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G = C2×C32⋊5Q16  order 288 = 25·32

Direct product of C2 and C32⋊5Q16

direct product, metabelian, supersoluble, monomial

Aliases: C2×C32⋊5Q16, C6⋊1Dic12, C24.75D6, C12.50D12, C62.93D4, (C3×C6)⋊5Q16, (C6×C24).8C2, (C2×C24).13S3, C3⋊2(C2×Dic12), C6.59(C2×D12), (C2×C6).41D12, C32⋊10(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), C32⋊4Q8.24C22, 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×C32⋊4Q8).6C2, (C2×C4).82(C2×C3⋊S3), SmallGroup(288,762)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C2×C32⋊5Q16
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4Q8 — C2×C32⋊4Q8 — C2×C32⋊5Q16
Lower central
C32 — C3×C6 — C3×C12 — C2×C32⋊5Q16
Upper central
C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C32⋊5Q16
 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, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C2×C6, C2×C8, Q16, C2×Q8, C3×C6, C3×C6, C24, Dic6, C2×Dic3, C2×C12, C2×Q16, C3⋊Dic3, C3×C12, C62, Dic12, C2×C24, C2×Dic6, C3×C24, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C2×Dic12, C32⋊5Q16, C6×C24, C2×C32⋊4Q8, C2×C32⋊5Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊S3, D12, C22×S3, C2×Q16, C2×C3⋊S3, Dic12, C2×D12, C12⋊S3, C22×C3⋊S3, C2×Dic12, C32⋊5Q16, C2×C12⋊S3, C2×C32⋊5Q16

Smallest permutation representation of C2×C32⋊5Q16
►Regular action on 288 points
Generators in S288
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 160)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 175)(24 176)(25 177)(26 178)(27 179)(28 180)(29 181)(30 182)(31 183)(32 184)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 65)(40 66)(49 220)(50 221)(51 222)(52 223)(53 224)(54 217)(55 218)(56 219)(57 237)(58 238)(59 239)(60 240)(61 233)(62 234)(63 235)(64 236)(73 198)(74 199)(75 200)(76 193)(77 194)(78 195)(79 196)(80 197)(81 188)(82 189)(83 190)(84 191)(85 192)(86 185)(87 186)(88 187)(89 205)(90 206)(91 207)(92 208)(93 201)(94 202)(95 203)(96 204)(97 213)(98 214)(99 215)(100 216)(101 209)(102 210)(103 211)(104 212)(105 149)(106 150)(107 151)(108 152)(109 145)(110 146)(111 147)(112 148)(113 274)(114 275)(115 276)(116 277)(117 278)(118 279)(119 280)(120 273)(121 272)(122 265)(123 266)(124 267)(125 268)(126 269)(127 270)(128 271)(129 262)(130 263)(131 264)(132 257)(133 258)(134 259)(135 260)(136 261)(137 230)(138 231)(139 232)(140 225)(141 226)(142 227)(143 228)(144 229)(161 247)(162 248)(163 241)(164 242)(165 243)(166 244)(167 245)(168 246)(249 284)(250 285)(251 286)(252 287)(253 288)(254 281)(255 282)(256 283)

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

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

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

78 conjugacy classes

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

78 irreducible representations

dim1111222222222
type+++++++++-++-
imageC1C2C2C2S3D4D4D6D6Q16D12D12Dic12
kernelC2×C32⋊5Q16C32⋊5Q16C6×C24C2×C32⋊4Q8C2×C24C3×C12C62C24C2×C12C3×C6C12C2×C6C6
# reps14124118448832

Matrix representation of C2×C32⋊5Q16 ►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×C32⋊5Q16 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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