direct product, metacyclic, nilpotent (class 3), monomial
Aliases: C32×C4.Q8, C24⋊6C12, C62.140D4, C8⋊2(C3×C12), (C3×C24)⋊14C4, C4.6(C6×C12), (C6×C24).26C2, (C2×C24).27C6, (C3×C12).29Q8, C12.13(C3×Q8), C12.55(C2×C12), C4.1(Q8×C32), (C2×C4).17C62, C6.15(C3×SD16), (C3×C6).33SD16, C2.3(C32×SD16), (C6×C12).364C22, C22.10(D4×C32), C6.18(C3×C4⋊C4), C4⋊C4.2(C3×C6), (C2×C8).6(C3×C6), (C3×C4⋊C4).19C6, (C2×C6).67(C3×D4), C2.3(C32×C4⋊C4), (C3×C6).47(C4⋊C4), (C3×C12).140(C2×C4), (C2×C12).151(C2×C6), (C32×C4⋊C4).16C2, SmallGroup(288,324)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4.Q8
G = < a,b,c,d,e | a3=b3=c4=1, d4=c2, e2=c-1d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >
Subgroups: 156 in 108 conjugacy classes, 84 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, C12, C12, C2×C6, C4⋊C4, C2×C8, C3×C6, C3×C6, C24, C2×C12, C2×C12, C4.Q8, C3×C12, C3×C12, C62, C3×C4⋊C4, C2×C24, C3×C24, C6×C12, C6×C12, C3×C4.Q8, C32×C4⋊C4, C6×C24, C32×C4.Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4⋊C4, SD16, C3×C6, C2×C12, C3×D4, C3×Q8, C4.Q8, C3×C12, C62, C3×C4⋊C4, C3×SD16, C6×C12, D4×C32, Q8×C32, C3×C4.Q8, C32×C4⋊C4, C32×SD16, C32×C4.Q8
►Regular action on 288 points
Generators in S288
(1 87 97)(2 88 98)(3 81 99)(4 82 100)(5 83 101)(6 84 102)(7 85 103)(8 86 104)(9 26 90)(10 27 91)(11 28 92)(12 29 93)(13 30 94)(14 31 95)(15 32 96)(16 25 89)(17 105 33)(18 106 34)(19 107 35)(20 108 36)(21 109 37)(22 110 38)(23 111 39)(24 112 40)(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)(73 207 191)(74 208 192)(75 201 185)(76 202 186)(77 203 187)(78 204 188)(79 205 189)(80 206 190)(153 248 232)(154 241 225)(155 242 226)(156 243 227)(157 244 228)(158 245 229)(159 246 230)(160 247 231)(161 253 177)(162 254 178)(163 255 179)(164 256 180)(165 249 181)(166 250 182)(167 251 183)(168 252 184)(169 237 217)(170 238 218)(171 239 219)(172 240 220)(173 233 221)(174 234 222)(175 235 223)(176 236 224)(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)(9 106 88)(10 107 81)(11 108 82)(12 109 83)(13 110 84)(14 111 85)(15 112 86)(16 105 87)(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)(73 283 215)(74 284 216)(75 285 209)(76 286 210)(77 287 211)(78 288 212)(79 281 213)(80 282 214)(153 220 256)(154 221 249)(155 222 250)(156 223 251)(157 224 252)(158 217 253)(159 218 254)(160 219 255)(161 229 237)(162 230 238)(163 231 239)(164 232 240)(165 225 233)(166 226 234)(167 227 235)(168 228 236)(169 177 245)(170 178 246)(171 179 247)(172 180 248)(173 181 241)(174 182 242)(175 183 243)(176 184 244)(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 41 5 45)(2 42 6 46)(3 43 7 47)(4 44 8 48)(9 138 13 142)(10 139 14 143)(11 140 15 144)(12 141 16 137)(17 113 21 117)(18 114 22 118)(19 115 23 119)(20 116 24 120)(25 121 29 125)(26 122 30 126)(27 123 31 127)(28 124 32 128)(33 129 37 133)(34 130 38 134)(35 131 39 135)(36 132 40 136)(49 93 53 89)(50 94 54 90)(51 95 55 91)(52 96 56 92)(57 101 61 97)(58 102 62 98)(59 103 63 99)(60 104 64 100)(65 109 69 105)(66 110 70 106)(67 111 71 107)(68 112 72 108)(73 154 77 158)(74 155 78 159)(75 156 79 160)(76 157 80 153)(81 151 85 147)(82 152 86 148)(83 145 87 149)(84 146 88 150)(161 259 165 263)(162 260 166 264)(163 261 167 257)(164 262 168 258)(169 267 173 271)(170 268 174 272)(171 269 175 265)(172 270 176 266)(177 275 181 279)(178 276 182 280)(179 277 183 273)(180 278 184 274)(185 227 189 231)(186 228 190 232)(187 229 191 225)(188 230 192 226)(193 235 197 239)(194 236 198 240)(195 237 199 233)(196 238 200 234)(201 243 205 247)(202 244 206 248)(203 245 207 241)(204 246 208 242)(209 251 213 255)(210 252 214 256)(211 253 215 249)(212 254 216 250)(217 283 221 287)(218 284 222 288)(219 285 223 281)(220 286 224 282)
(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)
(1 185 47 225)(2 188 48 228)(3 191 41 231)(4 186 42 226)(5 189 43 229)(6 192 44 232)(7 187 45 227)(8 190 46 230)(9 288 144 224)(10 283 137 219)(11 286 138 222)(12 281 139 217)(13 284 140 220)(14 287 141 223)(15 282 142 218)(16 285 143 221)(17 261 119 165)(18 264 120 168)(19 259 113 163)(20 262 114 166)(21 257 115 161)(22 260 116 164)(23 263 117 167)(24 258 118 162)(25 269 127 173)(26 272 128 176)(27 267 121 171)(28 270 122 174)(29 265 123 169)(30 268 124 172)(31 271 125 175)(32 266 126 170)(33 277 135 181)(34 280 136 184)(35 275 129 179)(36 278 130 182)(37 273 131 177)(38 276 132 180)(39 279 133 183)(40 274 134 178)(49 239 91 199)(50 234 92 194)(51 237 93 197)(52 240 94 200)(53 235 95 195)(54 238 96 198)(55 233 89 193)(56 236 90 196)(57 247 99 207)(58 242 100 202)(59 245 101 205)(60 248 102 208)(61 243 103 203)(62 246 104 206)(63 241 97 201)(64 244 98 204)(65 255 107 215)(66 250 108 210)(67 253 109 213)(68 256 110 216)(69 251 111 211)(70 254 112 214)(71 249 105 209)(72 252 106 212)(73 149 160 81)(74 152 153 84)(75 147 154 87)(76 150 155 82)(77 145 156 85)(78 148 157 88)(79 151 158 83)(80 146 159 86)
G:=sub<Sym(288)| (1,87,97)(2,88,98)(3,81,99)(4,82,100)(5,83,101)(6,84,102)(7,85,103)(8,86,104)(9,26,90)(10,27,91)(11,28,92)(12,29,93)(13,30,94)(14,31,95)(15,32,96)(16,25,89)(17,105,33)(18,106,34)(19,107,35)(20,108,36)(21,109,37)(22,110,38)(23,111,39)(24,112,40)(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)(73,207,191)(74,208,192)(75,201,185)(76,202,186)(77,203,187)(78,204,188)(79,205,189)(80,206,190)(153,248,232)(154,241,225)(155,242,226)(156,243,227)(157,244,228)(158,245,229)(159,246,230)(160,247,231)(161,253,177)(162,254,178)(163,255,179)(164,256,180)(165,249,181)(166,250,182)(167,251,183)(168,252,184)(169,237,217)(170,238,218)(171,239,219)(172,240,220)(173,233,221)(174,234,222)(175,235,223)(176,236,224)(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)(9,106,88)(10,107,81)(11,108,82)(12,109,83)(13,110,84)(14,111,85)(15,112,86)(16,105,87)(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)(73,283,215)(74,284,216)(75,285,209)(76,286,210)(77,287,211)(78,288,212)(79,281,213)(80,282,214)(153,220,256)(154,221,249)(155,222,250)(156,223,251)(157,224,252)(158,217,253)(159,218,254)(160,219,255)(161,229,237)(162,230,238)(163,231,239)(164,232,240)(165,225,233)(166,226,234)(167,227,235)(168,228,236)(169,177,245)(170,178,246)(171,179,247)(172,180,248)(173,181,241)(174,182,242)(175,183,243)(176,184,244)(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), (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,138,13,142)(10,139,14,143)(11,140,15,144)(12,141,16,137)(17,113,21,117)(18,114,22,118)(19,115,23,119)(20,116,24,120)(25,121,29,125)(26,122,30,126)(27,123,31,127)(28,124,32,128)(33,129,37,133)(34,130,38,134)(35,131,39,135)(36,132,40,136)(49,93,53,89)(50,94,54,90)(51,95,55,91)(52,96,56,92)(57,101,61,97)(58,102,62,98)(59,103,63,99)(60,104,64,100)(65,109,69,105)(66,110,70,106)(67,111,71,107)(68,112,72,108)(73,154,77,158)(74,155,78,159)(75,156,79,160)(76,157,80,153)(81,151,85,147)(82,152,86,148)(83,145,87,149)(84,146,88,150)(161,259,165,263)(162,260,166,264)(163,261,167,257)(164,262,168,258)(169,267,173,271)(170,268,174,272)(171,269,175,265)(172,270,176,266)(177,275,181,279)(178,276,182,280)(179,277,183,273)(180,278,184,274)(185,227,189,231)(186,228,190,232)(187,229,191,225)(188,230,192,226)(193,235,197,239)(194,236,198,240)(195,237,199,233)(196,238,200,234)(201,243,205,247)(202,244,206,248)(203,245,207,241)(204,246,208,242)(209,251,213,255)(210,252,214,256)(211,253,215,249)(212,254,216,250)(217,283,221,287)(218,284,222,288)(219,285,223,281)(220,286,224,282), (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), (1,185,47,225)(2,188,48,228)(3,191,41,231)(4,186,42,226)(5,189,43,229)(6,192,44,232)(7,187,45,227)(8,190,46,230)(9,288,144,224)(10,283,137,219)(11,286,138,222)(12,281,139,217)(13,284,140,220)(14,287,141,223)(15,282,142,218)(16,285,143,221)(17,261,119,165)(18,264,120,168)(19,259,113,163)(20,262,114,166)(21,257,115,161)(22,260,116,164)(23,263,117,167)(24,258,118,162)(25,269,127,173)(26,272,128,176)(27,267,121,171)(28,270,122,174)(29,265,123,169)(30,268,124,172)(31,271,125,175)(32,266,126,170)(33,277,135,181)(34,280,136,184)(35,275,129,179)(36,278,130,182)(37,273,131,177)(38,276,132,180)(39,279,133,183)(40,274,134,178)(49,239,91,199)(50,234,92,194)(51,237,93,197)(52,240,94,200)(53,235,95,195)(54,238,96,198)(55,233,89,193)(56,236,90,196)(57,247,99,207)(58,242,100,202)(59,245,101,205)(60,248,102,208)(61,243,103,203)(62,246,104,206)(63,241,97,201)(64,244,98,204)(65,255,107,215)(66,250,108,210)(67,253,109,213)(68,256,110,216)(69,251,111,211)(70,254,112,214)(71,249,105,209)(72,252,106,212)(73,149,160,81)(74,152,153,84)(75,147,154,87)(76,150,155,82)(77,145,156,85)(78,148,157,88)(79,151,158,83)(80,146,159,86)>;
G:=Group( (1,87,97)(2,88,98)(3,81,99)(4,82,100)(5,83,101)(6,84,102)(7,85,103)(8,86,104)(9,26,90)(10,27,91)(11,28,92)(12,29,93)(13,30,94)(14,31,95)(15,32,96)(16,25,89)(17,105,33)(18,106,34)(19,107,35)(20,108,36)(21,109,37)(22,110,38)(23,111,39)(24,112,40)(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)(73,207,191)(74,208,192)(75,201,185)(76,202,186)(77,203,187)(78,204,188)(79,205,189)(80,206,190)(153,248,232)(154,241,225)(155,242,226)(156,243,227)(157,244,228)(158,245,229)(159,246,230)(160,247,231)(161,253,177)(162,254,178)(163,255,179)(164,256,180)(165,249,181)(166,250,182)(167,251,183)(168,252,184)(169,237,217)(170,238,218)(171,239,219)(172,240,220)(173,233,221)(174,234,222)(175,235,223)(176,236,224)(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)(9,106,88)(10,107,81)(11,108,82)(12,109,83)(13,110,84)(14,111,85)(15,112,86)(16,105,87)(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)(73,283,215)(74,284,216)(75,285,209)(76,286,210)(77,287,211)(78,288,212)(79,281,213)(80,282,214)(153,220,256)(154,221,249)(155,222,250)(156,223,251)(157,224,252)(158,217,253)(159,218,254)(160,219,255)(161,229,237)(162,230,238)(163,231,239)(164,232,240)(165,225,233)(166,226,234)(167,227,235)(168,228,236)(169,177,245)(170,178,246)(171,179,247)(172,180,248)(173,181,241)(174,182,242)(175,183,243)(176,184,244)(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), (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,138,13,142)(10,139,14,143)(11,140,15,144)(12,141,16,137)(17,113,21,117)(18,114,22,118)(19,115,23,119)(20,116,24,120)(25,121,29,125)(26,122,30,126)(27,123,31,127)(28,124,32,128)(33,129,37,133)(34,130,38,134)(35,131,39,135)(36,132,40,136)(49,93,53,89)(50,94,54,90)(51,95,55,91)(52,96,56,92)(57,101,61,97)(58,102,62,98)(59,103,63,99)(60,104,64,100)(65,109,69,105)(66,110,70,106)(67,111,71,107)(68,112,72,108)(73,154,77,158)(74,155,78,159)(75,156,79,160)(76,157,80,153)(81,151,85,147)(82,152,86,148)(83,145,87,149)(84,146,88,150)(161,259,165,263)(162,260,166,264)(163,261,167,257)(164,262,168,258)(169,267,173,271)(170,268,174,272)(171,269,175,265)(172,270,176,266)(177,275,181,279)(178,276,182,280)(179,277,183,273)(180,278,184,274)(185,227,189,231)(186,228,190,232)(187,229,191,225)(188,230,192,226)(193,235,197,239)(194,236,198,240)(195,237,199,233)(196,238,200,234)(201,243,205,247)(202,244,206,248)(203,245,207,241)(204,246,208,242)(209,251,213,255)(210,252,214,256)(211,253,215,249)(212,254,216,250)(217,283,221,287)(218,284,222,288)(219,285,223,281)(220,286,224,282), (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), (1,185,47,225)(2,188,48,228)(3,191,41,231)(4,186,42,226)(5,189,43,229)(6,192,44,232)(7,187,45,227)(8,190,46,230)(9,288,144,224)(10,283,137,219)(11,286,138,222)(12,281,139,217)(13,284,140,220)(14,287,141,223)(15,282,142,218)(16,285,143,221)(17,261,119,165)(18,264,120,168)(19,259,113,163)(20,262,114,166)(21,257,115,161)(22,260,116,164)(23,263,117,167)(24,258,118,162)(25,269,127,173)(26,272,128,176)(27,267,121,171)(28,270,122,174)(29,265,123,169)(30,268,124,172)(31,271,125,175)(32,266,126,170)(33,277,135,181)(34,280,136,184)(35,275,129,179)(36,278,130,182)(37,273,131,177)(38,276,132,180)(39,279,133,183)(40,274,134,178)(49,239,91,199)(50,234,92,194)(51,237,93,197)(52,240,94,200)(53,235,95,195)(54,238,96,198)(55,233,89,193)(56,236,90,196)(57,247,99,207)(58,242,100,202)(59,245,101,205)(60,248,102,208)(61,243,103,203)(62,246,104,206)(63,241,97,201)(64,244,98,204)(65,255,107,215)(66,250,108,210)(67,253,109,213)(68,256,110,216)(69,251,111,211)(70,254,112,214)(71,249,105,209)(72,252,106,212)(73,149,160,81)(74,152,153,84)(75,147,154,87)(76,150,155,82)(77,145,156,85)(78,148,157,88)(79,151,158,83)(80,146,159,86) );
G=PermutationGroup([[(1,87,97),(2,88,98),(3,81,99),(4,82,100),(5,83,101),(6,84,102),(7,85,103),(8,86,104),(9,26,90),(10,27,91),(11,28,92),(12,29,93),(13,30,94),(14,31,95),(15,32,96),(16,25,89),(17,105,33),(18,106,34),(19,107,35),(20,108,36),(21,109,37),(22,110,38),(23,111,39),(24,112,40),(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),(73,207,191),(74,208,192),(75,201,185),(76,202,186),(77,203,187),(78,204,188),(79,205,189),(80,206,190),(153,248,232),(154,241,225),(155,242,226),(156,243,227),(157,244,228),(158,245,229),(159,246,230),(160,247,231),(161,253,177),(162,254,178),(163,255,179),(164,256,180),(165,249,181),(166,250,182),(167,251,183),(168,252,184),(169,237,217),(170,238,218),(171,239,219),(172,240,220),(173,233,221),(174,234,222),(175,235,223),(176,236,224),(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),(9,106,88),(10,107,81),(11,108,82),(12,109,83),(13,110,84),(14,111,85),(15,112,86),(16,105,87),(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),(73,283,215),(74,284,216),(75,285,209),(76,286,210),(77,287,211),(78,288,212),(79,281,213),(80,282,214),(153,220,256),(154,221,249),(155,222,250),(156,223,251),(157,224,252),(158,217,253),(159,218,254),(160,219,255),(161,229,237),(162,230,238),(163,231,239),(164,232,240),(165,225,233),(166,226,234),(167,227,235),(168,228,236),(169,177,245),(170,178,246),(171,179,247),(172,180,248),(173,181,241),(174,182,242),(175,183,243),(176,184,244),(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)], [(1,41,5,45),(2,42,6,46),(3,43,7,47),(4,44,8,48),(9,138,13,142),(10,139,14,143),(11,140,15,144),(12,141,16,137),(17,113,21,117),(18,114,22,118),(19,115,23,119),(20,116,24,120),(25,121,29,125),(26,122,30,126),(27,123,31,127),(28,124,32,128),(33,129,37,133),(34,130,38,134),(35,131,39,135),(36,132,40,136),(49,93,53,89),(50,94,54,90),(51,95,55,91),(52,96,56,92),(57,101,61,97),(58,102,62,98),(59,103,63,99),(60,104,64,100),(65,109,69,105),(66,110,70,106),(67,111,71,107),(68,112,72,108),(73,154,77,158),(74,155,78,159),(75,156,79,160),(76,157,80,153),(81,151,85,147),(82,152,86,148),(83,145,87,149),(84,146,88,150),(161,259,165,263),(162,260,166,264),(163,261,167,257),(164,262,168,258),(169,267,173,271),(170,268,174,272),(171,269,175,265),(172,270,176,266),(177,275,181,279),(178,276,182,280),(179,277,183,273),(180,278,184,274),(185,227,189,231),(186,228,190,232),(187,229,191,225),(188,230,192,226),(193,235,197,239),(194,236,198,240),(195,237,199,233),(196,238,200,234),(201,243,205,247),(202,244,206,248),(203,245,207,241),(204,246,208,242),(209,251,213,255),(210,252,214,256),(211,253,215,249),(212,254,216,250),(217,283,221,287),(218,284,222,288),(219,285,223,281),(220,286,224,282)], [(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)], [(1,185,47,225),(2,188,48,228),(3,191,41,231),(4,186,42,226),(5,189,43,229),(6,192,44,232),(7,187,45,227),(8,190,46,230),(9,288,144,224),(10,283,137,219),(11,286,138,222),(12,281,139,217),(13,284,140,220),(14,287,141,223),(15,282,142,218),(16,285,143,221),(17,261,119,165),(18,264,120,168),(19,259,113,163),(20,262,114,166),(21,257,115,161),(22,260,116,164),(23,263,117,167),(24,258,118,162),(25,269,127,173),(26,272,128,176),(27,267,121,171),(28,270,122,174),(29,265,123,169),(30,268,124,172),(31,271,125,175),(32,266,126,170),(33,277,135,181),(34,280,136,184),(35,275,129,179),(36,278,130,182),(37,273,131,177),(38,276,132,180),(39,279,133,183),(40,274,134,178),(49,239,91,199),(50,234,92,194),(51,237,93,197),(52,240,94,200),(53,235,95,195),(54,238,96,198),(55,233,89,193),(56,236,90,196),(57,247,99,207),(58,242,100,202),(59,245,101,205),(60,248,102,208),(61,243,103,203),(62,246,104,206),(63,241,97,201),(64,244,98,204),(65,255,107,215),(66,250,108,210),(67,253,109,213),(68,256,110,216),(69,251,111,211),(70,254,112,214),(71,249,105,209),(72,252,106,212),(73,149,160,81),(74,152,153,84),(75,147,154,87),(76,150,155,82),(77,145,156,85),(78,148,157,88),(79,151,158,83),(80,146,159,86)]])
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 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | Q8 | D4 | SD16 | C3×Q8 | C3×D4 | C3×SD16 |
| kernel | C32×C4.Q8 | C32×C4⋊C4 | C6×C24 | C3×C4.Q8 | C3×C24 | C3×C4⋊C4 | C2×C24 | C24 | C3×C12 | C62 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 1 | 1 | 4 | 8 | 8 | 32 |
Matrix representation of C32×C4.Q8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 0 | 6 | 61 |
| 46 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 38 | 12 |
| 0 | 0 | 44 | 35 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,61,61],[46,0,0,0,0,1,0,0,0,0,38,44,0,0,12,35] >;
C32×C4.Q8 in GAP, Magma, Sage, TeX
C_3^2\times C_4.Q_8
% in TeX
G:=Group("C3^2xC4.Q8"); // GroupNames label
G:=SmallGroup(288,324);
// by ID
G=gap.SmallGroup(288,324);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,260,6304,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=1,d^4=c^2,e^2=c^-1*d^2,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,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations