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## G = C32×Q8⋊C4order 288 = 25·32

### Direct product of C32 and Q8⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×Q8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×C12 — C32×C4⋊C4 — C32×Q8⋊C4
 Lower central C1 — C2 — C4 — C32×Q8⋊C4
 Upper central C1 — C62 — C6×C12 — C32×Q8⋊C4

Generators and relations for C32×Q8⋊C4
G = < a,b,c,d,e | a3=b3=c4=e4=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c-1d >

Subgroups: 180 in 126 conjugacy classes, 84 normal (24 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×3], C22, C6 [×12], C8, C2×C4, C2×C4 [×2], Q8 [×2], Q8, C32, C12 [×8], C12 [×12], C2×C6 [×4], C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C24 [×4], C2×C12 [×4], C2×C12 [×8], C3×Q8 [×8], C3×Q8 [×4], Q8⋊C4, C3×C12 [×2], C3×C12 [×3], C62, C3×C4⋊C4 [×4], C2×C24 [×4], C6×Q8 [×4], C3×C24, C6×C12, C6×C12 [×2], Q8×C32 [×2], Q8×C32, C3×Q8⋊C4 [×4], C32×C4⋊C4, C6×C24, Q8×C3×C6, C32×Q8⋊C4
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4 [×2], C32, C12 [×8], C2×C6 [×4], C22⋊C4, SD16, Q16, C3×C6 [×3], C2×C12 [×4], C3×D4 [×8], Q8⋊C4, C3×C12 [×2], C62, C3×C22⋊C4 [×4], C3×SD16 [×4], C3×Q16 [×4], C6×C12, D4×C32 [×2], C3×Q8⋊C4 [×4], C32×C22⋊C4, C32×SD16, C32×Q16, C32×Q8⋊C4

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

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

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

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

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 1 1 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 SD16 Q16 C3×D4 C3×D4 C3×SD16 C3×Q16 kernel C32×Q8⋊C4 C32×C4⋊C4 C6×C24 Q8×C3×C6 C3×Q8⋊C4 Q8×C32 C3×C4⋊C4 C2×C24 C6×Q8 C3×Q8 C3×C12 C62 C3×C6 C3×C6 C12 C2×C6 C6 C6 # reps 1 1 1 1 8 4 8 8 8 32 1 1 2 2 8 8 16 16

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

 1 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 0 1 0 0 0 0 67 6 0 0 6 6
,
 27 0 0 0 0 72 0 0 0 0 0 46 0 0 46 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,67,6,0,0,6,6],[27,0,0,0,0,72,0,0,0,0,0,46,0,0,46,0] >;

C32×Q8⋊C4 in GAP, Magma, Sage, TeX

C_3^2\times Q_8\rtimes C_4
% in TeX

G:=Group("C3^2xQ8:C4");
// GroupNames label

G:=SmallGroup(288,321);
// by ID

G=gap.SmallGroup(288,321);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,1016,6304,3161,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^4=e^4=1,d^2=c^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,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=c^-1*d>;
// generators/relations

׿
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𝔽