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G = C32×C2.D8order 288 = 25·32

Direct product of C32 and C2.D8

direct product, metacyclic, nilpotent (class 3), monomial

Aliases: C32×C2.D8, C243C12, C62.141D4, (C3×C24)⋊9C4, C81(C3×C12), C4.7(C6×C12), C6.20(C3×D8), (C3×C6).43D8, (C2×C24).16C6, (C6×C24).15C2, (C3×C6).19Q16, C6.10(C3×Q16), (C3×C12).30Q8, C12.14(C3×Q8), C2.2(C32×D8), C12.56(C2×C12), C4.2(Q8×C32), (C2×C4).18C62, C2.2(C32×Q16), (C6×C12).365C22, C22.11(D4×C32), C6.19(C3×C4⋊C4), C4⋊C4.3(C3×C6), (C2×C8).3(C3×C6), (C3×C4⋊C4).20C6, (C2×C6).68(C3×D4), C2.4(C32×C4⋊C4), (C3×C6).48(C4⋊C4), (C2×C12).152(C2×C6), (C3×C12).141(C2×C4), (C32×C4⋊C4).17C2, SmallGroup(288,325)

Series: Derived Chief Lower central Upper central

C1C4 — C32×C2.D8
C1C2C22C2×C4C2×C12C6×C12C32×C4⋊C4 — C32×C2.D8
C1C2C4 — C32×C2.D8
C1C62C6×C12 — C32×C2.D8

Generators and relations for C32×C2.D8
 G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 156 in 108 conjugacy classes, 84 normal (20 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], C32, C12 [×8], C12 [×8], C2×C6 [×4], C4⋊C4 [×2], C2×C8, C3×C6 [×3], C24 [×8], C2×C12 [×4], C2×C12 [×8], C2.D8, C3×C12 [×2], C3×C12 [×2], C62, C3×C4⋊C4 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C6×C12 [×2], C3×C2.D8 [×4], C32×C4⋊C4 [×2], C6×C24, C32×C2.D8
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4, Q8, C32, C12 [×8], C2×C6 [×4], C4⋊C4, D8, Q16, C3×C6 [×3], C2×C12 [×4], C3×D4 [×4], C3×Q8 [×4], C2.D8, C3×C12 [×2], C62, C3×C4⋊C4 [×4], C3×D8 [×4], C3×Q16 [×4], C6×C12, D4×C32, Q8×C32, C3×C2.D8 [×4], C32×C4⋊C4, C32×D8, C32×Q16, C32×C2.D8

Smallest permutation representation of C32×C2.D8
Regular action on 288 points
Generators in S288
(1 16 97)(2 9 98)(3 10 99)(4 11 100)(5 12 101)(6 13 102)(7 14 103)(8 15 104)(17 105 33)(18 106 34)(19 107 35)(20 108 36)(21 109 37)(22 110 38)(23 111 39)(24 112 40)(25 89 81)(26 90 82)(27 91 83)(28 92 84)(29 93 85)(30 94 86)(31 95 87)(32 96 88)(41 149 57)(42 150 58)(43 151 59)(44 152 60)(45 145 61)(46 146 62)(47 147 63)(48 148 64)(49 137 121)(50 138 122)(51 139 123)(52 140 124)(53 141 125)(54 142 126)(55 143 127)(56 144 128)(65 129 113)(66 130 114)(67 131 115)(68 132 116)(69 133 117)(70 134 118)(71 135 119)(72 136 120)(73 208 192)(74 201 185)(75 202 186)(76 203 187)(77 204 188)(78 205 189)(79 206 190)(80 207 191)(153 222 169)(154 223 170)(155 224 171)(156 217 172)(157 218 173)(158 219 174)(159 220 175)(160 221 176)(161 253 237)(162 254 238)(163 255 239)(164 256 240)(165 249 233)(166 250 234)(167 251 235)(168 252 236)(177 245 229)(178 246 230)(179 247 231)(180 248 232)(181 241 225)(182 242 226)(183 243 227)(184 244 228)(193 285 269)(194 286 270)(195 287 271)(196 288 272)(197 281 265)(198 282 266)(199 283 267)(200 284 268)(209 277 261)(210 278 262)(211 279 263)(212 280 264)(213 273 257)(214 274 258)(215 275 259)(216 276 260)
(1 89 17)(2 90 18)(3 91 19)(4 92 20)(5 93 21)(6 94 22)(7 95 23)(8 96 24)(9 82 106)(10 83 107)(11 84 108)(12 85 109)(13 86 110)(14 87 111)(15 88 112)(16 81 105)(25 33 97)(26 34 98)(27 35 99)(28 36 100)(29 37 101)(30 38 102)(31 39 103)(32 40 104)(41 49 113)(42 50 114)(43 51 115)(44 52 116)(45 53 117)(46 54 118)(47 55 119)(48 56 120)(57 121 129)(58 122 130)(59 123 131)(60 124 132)(61 125 133)(62 126 134)(63 127 135)(64 128 136)(65 149 137)(66 150 138)(67 151 139)(68 152 140)(69 145 141)(70 146 142)(71 147 143)(72 148 144)(73 284 216)(74 285 209)(75 286 210)(76 287 211)(77 288 212)(78 281 213)(79 282 214)(80 283 215)(153 161 229)(154 162 230)(155 163 231)(156 164 232)(157 165 225)(158 166 226)(159 167 227)(160 168 228)(169 237 245)(170 238 246)(171 239 247)(172 240 248)(173 233 241)(174 234 242)(175 235 243)(176 236 244)(177 222 253)(178 223 254)(179 224 255)(180 217 256)(181 218 249)(182 219 250)(183 220 251)(184 221 252)(185 193 261)(186 194 262)(187 195 263)(188 196 264)(189 197 257)(190 198 258)(191 199 259)(192 200 260)(201 269 277)(202 270 278)(203 271 279)(204 272 280)(205 265 273)(206 266 274)(207 267 275)(208 268 276)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 152)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 151)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 113)(24 114)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 121)(32 122)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 129)(40 130)(49 95)(50 96)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 103)(58 104)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 111)(66 112)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 221)(74 222)(75 223)(76 224)(77 217)(78 218)(79 219)(80 220)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 137)(88 138)(153 185)(154 186)(155 187)(156 188)(157 189)(158 190)(159 191)(160 192)(161 193)(162 194)(163 195)(164 196)(165 197)(166 198)(167 199)(168 200)(169 201)(170 202)(171 203)(172 204)(173 205)(174 206)(175 207)(176 208)(177 209)(178 210)(179 211)(180 212)(181 213)(182 214)(183 215)(184 216)(225 257)(226 258)(227 259)(228 260)(229 261)(230 262)(231 263)(232 264)(233 265)(234 266)(235 267)(236 268)(237 269)(238 270)(239 271)(240 272)(241 273)(242 274)(243 275)(244 276)(245 277)(246 278)(247 279)(248 280)(249 281)(250 282)(251 283)(252 284)(253 285)(254 286)(255 287)(256 288)
(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)
(1 192 43 160)(2 191 44 159)(3 190 45 158)(4 189 46 157)(5 188 47 156)(6 187 48 155)(7 186 41 154)(8 185 42 153)(9 80 152 220)(10 79 145 219)(11 78 146 218)(12 77 147 217)(13 76 148 224)(14 75 149 223)(15 74 150 222)(16 73 151 221)(17 260 115 228)(18 259 116 227)(19 258 117 226)(20 257 118 225)(21 264 119 232)(22 263 120 231)(23 262 113 230)(24 261 114 229)(25 268 123 236)(26 267 124 235)(27 266 125 234)(28 265 126 233)(29 272 127 240)(30 271 128 239)(31 270 121 238)(32 269 122 237)(33 276 131 244)(34 275 132 243)(35 274 133 242)(36 273 134 241)(37 280 135 248)(38 279 136 247)(39 278 129 246)(40 277 130 245)(49 162 95 194)(50 161 96 193)(51 168 89 200)(52 167 90 199)(53 166 91 198)(54 165 92 197)(55 164 93 196)(56 163 94 195)(57 170 103 202)(58 169 104 201)(59 176 97 208)(60 175 98 207)(61 174 99 206)(62 173 100 205)(63 172 101 204)(64 171 102 203)(65 178 111 210)(66 177 112 209)(67 184 105 216)(68 183 106 215)(69 182 107 214)(70 181 108 213)(71 180 109 212)(72 179 110 211)(81 284 139 252)(82 283 140 251)(83 282 141 250)(84 281 142 249)(85 288 143 256)(86 287 144 255)(87 286 137 254)(88 285 138 253)

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

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

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

126 conjugacy classes

class 1 2A2B2C3A···3H4A4B4C4D4E4F6A···6X8A8B8C8D12A···12P12Q···12AV24A···24AF
order12223···34444446···6888812···1212···1224···24
size11111···12244441···122222···24···42···2

126 irreducible representations

dim1111111122222222
type+++-++-
imageC1C2C2C3C4C6C6C12Q8D4D8Q16C3×Q8C3×D4C3×D8C3×Q16
kernelC32×C2.D8C32×C4⋊C4C6×C24C3×C2.D8C3×C24C3×C4⋊C4C2×C24C24C3×C12C62C3×C6C3×C6C12C2×C6C6C6
# reps12184168321122881616

Matrix representation of C32×C2.D8 in GL4(𝔽73) generated by

1000
06400
0010
0001
,
8000
0100
0010
0001
,
72000
0100
0010
0001
,
1000
07200
005716
005757
,
46000
0100
007049
00493
G:=sub<GL(4,GF(73))| [1,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,57,57,0,0,16,57],[46,0,0,0,0,1,0,0,0,0,70,49,0,0,49,3] >;

C32×C2.D8 in GAP, Magma, Sage, TeX

C_3^2\times C_2.D_8
% in TeX

G:=Group("C3^2xC2.D8");
// GroupNames label

G:=SmallGroup(288,325);
// by ID

G=gap.SmallGroup(288,325);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,1268,6304,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽