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G = C22×C32⋊4Q8  order 288 = 25·32

Direct product of C22 and C32⋊4Q8

direct product, metabelian, supersoluble, monomial

Aliases: C22×C32⋊4Q8, C62⋊12Q8, C62.274C23, C6⋊3(C2×Dic6), (C2×C6)⋊11Dic6, (C2×C12).390D6, C32⋊7(C22×Q8), (C3×C6).53C24, C6.54(S3×C23), C3⋊3(C22×Dic6), (C22×C12).25S3, (C22×C6).164D6, (C3×C12).157C23, (C6×C12).305C22, C12.195(C22×S3), C3⋊Dic3.45C23, (C2×C62).120C22, (C3×C6)⋊6(C2×Q8), (C2×C6×C12).10C2, C2.3(C23×C3⋊S3), C4.32(C22×C3⋊S3), C23.38(C2×C3⋊S3), (C22×C4).10(C3⋊S3), (C2×C6).283(C22×S3), C22.28(C22×C3⋊S3), (C22×C3⋊Dic3).16C2, (C2×C3⋊Dic3).181C22, (C2×C4).86(C2×C3⋊S3), SmallGroup(288,1003)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C22×C32⋊4Q8
Chief series
C1 — C3 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C22×C32⋊4Q8
Lower central
C32 — C3×C6 — C22×C32⋊4Q8

Subgroups: 1284 in 468 conjugacy classes, 213 normal (9 characteristic)
C1, C2, C2 [×6], C3 [×4], C4 [×4], C4 [×8], C22 [×7], C6 [×28], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C32, Dic3 [×32], C12 [×16], C2×C6 [×28], C22×C4, C22×C4 [×2], C2×Q8 [×12], C3×C6, C3×C6 [×6], Dic6 [×64], C2×Dic3 [×48], C2×C12 [×24], C22×C6 [×4], C22×Q8, C3⋊Dic3 [×8], C3×C12 [×4], C62 [×7], C2×Dic6 [×48], C22×Dic3 [×8], C22×C12 [×4], C32⋊4Q8 [×16], C2×C3⋊Dic3 [×12], C6×C12 [×6], C2×C62, C22×Dic6 [×4], C2×C32⋊4Q8 [×12], C22×C3⋊Dic3 [×2], C2×C6×C12, C22×C32⋊4Q8

Quotients:
C1, C2 [×15], C22 [×35], S3 [×4], Q8 [×4], C23 [×15], D6 [×28], C2×Q8 [×6], C24, C3⋊S3, Dic6 [×16], C22×S3 [×28], C22×Q8, C2×C3⋊S3 [×7], C2×Dic6 [×24], S3×C23 [×4], C32⋊4Q8 [×4], C22×C3⋊S3 [×7], C22×Dic6 [×4], C2×C32⋊4Q8 [×6], C23×C3⋊S3, C22×C32⋊4Q8

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=c-1, de=ed, fdf-1=d-1, fef-1=e-1 >

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

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

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

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

(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)

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
000010
000001
,
300000
090000
001000
000100
000030
000009
,
100000
010000
009000
000300
000010
000001
,
100000
010000
0012000
0001200
000080
000005
,
010000
100000
000100
001000
000005
000050

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

84 conjugacy classes

class 1 2A···2G3A3B3C3D4A4B4C4D4E···4L6A···6AB12A···12AF
order12···2333344444···46···612···12
size11···12222222218···182···22···2

84 irreducible representations

dim111122222
type+++++-++-
imageC1C2C2C2S3Q8D6D6Dic6
kernelC22×C32⋊4Q8C2×C32⋊4Q8C22×C3⋊Dic3C2×C6×C12C22×C12C62C2×C12C22×C6C2×C6
# reps112214424432

In GAP, Magma, Sage, TeX 

C_2^2\times C_3^2\rtimes_4Q_8
 
% in TeX
 
G:=Group("C2^2xC3^2:4Q8");
 
// GroupNames label
 
G:=SmallGroup(288,1003);
 
// by ID
 
G=gap.SmallGroup(288,1003);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,675,80,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=c^-1,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
 
// generators/relations
 

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×
⋊
ℤ
𝔽
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