Copied to
clipboard

## G = C5×C32⋊4Q8order 360 = 23·32·5

### Direct product of C5 and C32⋊4Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C5×C32⋊4Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — C5×C3⋊Dic3 — C5×C32⋊4Q8
 Lower central C32 — C3×C6 — C5×C32⋊4Q8
 Upper central C1 — C10 — C20

Generators and relations for C5×C324Q8
G = < a,b,c,d,e | a5=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 168 in 72 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C3, C4, C4, C5, C6, Q8, C32, C10, Dic3, C12, C15, C3×C6, C20, C20, Dic6, C30, C3⋊Dic3, C3×C12, C5×Q8, C3×C15, C5×Dic3, C60, C324Q8, C3×C30, C5×Dic6, C5×C3⋊Dic3, C3×C60, C5×C324Q8
Quotients: C1, C2, C22, C5, S3, Q8, C10, D6, C3⋊S3, C2×C10, Dic6, C5×S3, C2×C3⋊S3, C5×Q8, S3×C10, C324Q8, C5×C3⋊S3, C5×Dic6, C10×C3⋊S3, C5×C324Q8

Smallest permutation representation of C5×C324Q8
Regular action on 360 points
Generators in S360
(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 27 77)(2 28 78)(3 29 79)(4 30 80)(5 26 76)(6 112 62)(7 113 63)(8 114 64)(9 115 65)(10 111 61)(11 66 93)(12 67 94)(13 68 95)(14 69 91)(15 70 92)(16 325 355)(17 321 351)(18 322 352)(19 323 353)(20 324 354)(21 88 146)(22 89 147)(23 90 148)(24 86 149)(25 87 150)(31 81 46)(32 82 47)(33 83 48)(34 84 49)(35 85 50)(36 346 296)(37 347 297)(38 348 298)(39 349 299)(40 350 300)(41 98 58)(42 99 59)(43 100 60)(44 96 56)(45 97 57)(51 360 311)(52 356 312)(53 357 313)(54 358 314)(55 359 315)(71 301 331)(72 302 332)(73 303 333)(74 304 334)(75 305 335)(101 156 126)(102 157 127)(103 158 128)(104 159 129)(105 160 130)(106 161 136)(107 162 137)(108 163 138)(109 164 139)(110 165 140)(116 166 205)(117 167 201)(118 168 202)(119 169 203)(120 170 204)(121 186 151)(122 187 152)(123 188 153)(124 189 154)(125 190 155)(131 180 238)(132 176 239)(133 177 240)(134 178 236)(135 179 237)(141 181 171)(142 182 172)(143 183 173)(144 184 174)(145 185 175)(191 216 246)(192 217 247)(193 218 248)(194 219 249)(195 220 250)(196 226 251)(197 227 252)(198 228 253)(199 229 254)(200 230 255)(206 295 256)(207 291 257)(208 292 258)(209 293 259)(210 294 260)(211 241 276)(212 242 277)(213 243 278)(214 244 279)(215 245 280)(221 328 270)(222 329 266)(223 330 267)(224 326 268)(225 327 269)(231 261 271)(232 262 272)(233 263 273)(234 264 274)(235 265 275)(281 306 336)(282 307 337)(283 308 338)(284 309 339)(285 310 340)(286 316 341)(287 317 342)(288 318 343)(289 319 344)(290 320 345)
(1 92 9)(2 93 10)(3 94 6)(4 95 7)(5 91 8)(11 111 28)(12 112 29)(13 113 30)(14 114 26)(15 115 27)(16 307 349)(17 308 350)(18 309 346)(19 310 347)(20 306 348)(21 45 83)(22 41 84)(23 42 85)(24 43 81)(25 44 82)(31 149 60)(32 150 56)(33 146 57)(34 147 58)(35 148 59)(36 352 284)(37 353 285)(38 354 281)(39 355 282)(40 351 283)(46 86 100)(47 87 96)(48 88 97)(49 89 98)(50 90 99)(51 287 334)(52 288 335)(53 289 331)(54 290 332)(55 286 333)(61 78 66)(62 79 67)(63 80 68)(64 76 69)(65 77 70)(71 357 319)(72 358 320)(73 359 316)(74 360 317)(75 356 318)(101 174 202)(102 175 203)(103 171 204)(104 172 205)(105 173 201)(106 237 153)(107 238 154)(108 239 155)(109 240 151)(110 236 152)(116 159 142)(117 160 143)(118 156 144)(119 157 145)(120 158 141)(121 164 133)(122 165 134)(123 161 135)(124 162 131)(125 163 132)(126 184 168)(127 185 169)(128 181 170)(129 182 166)(130 183 167)(136 179 188)(137 180 189)(138 176 190)(139 177 186)(140 178 187)(191 292 264)(192 293 265)(193 294 261)(194 295 262)(195 291 263)(196 243 327)(197 244 328)(198 245 329)(199 241 330)(200 242 326)(206 232 249)(207 233 250)(208 234 246)(209 235 247)(210 231 248)(211 223 254)(212 224 255)(213 225 251)(214 221 252)(215 222 253)(216 258 274)(217 259 275)(218 260 271)(219 256 272)(220 257 273)(226 278 269)(227 279 270)(228 280 266)(229 276 267)(230 277 268)(296 322 339)(297 323 340)(298 324 336)(299 325 337)(300 321 338)(301 313 344)(302 314 345)(303 315 341)(304 311 342)(305 312 343)
(1 134 21 129)(2 135 22 130)(3 131 23 126)(4 132 24 127)(5 133 25 128)(6 162 85 168)(7 163 81 169)(8 164 82 170)(9 165 83 166)(10 161 84 167)(11 188 98 173)(12 189 99 174)(13 190 100 175)(14 186 96 171)(15 187 97 172)(16 215 305 275)(17 211 301 271)(18 212 302 272)(19 213 303 273)(20 214 304 274)(26 177 87 103)(27 178 88 104)(28 179 89 105)(29 180 90 101)(30 176 86 102)(31 119 63 108)(32 120 64 109)(33 116 65 110)(34 117 61 106)(35 118 62 107)(36 230 320 295)(37 226 316 291)(38 227 317 292)(39 228 318 293)(40 229 319 294)(41 183 93 123)(42 184 94 124)(43 185 95 125)(44 181 91 121)(45 182 92 122)(46 203 113 138)(47 204 114 139)(48 205 115 140)(49 201 111 136)(50 202 112 137)(51 246 336 328)(52 247 337 329)(53 248 338 330)(54 249 339 326)(55 250 340 327)(56 141 69 151)(57 142 70 152)(58 143 66 153)(59 144 67 154)(60 145 68 155)(71 261 351 276)(72 262 352 277)(73 263 353 278)(74 264 354 279)(75 265 355 280)(76 240 150 158)(77 236 146 159)(78 237 147 160)(79 238 148 156)(80 239 149 157)(191 281 270 360)(192 282 266 356)(193 283 267 357)(194 284 268 358)(195 285 269 359)(196 286 207 297)(197 287 208 298)(198 288 209 299)(199 289 210 300)(200 290 206 296)(216 306 221 311)(217 307 222 312)(218 308 223 313)(219 309 224 314)(220 310 225 315)(231 321 241 331)(232 322 242 332)(233 323 243 333)(234 324 244 334)(235 325 245 335)(251 341 257 347)(252 342 258 348)(253 343 259 349)(254 344 260 350)(255 345 256 346)
(1 224 21 219)(2 225 22 220)(3 221 23 216)(4 222 24 217)(5 223 25 218)(6 252 85 258)(7 253 81 259)(8 254 82 260)(9 255 83 256)(10 251 84 257)(11 278 98 263)(12 279 99 264)(13 280 100 265)(14 276 96 261)(15 277 97 262)(16 185 305 125)(17 181 301 121)(18 182 302 122)(19 183 303 123)(20 184 304 124)(26 267 87 193)(27 268 88 194)(28 269 89 195)(29 270 90 191)(30 266 86 192)(31 209 63 198)(32 210 64 199)(33 206 65 200)(34 207 61 196)(35 208 62 197)(36 205 320 140)(37 201 316 136)(38 202 317 137)(39 203 318 138)(40 204 319 139)(41 273 93 213)(42 274 94 214)(43 275 95 215)(44 271 91 211)(45 272 92 212)(46 293 113 228)(47 294 114 229)(48 295 115 230)(49 291 111 226)(50 292 112 227)(51 238 336 156)(52 239 337 157)(53 240 338 158)(54 236 339 159)(55 237 340 160)(56 231 69 241)(57 232 70 242)(58 233 66 243)(59 234 67 244)(60 235 68 245)(71 186 351 171)(72 187 352 172)(73 188 353 173)(74 189 354 174)(75 190 355 175)(76 330 150 248)(77 326 146 249)(78 327 147 250)(79 328 148 246)(80 329 149 247)(101 360 180 281)(102 356 176 282)(103 357 177 283)(104 358 178 284)(105 359 179 285)(106 297 117 286)(107 298 118 287)(108 299 119 288)(109 300 120 289)(110 296 116 290)(126 311 131 306)(127 312 132 307)(128 313 133 308)(129 314 134 309)(130 315 135 310)(141 331 151 321)(142 332 152 322)(143 333 153 323)(144 334 154 324)(145 335 155 325)(161 347 167 341)(162 348 168 342)(163 349 169 343)(164 350 170 344)(165 346 166 345)

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

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

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

105 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 12A ··· 12H 15A ··· 15P 20A 20B 20C 20D 20E ··· 20L 30A ··· 30P 60A ··· 60AF order 1 2 3 3 3 3 4 4 4 5 5 5 5 6 6 6 6 10 10 10 10 12 ··· 12 15 ··· 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 2 2 2 2 2 18 18 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 2 ··· 2 2 2 2 2 18 ··· 18 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C5 C10 C10 S3 Q8 D6 Dic6 C5×S3 C5×Q8 S3×C10 C5×Dic6 kernel C5×C32⋊4Q8 C5×C3⋊Dic3 C3×C60 C32⋊4Q8 C3⋊Dic3 C3×C12 C60 C3×C15 C30 C15 C12 C32 C6 C3 # reps 1 2 1 4 8 4 4 1 4 8 16 4 16 32

Matrix representation of C5×C324Q8 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 60
,
 0 1 0 0 60 60 0 0 0 0 0 1 0 0 60 60
,
 23 46 0 0 15 38 0 0 0 0 1 0 0 0 0 1
,
 30 34 0 0 4 31 0 0 0 0 49 19 0 0 31 12
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,60],[0,60,0,0,1,60,0,0,0,0,0,60,0,0,1,60],[23,15,0,0,46,38,0,0,0,0,1,0,0,0,0,1],[30,4,0,0,34,31,0,0,0,0,49,31,0,0,19,12] >;

C5×C324Q8 in GAP, Magma, Sage, TeX

C_5\times C_3^2\rtimes_4Q_8
% in TeX

G:=Group("C5xC3^2:4Q8");
// GroupNames label

G:=SmallGroup(360,105);
// by ID

G=gap.SmallGroup(360,105);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,265,127,2404,8645]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽