,267,179),(132,268,180),(133,269,166),(134,270,167),(135,256,168)], [(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),(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345),(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)], [(1,225,347,319),(2,211,348,320),(3,212,349,321),(4,213,350,322),(5,214,351,323),(6,215,352,324),(7,216,353,325),(8,217,354,326),(9,218,355,327),(10,219,356,328),(11,220,357,329),(12,221,358,330),(13,222,359,316),(14,223,360,317),(15,224,346,318),(16,231,191,307),(17,232,192,308),(18,233,193,309),(19,234,194,310),(20,235,195,311),(21,236,181,312),(22,237,182,313),(23,238,183,314),(24,239,184,315),(25,240,185,301),(26,226,186,302),(27,227,187,303),(28,228,188,304),(29,229,189,305),(30,230,190,306),(31,48,115,250),(32,49,116,251),(33,50,117,252),(34,51,118,253),(35,52,119,254),(36,53,120,255),(37,54,106,241),(38,55,107,242),(39,56,108,243),(40,57,109,244),(41,58,110,245),(42,59,111,246),(43,60,112,247),(44,46,113,248),(45,47,114,249),(61,151,121,101),(62,152,122,102),(63,153,123,103),(64,154,124,104),(65,155,125,105),(66,156,126,91),(67,157,127,92),(68,158,128,93),(69,159,129,94),(70,160,130,95),(71,161,131,96),(72,162,132,97),(73,163,133,98),(74,164,134,99),(75,165,135,100),(76,264,339,289),(77,265,340,290),(78,266,341,291),(79,267,342,292),(80,268,343,293),(81,269,344,294),(82,270,345,295),(83,256,331,296),(84,257,332,297),(85,258,333,298),(86,259,334,299),(87,260,335,300),(88,261,336,286),(89,262,337,287),(90,263,338,288),(136,203,167,280),(137,204,168,281),(138,205,169,282),(139,206,170,283),(140,207,171,284),(141,208,172,285),(142,209,173,271),(143,210,174,272),(144,196,175,273),(145,197,176,274),(146,198,177,275),(147,199,178,276),(148,200,179,277),(149,201,180,278),(150,202,166,279)], [(1,132,347,72),(2,133,348,73),(3,134,349,74),(4,135,350,75),(5,121,351,61),(6,122,352,62),(7,123,353,63),(8,124,354,64),(9,125,355,65),(10,126,356,66),(11,127,357,67),(12,128,358,68),(13,129,359,69),(14,130,360,70),(15,131,346,71),(16,259,191,299),(17,260,192,300),(18,261,193,286),(19,262,194,287),(20,263,195,288),(21,264,181,289),(22,265,182,290),(23,266,183,291),(24,267,184,292),(25,268,185,293),(26,269,186,294),(27,270,187,295),(28,256,188,296),(29,257,189,297),(30,258,190,298),(31,198,115,275),(32,199,116,276),(33,200,117,277),(34,201,118,278),(35,202,119,279),(36,203,120,280),(37,204,106,281),(38,205,107,282),(39,206,108,283),(40,207,109,284),(41,208,110,285),(42,209,111,271),(43,210,112,272),(44,196,113,273),(45,197,114,274),(46,144,248,175),(47,145,249,176),(48,146,250,177),(49,147,251,178),(50,148,252,179),(51,149,253,180),(52,150,254,166),(53,136,255,167),(54,137,241,168),(55,138,242,169),(56,139,243,170),(57,140,244,171),(58,141,245,172),(59,142,246,173),(60,143,247,174),(76,312,339,236),(77,313,340,237),(78,314,341,238),(79,315,342,239),(80,301,343,240),(81,302,344,226),(82,303,345,227),(83,304,331,228),(84,305,332,229),(85,306,333,230),(86,307,334,231),(87,308,335,232),(88,309,336,233),(89,310,337,234),(90,311,338,235),(91,219,156,328),(92,220,157,329),(93,221,158,330),(94,222,159,316),(95,223,160,317),(96,224,161,318),(97,225,162,319),(98,211,163,320),(99,212,164,321),(100,213,165,322),(101,214,151,323),(102,215,152,324),(103,216,153,325),(104,217,154,326),(105,218,155,327)]])

225 conjugacy classes

 class 1 2 3A ··· 3H 4A 4B 4C 5A 5B 5C 5D 6A ··· 6H 10A 10B 10C 10D 12A ··· 12X 15A ··· 15AF 20A ··· 20L 30A ··· 30AF 60A ··· 60CR order 1 2 3 ··· 3 4 4 4 5 5 5 5 6 ··· 6 10 10 10 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 ··· 1 2 2 2 1 1 1 1 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

225 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + - image C1 C2 C3 C5 C6 C10 C15 C30 Q8 C3×Q8 C5×Q8 Q8×C15 kernel Q8×C3×C15 C3×C60 Q8×C15 Q8×C32 C60 C3×C12 C3×Q8 C12 C3×C15 C15 C32 C3 # reps 1 3 8 4 24 12 32 96 1 8 4 32

Matrix representation of Q8×C3×C15 in GL3(𝔽61) generated by

 47 0 0 0 47 0 0 0 47
,
 13 0 0 0 25 0 0 0 25
,
 1 0 0 0 0 1 0 60 0
,
 60 0 0 0 42 2 0 2 19
G:=sub<GL(3,GF(61))| [47,0,0,0,47,0,0,0,47],[13,0,0,0,25,0,0,0,25],[1,0,0,0,0,60,0,1,0],[60,0,0,0,42,2,0,2,19] >;

Q8×C3×C15 in GAP, Magma, Sage, TeX

Q_8\times C_3\times C_{15}
% in TeX

G:=Group("Q8xC3xC15");
// GroupNames label

G:=SmallGroup(360,117);
// by ID

G=gap.SmallGroup(360,117);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-5,-2,1080,2185,1087]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^15=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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𝔽