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G = C32×C4.Q8order 288 = 25·32

Direct product of C32 and C4.Q8

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

Aliases: C32×C4.Q8, C246C12, C62.140D4, C82(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

C1C4 — C32×C4.Q8
C1C2C22C2×C4C2×C12C6×C12C32×C4⋊C4 — C32×C4.Q8
C1C2C4 — C32×C4.Q8
C1C62C6×C12 — C32×C4.Q8

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 [×2], 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, C3×C6 [×2], C24 [×8], C2×C12 [×4], C2×C12 [×8], C4.Q8, C3×C12 [×2], C3×C12 [×2], C62, C3×C4⋊C4 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C6×C12 [×2], C3×C4.Q8 [×4], C32×C4⋊C4 [×2], C6×C24, C32×C4.Q8
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4, Q8, C32, C12 [×8], C2×C6 [×4], C4⋊C4, SD16 [×2], C3×C6 [×3], C2×C12 [×4], C3×D4 [×4], C3×Q8 [×4], C4.Q8, C3×C12 [×2], C62, C3×C4⋊C4 [×4], C3×SD16 [×8], C6×C12, D4×C32, Q8×C32, C3×C4.Q8 [×4], C32×C4⋊C4, C32×SD16 [×2], C32×C4.Q8

Smallest permutation representation of C32×C4.Q8
Regular action on 288 points
Generators in S288
(1 14 103)(2 15 104)(3 16 97)(4 9 98)(5 10 99)(6 11 100)(7 12 101)(8 13 102)(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)(73 207 191)(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)(169 237 218)(170 238 219)(171 239 220)(172 240 221)(173 233 222)(174 234 223)(175 235 224)(176 236 217)(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 95 23)(2 96 24)(3 89 17)(4 90 18)(5 91 19)(6 92 20)(7 93 21)(8 94 22)(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)(73 283 215)(74 284 216)(75 285 209)(76 286 210)(77 287 211)(78 288 212)(79 281 213)(80 282 214)(153 221 256)(154 222 249)(155 223 250)(156 224 251)(157 217 252)(158 218 253)(159 219 254)(160 220 255)(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)(9 152 13 148)(10 145 14 149)(11 146 15 150)(12 147 16 151)(17 115 21 119)(18 116 22 120)(19 117 23 113)(20 118 24 114)(25 123 29 127)(26 124 30 128)(27 125 31 121)(28 126 32 122)(33 131 37 135)(34 132 38 136)(35 133 39 129)(36 134 40 130)(49 91 53 95)(50 92 54 96)(51 93 55 89)(52 94 56 90)(57 99 61 103)(58 100 62 104)(59 101 63 97)(60 102 64 98)(65 107 69 111)(66 108 70 112)(67 109 71 105)(68 110 72 106)(73 154 77 158)(74 155 78 159)(75 156 79 160)(76 157 80 153)(81 139 85 143)(82 140 86 144)(83 141 87 137)(84 142 88 138)(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)(217 282 221 286)(218 283 222 287)(219 284 223 288)(220 285 224 281)
(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 76 150 155)(10 79 151 158)(11 74 152 153)(12 77 145 156)(13 80 146 159)(14 75 147 154)(15 78 148 157)(16 73 149 160)(17 259 113 163)(18 262 114 166)(19 257 115 161)(20 260 116 164)(21 263 117 167)(22 258 118 162)(23 261 119 165)(24 264 120 168)(25 267 121 171)(26 270 122 174)(27 265 123 169)(28 268 124 172)(29 271 125 175)(30 266 126 170)(31 269 127 173)(32 272 128 176)(33 275 129 179)(34 278 130 182)(35 273 131 177)(36 276 132 180)(37 279 133 183)(38 274 134 178)(39 277 135 181)(40 280 136 184)(49 239 89 199)(50 234 90 194)(51 237 91 197)(52 240 92 200)(53 235 93 195)(54 238 94 198)(55 233 95 193)(56 236 96 196)(57 247 97 207)(58 242 98 202)(59 245 99 205)(60 248 100 208)(61 243 101 203)(62 246 102 206)(63 241 103 201)(64 244 104 204)(65 255 105 215)(66 250 106 210)(67 253 107 213)(68 256 108 216)(69 251 109 211)(70 254 110 214)(71 249 111 209)(72 252 112 212)(81 283 137 220)(82 286 138 223)(83 281 139 218)(84 284 140 221)(85 287 141 224)(86 282 142 219)(87 285 143 222)(88 288 144 217)

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

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

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

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

dim11111111222222
type+++-+
imageC1C2C2C3C4C6C6C12Q8D4SD16C3×Q8C3×D4C3×SD16
kernelC32×C4.Q8C32×C4⋊C4C6×C24C3×C4.Q8C3×C24C3×C4⋊C4C2×C24C24C3×C12C62C3×C6C12C2×C6C6
# reps12184168321148832

Matrix representation of C32×C4.Q8 in GL4(𝔽73) generated by

1000
06400
0010
0001
,
64000
06400
00640
00064
,
72000
0100
00171
00172
,
1000
0100
00061
00661
,
46000
0100
003812
004435
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

׿
×
𝔽