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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
Upper central
C1 — C23 — C22×C4

Generators and relations for C22×C32⋊4Q8
 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 >

Subgroups: 1284 in 468 conjugacy classes, 213 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C2×C6, C22×C4, C22×C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C22×C6, C22×Q8, C3⋊Dic3, C3×C12, C62, C2×Dic6, C22×Dic3, C22×C12, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C2×C62, C22×Dic6, C2×C32⋊4Q8, C22×C3⋊Dic3, C2×C6×C12, C22×C32⋊4Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C3⋊S3, Dic6, C22×S3, C22×Q8, C2×C3⋊S3, C2×Dic6, S3×C23, C32⋊4Q8, C22×C3⋊S3, C22×Dic6, C2×C32⋊4Q8, C23×C3⋊S3, C22×C32⋊4Q8

Smallest permutation representation of C22×C32⋊4Q8
►Regular action on 288 points
Generators in S288
(1 166)(2 167)(3 168)(4 165)(5 88)(6 85)(7 86)(8 87)(9 256)(10 253)(11 254)(12 255)(13 82)(14 83)(15 84)(16 81)(17 91)(18 92)(19 89)(20 90)(21 99)(22 100)(23 97)(24 98)(25 79)(26 80)(27 77)(28 78)(29 107)(30 108)(31 105)(32 106)(33 74)(34 75)(35 76)(36 73)(37 115)(38 116)(39 113)(40 114)(41 127)(42 128)(43 125)(44 126)(45 123)(46 124)(47 121)(48 122)(49 130)(50 131)(51 132)(52 129)(53 237)(54 238)(55 239)(56 240)(57 140)(58 137)(59 138)(60 139)(61 245)(62 246)(63 247)(64 248)(65 134)(66 135)(67 136)(68 133)(69 118)(70 119)(71 120)(72 117)(93 176)(94 173)(95 174)(96 175)(101 143)(102 144)(103 141)(104 142)(109 183)(110 184)(111 181)(112 182)(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 44)(2 41)(3 42)(4 43)(5 71)(6 72)(7 69)(8 70)(9 231)(10 232)(11 229)(12 230)(13 40)(14 37)(15 38)(16 39)(17 45)(18 46)(19 47)(20 48)(21 66)(22 67)(23 68)(24 65)(25 109)(26 110)(27 111)(28 112)(29 60)(30 57)(31 58)(32 59)(33 101)(34 102)(35 103)(36 104)(49 96)(50 93)(51 94)(52 95)(53 217)(54 218)(55 219)(56 220)(61 274)(62 275)(63 276)(64 273)(73 142)(74 143)(75 144)(76 141)(77 181)(78 182)(79 183)(80 184)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(91 123)(92 124)(97 133)(98 134)(99 135)(100 136)(105 137)(106 138)(107 139)(108 140)(125 165)(126 166)(127 167)(128 168)(129 174)(130 175)(131 176)(132 173)(145 239)(146 240)(147 237)(148 238)(149 190)(150 191)(151 192)(152 189)(153 194)(154 195)(155 196)(156 193)(157 256)(158 253)(159 254)(160 255)(161 188)(162 185)(163 186)(164 187)(169 210)(170 211)(171 212)(172 209)(177 208)(178 205)(179 206)(180 207)(197 216)(198 213)(199 214)(200 215)(201 246)(202 247)(203 248)(204 245)(221 262)(222 263)(223 264)(224 261)(225 266)(226 267)(227 268)(228 265)(233 260)(234 257)(235 258)(236 259)(241 282)(242 283)(243 284)(244 281)(249 280)(250 277)(251 278)(252 279)(269 288)(270 285)(271 286)(272 287)

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

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

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

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

Matrix representation of C22×C32⋊4Q8 ►in 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] >;
 

C22×C32⋊4Q8 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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