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G = C2×C32⋊7Q16  order 288 = 25·32

Direct product of C2 and C32⋊7Q16

direct product, metabelian, supersoluble, monomial

Aliases: C2×C32⋊7Q16, C62.135D4, (C3×C6)⋊7Q16, C6⋊3(C3⋊Q16), (C6×Q8).23S3, (C3×Q8).66D6, C32⋊14(C2×Q16), (C3×C12).103D4, (C2×C12).158D6, C12.62(C3⋊D4), C4.9(C32⋊7D4), (C3×C12).106C23, C12.102(C22×S3), (C6×C12).149C22, C32⋊4C8.30C22, (Q8×C32).28C22, C32⋊4Q8.31C22, C22.24(C32⋊7D4), (Q8×C3×C6).8C2, C3⋊4(C2×C3⋊Q16), Q8.12(C2×C3⋊S3), (C3×C6).289(C2×D4), C6.130(C2×C3⋊D4), C4.16(C22×C3⋊S3), (C2×Q8).5(C3⋊S3), C2.19(C2×C32⋊7D4), (C2×C6).103(C3⋊D4), (C2×C32⋊4C8).11C2, (C2×C32⋊4Q8).15C2, (C2×C4).53(C2×C3⋊S3), SmallGroup(288,800)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C2×C32⋊7Q16
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4Q8 — C2×C32⋊4Q8 — C2×C32⋊7Q16
Lower central
C32 — C3×C6 — C3×C12 — C2×C32⋊7Q16
Upper central
C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×C32⋊7Q16
 G = < a,b,c,d,e | a2=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 532 in 180 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C32, Dic3, C12, C12, C2×C6, C2×C8, Q16, C2×Q8, C2×Q8, C3×C6, C3×C6, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C2×Q16, C3⋊Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C3⋊Q16, C2×Dic6, C6×Q8, C32⋊4C8, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6×C12, Q8×C32, Q8×C32, C2×C3⋊Q16, C2×C32⋊4C8, C32⋊7Q16, C2×C32⋊4Q8, Q8×C3×C6, C2×C32⋊7Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C2×Q16, C2×C3⋊S3, C3⋊Q16, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C2×C3⋊Q16, C32⋊7Q16, C2×C32⋊7D4, C2×C32⋊7Q16

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

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

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

(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)

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

54 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A···6L8A8B8C8D12A···12X
order122233334444446···6888812···12
size11112222224436362···2181818184···4

54 irreducible representations

dim11111222222224
type++++++++++--
imageC1C2C2C2C2S3D4D4D6D6Q16C3⋊D4C3⋊D4C3⋊Q16
kernelC2×C32⋊7Q16C2×C32⋊4C8C32⋊7Q16C2×C32⋊4Q8Q8×C3×C6C6×Q8C3×C12C62C2×C12C3×Q8C3×C6C12C2×C6C6
# reps11411411484888

Matrix representation of C2×C32⋊7Q16 ►in GL6(𝔽73)

100000
010000
0072000
0007200
000010
000001
,
100000
010000
000100
00727200
000010
000001
,
7210000
7200000
001000
000100
000010
000001
,
36620000
25370000
00464600
0002700
00003232
0000570
,
43600000
13300000
00431300
00603000
00003335
00002140

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,25,0,0,0,0,62,37,0,0,0,0,0,0,46,0,0,0,0,0,46,27,0,0,0,0,0,0,32,57,0,0,0,0,32,0],[43,13,0,0,0,0,60,30,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,0,0,0,0,33,21,0,0,0,0,35,40] >;
 

C2×C32⋊7Q16 in GAP, Magma, Sage, TeX 

C_2\times C_3^2\rtimes_7Q_{16}
 
% in TeX
 
G:=Group("C2xC3^2:7Q16");
 
// GroupNames label
 
G:=SmallGroup(288,800);
 
// by ID
 
G=gap.SmallGroup(288,800);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,100,675,185,80,2693,9414]);
 
// 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,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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