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G = C2×C325Q16order 288 = 25·32

Direct product of C2 and C325Q16

direct product, metabelian, supersoluble, monomial

Aliases: C2×C325Q16, C61Dic12, C24.75D6, C12.50D12, C62.93D4, (C3×C6)⋊5Q16, (C6×C24).8C2, (C2×C24).13S3, C32(C2×Dic12), C6.59(C2×D12), (C2×C6).41D12, C3210(C2×Q16), (C2×C12).386D6, (C3×C12).125D4, C4.8(C12⋊S3), (C3×C24).53C22, (C3×C12).154C23, C12.192(C22×S3), (C6×C12).302C22, C22.14(C12⋊S3), C324Q8.24C22, C8.17(C2×C3⋊S3), (C2×C8).4(C3⋊S3), (C3×C6).199(C2×D4), C4.29(C22×C3⋊S3), C2.14(C2×C12⋊S3), (C2×C324Q8).6C2, (C2×C4).82(C2×C3⋊S3), SmallGroup(288,762)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C2×C325Q16
C1C3C32C3×C6C3×C12C324Q8C2×C324Q8 — C2×C325Q16
C32C3×C6C3×C12 — C2×C325Q16
C1C22C2×C4C2×C8

Generators and relations for C2×C325Q16
 G = < a,b,c,d,e | a2=b3=c3=d8=1, e2=d4, 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: 660 in 180 conjugacy classes, 77 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C32, Dic3 [×16], C12 [×8], C2×C6 [×4], C2×C8, Q16 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], C24 [×8], Dic6 [×24], C2×Dic3 [×8], C2×C12 [×4], C2×Q16, C3⋊Dic3 [×4], C3×C12 [×2], C62, Dic12 [×16], C2×C24 [×4], C2×Dic6 [×8], C3×C24 [×2], C324Q8 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C2×Dic12 [×4], C325Q16 [×4], C6×C24, C2×C324Q8 [×2], C2×C325Q16
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], Q16 [×2], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C2×Q16, C2×C3⋊S3 [×3], Dic12 [×8], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C2×Dic12 [×4], C325Q16 [×2], C2×C12⋊S3, C2×C325Q16

Smallest permutation representation of C2×C325Q16
Regular action on 288 points
Generators in S288
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 170)(10 171)(11 172)(12 173)(13 174)(14 175)(15 176)(16 169)(17 178)(18 179)(19 180)(20 181)(21 182)(22 183)(23 184)(24 177)(25 267)(26 268)(27 269)(28 270)(29 271)(30 272)(31 265)(32 266)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(49 143)(50 144)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(65 236)(66 237)(67 238)(68 239)(69 240)(70 233)(71 234)(72 235)(73 189)(74 190)(75 191)(76 192)(77 185)(78 186)(79 187)(80 188)(81 225)(82 226)(83 227)(84 228)(85 229)(86 230)(87 231)(88 232)(89 196)(90 197)(91 198)(92 199)(93 200)(94 193)(95 194)(96 195)(97 213)(98 214)(99 215)(100 216)(101 209)(102 210)(103 211)(104 212)(105 275)(106 276)(107 277)(108 278)(109 279)(110 280)(111 273)(112 274)(113 156)(114 157)(115 158)(116 159)(117 160)(118 153)(119 154)(120 155)(121 263)(122 264)(123 257)(124 258)(125 259)(126 260)(127 261)(128 262)(129 254)(130 255)(131 256)(132 249)(133 250)(134 251)(135 252)(136 253)(145 288)(146 281)(147 282)(148 283)(149 284)(150 285)(151 286)(152 287)(161 247)(162 248)(163 241)(164 242)(165 243)(166 244)(167 245)(168 246)(201 224)(202 217)(203 218)(204 219)(205 220)(206 221)(207 222)(208 223)
(1 54 240)(2 55 233)(3 56 234)(4 49 235)(5 50 236)(6 51 237)(7 52 238)(8 53 239)(9 149 25)(10 150 26)(11 151 27)(12 152 28)(13 145 29)(14 146 30)(15 147 31)(16 148 32)(17 156 245)(18 157 246)(19 158 247)(20 159 248)(21 160 241)(22 153 242)(23 154 243)(24 155 244)(33 193 82)(34 194 83)(35 195 84)(36 196 85)(37 197 86)(38 198 87)(39 199 88)(40 200 81)(41 137 66)(42 138 67)(43 139 68)(44 140 69)(45 141 70)(46 142 71)(47 143 72)(48 144 65)(57 96 228)(58 89 229)(59 90 230)(60 91 231)(61 92 232)(62 93 225)(63 94 226)(64 95 227)(73 98 208)(74 99 201)(75 100 202)(76 101 203)(77 102 204)(78 103 205)(79 104 206)(80 97 207)(105 256 125)(106 249 126)(107 250 127)(108 251 128)(109 252 121)(110 253 122)(111 254 123)(112 255 124)(113 167 178)(114 168 179)(115 161 180)(116 162 181)(117 163 182)(118 164 183)(119 165 184)(120 166 177)(129 257 273)(130 258 274)(131 259 275)(132 260 276)(133 261 277)(134 262 278)(135 263 279)(136 264 280)(169 283 266)(170 284 267)(171 285 268)(172 286 269)(173 287 270)(174 288 271)(175 281 272)(176 282 265)(185 210 219)(186 211 220)(187 212 221)(188 213 222)(189 214 223)(190 215 224)(191 216 217)(192 209 218)
(1 190 229)(2 191 230)(3 192 231)(4 185 232)(5 186 225)(6 187 226)(7 188 227)(8 189 228)(9 159 255)(10 160 256)(11 153 249)(12 154 250)(13 155 251)(14 156 252)(15 157 253)(16 158 254)(17 109 30)(18 110 31)(19 111 32)(20 112 25)(21 105 26)(22 106 27)(23 107 28)(24 108 29)(33 137 104)(34 138 97)(35 139 98)(36 140 99)(37 141 100)(38 142 101)(39 143 102)(40 144 103)(41 79 82)(42 80 83)(43 73 84)(44 74 85)(45 75 86)(46 76 87)(47 77 88)(48 78 81)(49 210 61)(50 211 62)(51 212 63)(52 213 64)(53 214 57)(54 215 58)(55 216 59)(56 209 60)(65 205 200)(66 206 193)(67 207 194)(68 208 195)(69 201 196)(70 202 197)(71 203 198)(72 204 199)(89 240 224)(90 233 217)(91 234 218)(92 235 219)(93 236 220)(94 237 221)(95 238 222)(96 239 223)(113 135 175)(114 136 176)(115 129 169)(116 130 170)(117 131 171)(118 132 172)(119 133 173)(120 134 174)(121 146 245)(122 147 246)(123 148 247)(124 149 248)(125 150 241)(126 151 242)(127 152 243)(128 145 244)(161 257 283)(162 258 284)(163 259 285)(164 260 286)(165 261 287)(166 262 288)(167 263 281)(168 264 282)(177 278 271)(178 279 272)(179 280 265)(180 273 266)(181 274 267)(182 275 268)(183 276 269)(184 277 270)
(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)
(1 29 5 25)(2 28 6 32)(3 27 7 31)(4 26 8 30)(9 240 13 236)(10 239 14 235)(11 238 15 234)(12 237 16 233)(17 232 21 228)(18 231 22 227)(19 230 23 226)(20 229 24 225)(33 161 37 165)(34 168 38 164)(35 167 39 163)(36 166 40 162)(41 266 45 270)(42 265 46 269)(43 272 47 268)(44 271 48 267)(49 150 53 146)(50 149 54 145)(51 148 55 152)(52 147 56 151)(57 245 61 241)(58 244 62 248)(59 243 63 247)(60 242 64 246)(65 170 69 174)(66 169 70 173)(67 176 71 172)(68 175 72 171)(73 279 77 275)(74 278 78 274)(75 277 79 273)(76 276 80 280)(81 181 85 177)(82 180 86 184)(83 179 87 183)(84 178 88 182)(89 155 93 159)(90 154 94 158)(91 153 95 157)(92 160 96 156)(97 264 101 260)(98 263 102 259)(99 262 103 258)(100 261 104 257)(105 189 109 185)(106 188 110 192)(107 187 111 191)(108 186 112 190)(113 199 117 195)(114 198 118 194)(115 197 119 193)(116 196 120 200)(121 210 125 214)(122 209 126 213)(123 216 127 212)(124 215 128 211)(129 202 133 206)(130 201 134 205)(131 208 135 204)(132 207 136 203)(137 283 141 287)(138 282 142 286)(139 281 143 285)(140 288 144 284)(217 250 221 254)(218 249 222 253)(219 256 223 252)(220 255 224 251)

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

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

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

78 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A···6L8A8B8C8D12A···12P24A···24AF
order122233334444446···6888812···1224···24
size1111222222363636362···222222···22···2

78 irreducible representations

dim1111222222222
type+++++++++-++-
imageC1C2C2C2S3D4D4D6D6Q16D12D12Dic12
kernelC2×C325Q16C325Q16C6×C24C2×C324Q8C2×C24C3×C12C62C24C2×C12C3×C6C12C2×C6C6
# reps14124118448832

Matrix representation of C2×C325Q16 in GL6(𝔽73)

7200000
0720000
0072000
0007200
000010
000001
,
0720000
1720000
00727200
001000
000001
00007272
,
7210000
7200000
000100
00727200
000001
00007272
,
23680000
5180000
001000
000100
00006659
0000147
,
5320000
55200000
00727200
000100
00004151
00001032

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[23,5,0,0,0,0,68,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,14,0,0,0,0,59,7],[53,55,0,0,0,0,2,20,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,41,10,0,0,0,0,51,32] >;

C2×C325Q16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_5Q_{16}
% in TeX

G:=Group("C2xC3^2:5Q16");
// GroupNames label

G:=SmallGroup(288,762);
// by ID

G=gap.SmallGroup(288,762);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,142,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=1,e^2=d^4,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

׿
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𝔽