direct product, metacyclic, nilpotent (class 3), monomial
Aliases: C32×C2.D8, C24⋊3C12, C62.141D4, (C3×C24)⋊9C4, C8⋊1(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
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, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, C12, C12, C2×C6, C4⋊C4, C2×C8, C3×C6, C24, C2×C12, C2×C12, C2.D8, C3×C12, C3×C12, C62, C3×C4⋊C4, C2×C24, C3×C24, C6×C12, C6×C12, C3×C2.D8, C32×C4⋊C4, C6×C24, C32×C2.D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4⋊C4, D8, Q16, C3×C6, C2×C12, C3×D4, C3×Q8, C2.D8, C3×C12, C62, C3×C4⋊C4, C3×D8, C3×Q16, C6×C12, D4×C32, Q8×C32, C3×C2.D8, C32×C4⋊C4, C32×D8, C32×Q16, C32×C2.D8
►Regular action on 288 points
Generators in S288
(1 17 37)(2 18 38)(3 19 39)(4 20 40)(5 21 33)(6 22 34)(7 23 35)(8 24 36)(9 109 93)(10 110 94)(11 111 95)(12 112 96)(13 105 89)(14 106 90)(15 107 91)(16 108 92)(25 82 97)(26 83 98)(27 84 99)(28 85 100)(29 86 101)(30 87 102)(31 88 103)(32 81 104)(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 29 93)(2 30 94)(3 31 95)(4 32 96)(5 25 89)(6 26 90)(7 27 91)(8 28 92)(9 17 86)(10 18 87)(11 19 88)(12 20 81)(13 21 82)(14 22 83)(15 23 84)(16 24 85)(33 97 105)(34 98 106)(35 99 107)(36 100 108)(37 101 109)(38 102 110)(39 103 111)(40 104 112)(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 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 65)(16 66)(17 151)(18 152)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(73 223)(74 224)(75 217)(76 218)(77 219)(78 220)(79 221)(80 222)(81 142)(82 143)(83 144)(84 137)(85 138)(86 139)(87 140)(88 141)(89 119)(90 120)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 127)(98 128)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 135)(106 136)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(153 191)(154 192)(155 185)(156 186)(157 187)(158 188)(159 189)(160 190)(161 199)(162 200)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 207)(170 208)(171 201)(172 202)(173 203)(174 204)(175 205)(176 206)(177 215)(178 216)(179 209)(180 210)(181 211)(182 212)(183 213)(184 214)(225 263)(226 264)(227 257)(228 258)(229 259)(230 260)(231 261)(232 262)(233 271)(234 272)(235 265)(236 266)(237 267)(238 268)(239 269)(240 270)(241 279)(242 280)(243 273)(244 274)(245 275)(246 276)(247 277)(248 278)(249 287)(250 288)(251 281)(252 282)(253 283)(254 284)(255 285)(256 286)
(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 43 154)(2 191 44 153)(3 190 45 160)(4 189 46 159)(5 188 47 158)(6 187 48 157)(7 186 41 156)(8 185 42 155)(9 216 67 178)(10 215 68 177)(11 214 69 184)(12 213 70 183)(13 212 71 182)(14 211 72 181)(15 210 65 180)(16 209 66 179)(17 73 151 223)(18 80 152 222)(19 79 145 221)(20 78 146 220)(21 77 147 219)(22 76 148 218)(23 75 149 217)(24 74 150 224)(25 196 55 166)(26 195 56 165)(27 194 49 164)(28 193 50 163)(29 200 51 162)(30 199 52 161)(31 198 53 168)(32 197 54 167)(33 204 63 174)(34 203 64 173)(35 202 57 172)(36 201 58 171)(37 208 59 170)(38 207 60 169)(39 206 61 176)(40 205 62 175)(81 281 142 251)(82 288 143 250)(83 287 144 249)(84 286 137 256)(85 285 138 255)(86 284 139 254)(87 283 140 253)(88 282 141 252)(89 264 119 226)(90 263 120 225)(91 262 113 232)(92 261 114 231)(93 260 115 230)(94 259 116 229)(95 258 117 228)(96 257 118 227)(97 272 127 234)(98 271 128 233)(99 270 121 240)(100 269 122 239)(101 268 123 238)(102 267 124 237)(103 266 125 236)(104 265 126 235)(105 280 135 242)(106 279 136 241)(107 278 129 248)(108 277 130 247)(109 276 131 246)(110 275 132 245)(111 274 133 244)(112 273 134 243)
G:=sub<Sym(288)| (1,17,37)(2,18,38)(3,19,39)(4,20,40)(5,21,33)(6,22,34)(7,23,35)(8,24,36)(9,109,93)(10,110,94)(11,111,95)(12,112,96)(13,105,89)(14,106,90)(15,107,91)(16,108,92)(25,82,97)(26,83,98)(27,84,99)(28,85,100)(29,86,101)(30,87,102)(31,88,103)(32,81,104)(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), (1,29,93)(2,30,94)(3,31,95)(4,32,96)(5,25,89)(6,26,90)(7,27,91)(8,28,92)(9,17,86)(10,18,87)(11,19,88)(12,20,81)(13,21,82)(14,22,83)(15,23,84)(16,24,85)(33,97,105)(34,98,106)(35,99,107)(36,100,108)(37,101,109)(38,102,110)(39,103,111)(40,104,112)(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)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,151)(18,152)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(73,223)(74,224)(75,217)(76,218)(77,219)(78,220)(79,221)(80,222)(81,142)(82,143)(83,144)(84,137)(85,138)(86,139)(87,140)(88,141)(89,119)(90,120)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,127)(98,128)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(153,191)(154,192)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190)(161,199)(162,200)(163,193)(164,194)(165,195)(166,196)(167,197)(168,198)(169,207)(170,208)(171,201)(172,202)(173,203)(174,204)(175,205)(176,206)(177,215)(178,216)(179,209)(180,210)(181,211)(182,212)(183,213)(184,214)(225,263)(226,264)(227,257)(228,258)(229,259)(230,260)(231,261)(232,262)(233,271)(234,272)(235,265)(236,266)(237,267)(238,268)(239,269)(240,270)(241,279)(242,280)(243,273)(244,274)(245,275)(246,276)(247,277)(248,278)(249,287)(250,288)(251,281)(252,282)(253,283)(254,284)(255,285)(256,286), (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), (1,192,43,154)(2,191,44,153)(3,190,45,160)(4,189,46,159)(5,188,47,158)(6,187,48,157)(7,186,41,156)(8,185,42,155)(9,216,67,178)(10,215,68,177)(11,214,69,184)(12,213,70,183)(13,212,71,182)(14,211,72,181)(15,210,65,180)(16,209,66,179)(17,73,151,223)(18,80,152,222)(19,79,145,221)(20,78,146,220)(21,77,147,219)(22,76,148,218)(23,75,149,217)(24,74,150,224)(25,196,55,166)(26,195,56,165)(27,194,49,164)(28,193,50,163)(29,200,51,162)(30,199,52,161)(31,198,53,168)(32,197,54,167)(33,204,63,174)(34,203,64,173)(35,202,57,172)(36,201,58,171)(37,208,59,170)(38,207,60,169)(39,206,61,176)(40,205,62,175)(81,281,142,251)(82,288,143,250)(83,287,144,249)(84,286,137,256)(85,285,138,255)(86,284,139,254)(87,283,140,253)(88,282,141,252)(89,264,119,226)(90,263,120,225)(91,262,113,232)(92,261,114,231)(93,260,115,230)(94,259,116,229)(95,258,117,228)(96,257,118,227)(97,272,127,234)(98,271,128,233)(99,270,121,240)(100,269,122,239)(101,268,123,238)(102,267,124,237)(103,266,125,236)(104,265,126,235)(105,280,135,242)(106,279,136,241)(107,278,129,248)(108,277,130,247)(109,276,131,246)(110,275,132,245)(111,274,133,244)(112,273,134,243)>;
G:=Group( (1,17,37)(2,18,38)(3,19,39)(4,20,40)(5,21,33)(6,22,34)(7,23,35)(8,24,36)(9,109,93)(10,110,94)(11,111,95)(12,112,96)(13,105,89)(14,106,90)(15,107,91)(16,108,92)(25,82,97)(26,83,98)(27,84,99)(28,85,100)(29,86,101)(30,87,102)(31,88,103)(32,81,104)(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), (1,29,93)(2,30,94)(3,31,95)(4,32,96)(5,25,89)(6,26,90)(7,27,91)(8,28,92)(9,17,86)(10,18,87)(11,19,88)(12,20,81)(13,21,82)(14,22,83)(15,23,84)(16,24,85)(33,97,105)(34,98,106)(35,99,107)(36,100,108)(37,101,109)(38,102,110)(39,103,111)(40,104,112)(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)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,151)(18,152)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(73,223)(74,224)(75,217)(76,218)(77,219)(78,220)(79,221)(80,222)(81,142)(82,143)(83,144)(84,137)(85,138)(86,139)(87,140)(88,141)(89,119)(90,120)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,127)(98,128)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(153,191)(154,192)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190)(161,199)(162,200)(163,193)(164,194)(165,195)(166,196)(167,197)(168,198)(169,207)(170,208)(171,201)(172,202)(173,203)(174,204)(175,205)(176,206)(177,215)(178,216)(179,209)(180,210)(181,211)(182,212)(183,213)(184,214)(225,263)(226,264)(227,257)(228,258)(229,259)(230,260)(231,261)(232,262)(233,271)(234,272)(235,265)(236,266)(237,267)(238,268)(239,269)(240,270)(241,279)(242,280)(243,273)(244,274)(245,275)(246,276)(247,277)(248,278)(249,287)(250,288)(251,281)(252,282)(253,283)(254,284)(255,285)(256,286), (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), (1,192,43,154)(2,191,44,153)(3,190,45,160)(4,189,46,159)(5,188,47,158)(6,187,48,157)(7,186,41,156)(8,185,42,155)(9,216,67,178)(10,215,68,177)(11,214,69,184)(12,213,70,183)(13,212,71,182)(14,211,72,181)(15,210,65,180)(16,209,66,179)(17,73,151,223)(18,80,152,222)(19,79,145,221)(20,78,146,220)(21,77,147,219)(22,76,148,218)(23,75,149,217)(24,74,150,224)(25,196,55,166)(26,195,56,165)(27,194,49,164)(28,193,50,163)(29,200,51,162)(30,199,52,161)(31,198,53,168)(32,197,54,167)(33,204,63,174)(34,203,64,173)(35,202,57,172)(36,201,58,171)(37,208,59,170)(38,207,60,169)(39,206,61,176)(40,205,62,175)(81,281,142,251)(82,288,143,250)(83,287,144,249)(84,286,137,256)(85,285,138,255)(86,284,139,254)(87,283,140,253)(88,282,141,252)(89,264,119,226)(90,263,120,225)(91,262,113,232)(92,261,114,231)(93,260,115,230)(94,259,116,229)(95,258,117,228)(96,257,118,227)(97,272,127,234)(98,271,128,233)(99,270,121,240)(100,269,122,239)(101,268,123,238)(102,267,124,237)(103,266,125,236)(104,265,126,235)(105,280,135,242)(106,279,136,241)(107,278,129,248)(108,277,130,247)(109,276,131,246)(110,275,132,245)(111,274,133,244)(112,273,134,243) );
G=PermutationGroup([[(1,17,37),(2,18,38),(3,19,39),(4,20,40),(5,21,33),(6,22,34),(7,23,35),(8,24,36),(9,109,93),(10,110,94),(11,111,95),(12,112,96),(13,105,89),(14,106,90),(15,107,91),(16,108,92),(25,82,97),(26,83,98),(27,84,99),(28,85,100),(29,86,101),(30,87,102),(31,88,103),(32,81,104),(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)], [(1,29,93),(2,30,94),(3,31,95),(4,32,96),(5,25,89),(6,26,90),(7,27,91),(8,28,92),(9,17,86),(10,18,87),(11,19,88),(12,20,81),(13,21,82),(14,22,83),(15,23,84),(16,24,85),(33,97,105),(34,98,106),(35,99,107),(36,100,108),(37,101,109),(38,102,110),(39,103,111),(40,104,112),(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),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,65),(16,66),(17,151),(18,152),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(73,223),(74,224),(75,217),(76,218),(77,219),(78,220),(79,221),(80,222),(81,142),(82,143),(83,144),(84,137),(85,138),(86,139),(87,140),(88,141),(89,119),(90,120),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,127),(98,128),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,135),(106,136),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(153,191),(154,192),(155,185),(156,186),(157,187),(158,188),(159,189),(160,190),(161,199),(162,200),(163,193),(164,194),(165,195),(166,196),(167,197),(168,198),(169,207),(170,208),(171,201),(172,202),(173,203),(174,204),(175,205),(176,206),(177,215),(178,216),(179,209),(180,210),(181,211),(182,212),(183,213),(184,214),(225,263),(226,264),(227,257),(228,258),(229,259),(230,260),(231,261),(232,262),(233,271),(234,272),(235,265),(236,266),(237,267),(238,268),(239,269),(240,270),(241,279),(242,280),(243,273),(244,274),(245,275),(246,276),(247,277),(248,278),(249,287),(250,288),(251,281),(252,282),(253,283),(254,284),(255,285),(256,286)], [(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)], [(1,192,43,154),(2,191,44,153),(3,190,45,160),(4,189,46,159),(5,188,47,158),(6,187,48,157),(7,186,41,156),(8,185,42,155),(9,216,67,178),(10,215,68,177),(11,214,69,184),(12,213,70,183),(13,212,71,182),(14,211,72,181),(15,210,65,180),(16,209,66,179),(17,73,151,223),(18,80,152,222),(19,79,145,221),(20,78,146,220),(21,77,147,219),(22,76,148,218),(23,75,149,217),(24,74,150,224),(25,196,55,166),(26,195,56,165),(27,194,49,164),(28,193,50,163),(29,200,51,162),(30,199,52,161),(31,198,53,168),(32,197,54,167),(33,204,63,174),(34,203,64,173),(35,202,57,172),(36,201,58,171),(37,208,59,170),(38,207,60,169),(39,206,61,176),(40,205,62,175),(81,281,142,251),(82,288,143,250),(83,287,144,249),(84,286,137,256),(85,285,138,255),(86,284,139,254),(87,283,140,253),(88,282,141,252),(89,264,119,226),(90,263,120,225),(91,262,113,232),(92,261,114,231),(93,260,115,230),(94,259,116,229),(95,258,117,228),(96,257,118,227),(97,272,127,234),(98,271,128,233),(99,270,121,240),(100,269,122,239),(101,268,123,238),(102,267,124,237),(103,266,125,236),(104,265,126,235),(105,280,135,242),(106,279,136,241),(107,278,129,248),(108,277,130,247),(109,276,131,246),(110,275,132,245),(111,274,133,244),(112,273,134,243)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6X | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12AV | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | Q8 | D4 | D8 | Q16 | C3×Q8 | C3×D4 | C3×D8 | C3×Q16 |
| kernel | C32×C2.D8 | C32×C4⋊C4 | C6×C24 | C3×C2.D8 | C3×C24 | C3×C4⋊C4 | C2×C24 | C24 | C3×C12 | C62 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 1 | 1 | 2 | 2 | 8 | 8 | 16 | 16 |
Matrix representation of C32×C2.D8 ►in GL4(𝔽73) generated by
| 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 | 16 |
| 0 | 0 | 57 | 57 |
| 46 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 49 |
| 0 | 0 | 49 | 3 |
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