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## G = C32×C2.C42order 288 = 25·32

### Direct product of C32 and C2.C42

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C2×C62 — C2×C6×C12 — C32×C2.C42
 Lower central C1 — C2 — C32×C2.C42
 Upper central C1 — C2×C62 — C32×C2.C42

Generators and relations for C32×C2.C42
G = < a,b,c,d,e | a3=b3=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec >

Subgroups: 300 in 228 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2 [×6], C3 [×4], C4 [×6], C22, C22 [×6], C6 [×28], C2×C4 [×6], C2×C4 [×6], C23, C32, C12 [×24], C2×C6 [×28], C22×C4 [×3], C3×C6, C3×C6 [×6], C2×C12 [×24], C2×C12 [×24], C22×C6 [×4], C2.C42, C3×C12 [×6], C62, C62 [×6], C22×C12 [×12], C6×C12 [×6], C6×C12 [×6], C2×C62, C3×C2.C42 [×4], C2×C6×C12 [×3], C32×C2.C42
Quotients: C1, C2 [×3], C3 [×4], C4 [×6], C22, C6 [×12], C2×C4 [×3], D4 [×3], Q8, C32, C12 [×24], C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×C6 [×3], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], C2.C42, C3×C12 [×6], C62, C4×C12 [×4], C3×C22⋊C4 [×12], C3×C4⋊C4 [×12], C6×C12 [×3], D4×C32 [×3], Q8×C32, C3×C2.C42 [×4], C122, C32×C22⋊C4 [×3], C32×C4⋊C4 [×3], C32×C2.C42

Smallest permutation representation of C32×C2.C42
Regular action on 288 points
Generators in S288
(1 39 23)(2 40 24)(3 37 21)(4 38 22)(5 201 185)(6 202 186)(7 203 187)(8 204 188)(9 29 25)(10 30 26)(11 31 27)(12 32 28)(13 33 17)(14 34 18)(15 35 19)(16 36 20)(41 73 57)(42 74 58)(43 75 59)(44 76 60)(45 65 61)(46 66 62)(47 67 63)(48 68 64)(49 69 53)(50 70 54)(51 71 55)(52 72 56)(77 109 93)(78 110 94)(79 111 95)(80 112 96)(81 101 97)(82 102 98)(83 103 99)(84 104 100)(85 105 89)(86 106 90)(87 107 91)(88 108 92)(113 148 129)(114 145 130)(115 146 131)(116 147 132)(117 137 133)(118 138 134)(119 139 135)(120 140 136)(121 141 125)(122 142 126)(123 143 127)(124 144 128)(149 181 165)(150 182 166)(151 183 167)(152 184 168)(153 173 169)(154 174 170)(155 175 171)(156 176 172)(157 177 161)(158 178 162)(159 179 163)(160 180 164)(189 209 205)(190 210 206)(191 211 207)(192 212 208)(193 213 197)(194 214 198)(195 215 199)(196 216 200)(217 249 233)(218 250 234)(219 251 235)(220 252 236)(221 241 237)(222 242 238)(223 243 239)(224 244 240)(225 245 229)(226 246 230)(227 247 231)(228 248 232)(253 285 269)(254 286 270)(255 287 271)(256 288 272)(257 277 273)(258 278 274)(259 279 275)(260 280 276)(261 281 265)(262 282 266)(263 283 267)(264 284 268)
(1 15 11)(2 16 12)(3 13 9)(4 14 10)(5 213 209)(6 214 210)(7 215 211)(8 216 212)(17 25 21)(18 26 22)(19 27 23)(20 28 24)(29 37 33)(30 38 34)(31 39 35)(32 40 36)(41 49 45)(42 50 46)(43 51 47)(44 52 48)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(65 73 69)(66 74 70)(67 75 71)(68 76 72)(77 85 81)(78 86 82)(79 87 83)(80 88 84)(89 97 93)(90 98 94)(91 99 95)(92 100 96)(101 109 105)(102 110 106)(103 111 107)(104 112 108)(113 121 117)(114 122 118)(115 123 119)(116 124 120)(125 133 129)(126 134 130)(127 135 131)(128 136 132)(137 148 141)(138 145 142)(139 146 143)(140 147 144)(149 157 153)(150 158 154)(151 159 155)(152 160 156)(161 169 165)(162 170 166)(163 171 167)(164 172 168)(173 181 177)(174 182 178)(175 183 179)(176 184 180)(185 193 189)(186 194 190)(187 195 191)(188 196 192)(197 205 201)(198 206 202)(199 207 203)(200 208 204)(217 225 221)(218 226 222)(219 227 223)(220 228 224)(229 237 233)(230 238 234)(231 239 235)(232 240 236)(241 249 245)(242 250 246)(243 251 247)(244 252 248)(253 261 257)(254 262 258)(255 263 259)(256 264 260)(265 273 269)(266 274 270)(267 275 271)(268 276 272)(277 285 281)(278 286 282)(279 287 283)(280 288 284)
(1 151)(2 152)(3 149)(4 150)(5 73)(6 74)(7 75)(8 76)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 161)(18 162)(19 163)(20 164)(21 165)(22 166)(23 167)(24 168)(25 169)(26 170)(27 171)(28 172)(29 173)(30 174)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 181)(38 182)(39 183)(40 184)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 191)(48 192)(49 193)(50 194)(51 195)(52 196)(53 197)(54 198)(55 199)(56 200)(57 201)(58 202)(59 203)(60 204)(61 205)(62 206)(63 207)(64 208)(65 209)(66 210)(67 211)(68 212)(69 213)(70 214)(71 215)(72 216)(77 217)(78 218)(79 219)(80 220)(81 221)(82 222)(83 223)(84 224)(85 225)(86 226)(87 227)(88 228)(89 229)(90 230)(91 231)(92 232)(93 233)(94 234)(95 235)(96 236)(97 237)(98 238)(99 239)(100 240)(101 241)(102 242)(103 243)(104 244)(105 245)(106 246)(107 247)(108 248)(109 249)(110 250)(111 251)(112 252)(113 253)(114 254)(115 255)(116 256)(117 257)(118 258)(119 259)(120 260)(121 261)(122 262)(123 263)(124 264)(125 265)(126 266)(127 267)(128 268)(129 269)(130 270)(131 271)(132 272)(133 273)(134 274)(135 275)(136 276)(137 277)(138 278)(139 279)(140 280)(141 281)(142 282)(143 283)(144 284)(145 286)(146 287)(147 288)(148 285)
(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)
(1 255 219 43)(2 116 220 188)(3 253 217 41)(4 114 218 186)(5 181 148 109)(6 38 145 250)(7 183 146 111)(8 40 147 252)(9 257 221 45)(10 118 222 190)(11 259 223 47)(12 120 224 192)(13 261 225 49)(14 122 226 194)(15 263 227 51)(16 124 228 196)(17 265 229 53)(18 126 230 198)(19 267 231 55)(20 128 232 200)(21 269 233 57)(22 130 234 202)(23 271 235 59)(24 132 236 204)(25 273 237 61)(26 134 238 206)(27 275 239 63)(28 136 240 208)(29 277 241 65)(30 138 242 210)(31 279 243 67)(32 140 244 212)(33 281 245 69)(34 142 246 214)(35 283 247 71)(36 144 248 216)(37 285 249 73)(39 287 251 75)(42 150 254 78)(44 152 256 80)(46 154 258 82)(48 156 260 84)(50 158 262 86)(52 160 264 88)(54 162 266 90)(56 164 268 92)(58 166 270 94)(60 168 272 96)(62 170 274 98)(64 172 276 100)(66 174 278 102)(68 176 280 104)(70 178 282 106)(72 180 284 108)(74 182 286 110)(76 184 288 112)(77 185 149 113)(79 187 151 115)(81 189 153 117)(83 191 155 119)(85 193 157 121)(87 195 159 123)(89 197 161 125)(91 199 163 127)(93 201 165 129)(95 203 167 131)(97 205 169 133)(99 207 171 135)(101 209 173 137)(103 211 175 139)(105 213 177 141)(107 215 179 143)

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 3A ··· 3H 4A ··· 4L 6A ··· 6BD 12A ··· 12CR order 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 D4 Q8 C3×D4 C3×Q8 kernel C32×C2.C42 C2×C6×C12 C3×C2.C42 C6×C12 C22×C12 C2×C12 C62 C62 C2×C6 C2×C6 # reps 1 3 8 12 24 96 3 1 24 8

Matrix representation of C32×C2.C42 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 3 0 0 0 0 0 3
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 2 10 0 0 0 6 11
,
 8 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 5 1 0 0 0 0 8

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,2,6,0,0,0,10,11],[8,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,1,8] >;

C32×C2.C42 in GAP, Magma, Sage, TeX

C_3^2\times C_2.C_4^2
% in TeX

G:=Group("C3^2xC2.C4^2");
// GroupNames label

G:=SmallGroup(288,313);
// by ID

G=gap.SmallGroup(288,313);
# by ID

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

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

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