direct product, non-abelian, soluble
Aliases: C22×Q8⋊C9, Q8⋊(C2×C18), (C2×Q8)⋊2C18, (C6×Q8).5C6, (C22×Q8)⋊1C9, (C22×C6).14A4, C6.14(C22×A4), C23.6(C3.A4), C6.3(C2×SL2(𝔽3)), (C2×C6).4SL2(𝔽3), C3.(C22×SL2(𝔽3)), (Q8×C2×C6).1C3, (C2×C6).24(C2×A4), (C3×Q8).11(C2×C6), C22.8(C2×C3.A4), C2.3(C22×C3.A4), SmallGroup(288,345)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C22×Q8⋊C9 |
Generators and relations for C22×Q8⋊C9
G = < a,b,c,d,e | a2=b2=c4=e9=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Subgroups: 225 in 101 conjugacy classes, 47 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, Q8, Q8, C23, C9, C12, C2×C6, C22×C4, C2×Q8, C2×Q8, C18, C2×C12, C3×Q8, C3×Q8, C22×C6, C22×Q8, C2×C18, C22×C12, C6×Q8, C6×Q8, Q8⋊C9, C22×C18, Q8×C2×C6, C2×Q8⋊C9, C22×Q8⋊C9
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, SL2(𝔽3), C2×A4, C3.A4, C2×C18, C2×SL2(𝔽3), C22×A4, Q8⋊C9, C2×C3.A4, C22×SL2(𝔽3), C2×Q8⋊C9, C22×C3.A4, C22×Q8⋊C9
►Regular action on 288 points
Generators in S288
(1 253)(2 254)(3 255)(4 256)(5 257)(6 258)(7 259)(8 260)(9 261)(10 156)(11 157)(12 158)(13 159)(14 160)(15 161)(16 162)(17 154)(18 155)(19 166)(20 167)(21 168)(22 169)(23 170)(24 171)(25 163)(26 164)(27 165)(28 201)(29 202)(30 203)(31 204)(32 205)(33 206)(34 207)(35 199)(36 200)(37 153)(38 145)(39 146)(40 147)(41 148)(42 149)(43 150)(44 151)(45 152)(46 192)(47 193)(48 194)(49 195)(50 196)(51 197)(52 198)(53 190)(54 191)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 136)(62 137)(63 138)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 237)(74 238)(75 239)(76 240)(77 241)(78 242)(79 243)(80 235)(81 236)(82 272)(83 273)(84 274)(85 275)(86 276)(87 277)(88 278)(89 279)(90 271)(91 224)(92 225)(93 217)(94 218)(95 219)(96 220)(97 221)(98 222)(99 223)(100 263)(101 264)(102 265)(103 266)(104 267)(105 268)(106 269)(107 270)(108 262)(109 210)(110 211)(111 212)(112 213)(113 214)(114 215)(115 216)(116 208)(117 209)(118 252)(119 244)(120 245)(121 246)(122 247)(123 248)(124 249)(125 250)(126 251)(127 234)(128 226)(129 227)(130 228)(131 229)(132 230)(133 231)(134 232)(135 233)(181 282)(182 283)(183 284)(184 285)(185 286)(186 287)(187 288)(188 280)(189 281)
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 136)(9 137)(10 274)(11 275)(12 276)(13 277)(14 278)(15 279)(16 271)(17 272)(18 273)(19 218)(20 219)(21 220)(22 221)(23 222)(24 223)(25 224)(26 225)(27 217)(28 227)(29 228)(30 229)(31 230)(32 231)(33 232)(34 233)(35 234)(36 226)(37 236)(38 237)(39 238)(40 239)(41 240)(42 241)(43 242)(44 243)(45 235)(46 245)(47 246)(48 247)(49 248)(50 249)(51 250)(52 251)(53 252)(54 244)(55 254)(56 255)(57 256)(58 257)(59 258)(60 259)(61 260)(62 261)(63 253)(64 263)(65 264)(66 265)(67 266)(68 267)(69 268)(70 269)(71 270)(72 262)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 161)(90 162)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 169)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(105 177)(106 178)(107 179)(108 180)(109 181)(110 182)(111 183)(112 184)(113 185)(114 186)(115 187)(116 188)(117 189)(118 190)(119 191)(120 192)(121 193)(122 194)(123 195)(124 196)(125 197)(126 198)(127 199)(128 200)(129 201)(130 202)(131 203)(132 204)(133 205)(134 206)(135 207)(208 280)(209 281)(210 282)(211 283)(212 284)(213 285)(214 286)(215 287)(216 288)
(1 129 109 83)(2 79 110 99)(3 102 111 121)(4 132 112 86)(5 73 113 93)(6 105 114 124)(7 135 115 89)(8 76 116 96)(9 108 117 118)(10 46 29 65)(11 45 30 25)(12 57 31 285)(13 49 32 68)(14 39 33 19)(15 60 34 288)(16 52 35 71)(17 42 36 22)(18 63 28 282)(20 70 40 51)(21 61 41 280)(23 64 43 54)(24 55 44 283)(26 67 37 48)(27 58 38 286)(47 56 66 284)(50 59 69 287)(53 62 72 281)(74 134 94 88)(75 125 95 106)(77 128 97 82)(78 119 98 100)(80 131 91 85)(81 122 92 103)(84 120 130 101)(87 123 133 104)(90 126 127 107)(136 148 188 168)(137 180 189 190)(138 201 181 155)(139 151 182 171)(140 174 183 193)(141 204 184 158)(142 145 185 165)(143 177 186 196)(144 207 187 161)(146 206 166 160)(147 197 167 178)(149 200 169 154)(150 191 170 172)(152 203 163 157)(153 194 164 175)(156 192 202 173)(159 195 205 176)(162 198 199 179)(208 220 260 240)(209 252 261 262)(210 273 253 227)(211 223 254 243)(212 246 255 265)(213 276 256 230)(214 217 257 237)(215 249 258 268)(216 279 259 233)(218 278 238 232)(219 269 239 250)(221 272 241 226)(222 263 242 244)(224 275 235 229)(225 266 236 247)(228 264 274 245)(231 267 277 248)(234 270 271 251)
(1 78 109 98)(2 101 110 120)(3 131 111 85)(4 81 112 92)(5 104 113 123)(6 134 114 88)(7 75 115 95)(8 107 116 126)(9 128 117 82)(10 44 29 24)(11 56 30 284)(12 48 31 67)(13 38 32 27)(14 59 33 287)(15 51 34 70)(16 41 35 21)(17 62 36 281)(18 54 28 64)(19 69 39 50)(20 60 40 288)(22 72 42 53)(23 63 43 282)(25 66 45 47)(26 57 37 285)(46 55 65 283)(49 58 68 286)(52 61 71 280)(73 133 93 87)(74 124 94 105)(76 127 96 90)(77 118 97 108)(79 130 99 84)(80 121 91 102)(83 119 129 100)(86 122 132 103)(89 125 135 106)(136 179 188 198)(137 200 189 154)(138 150 181 170)(139 173 182 192)(140 203 183 157)(141 153 184 164)(142 176 185 195)(143 206 186 160)(144 147 187 167)(145 205 165 159)(146 196 166 177)(148 199 168 162)(149 190 169 180)(151 202 171 156)(152 193 163 174)(155 191 201 172)(158 194 204 175)(161 197 207 178)(208 251 260 270)(209 272 261 226)(210 222 253 242)(211 245 254 264)(212 275 255 229)(213 225 256 236)(214 248 257 267)(215 278 258 232)(216 219 259 239)(217 277 237 231)(218 268 238 249)(220 271 240 234)(221 262 241 252)(223 274 243 228)(224 265 235 246)(227 263 273 244)(230 266 276 247)(233 269 279 250)
(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)
G:=sub<Sym(288)| (1,253)(2,254)(3,255)(4,256)(5,257)(6,258)(7,259)(8,260)(9,261)(10,156)(11,157)(12,158)(13,159)(14,160)(15,161)(16,162)(17,154)(18,155)(19,166)(20,167)(21,168)(22,169)(23,170)(24,171)(25,163)(26,164)(27,165)(28,201)(29,202)(30,203)(31,204)(32,205)(33,206)(34,207)(35,199)(36,200)(37,153)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,192)(47,193)(48,194)(49,195)(50,196)(51,197)(52,198)(53,190)(54,191)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,136)(62,137)(63,138)(64,172)(65,173)(66,174)(67,175)(68,176)(69,177)(70,178)(71,179)(72,180)(73,237)(74,238)(75,239)(76,240)(77,241)(78,242)(79,243)(80,235)(81,236)(82,272)(83,273)(84,274)(85,275)(86,276)(87,277)(88,278)(89,279)(90,271)(91,224)(92,225)(93,217)(94,218)(95,219)(96,220)(97,221)(98,222)(99,223)(100,263)(101,264)(102,265)(103,266)(104,267)(105,268)(106,269)(107,270)(108,262)(109,210)(110,211)(111,212)(112,213)(113,214)(114,215)(115,216)(116,208)(117,209)(118,252)(119,244)(120,245)(121,246)(122,247)(123,248)(124,249)(125,250)(126,251)(127,234)(128,226)(129,227)(130,228)(131,229)(132,230)(133,231)(134,232)(135,233)(181,282)(182,283)(183,284)(184,285)(185,286)(186,287)(187,288)(188,280)(189,281), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,136)(9,137)(10,274)(11,275)(12,276)(13,277)(14,278)(15,279)(16,271)(17,272)(18,273)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,224)(26,225)(27,217)(28,227)(29,228)(30,229)(31,230)(32,231)(33,232)(34,233)(35,234)(36,226)(37,236)(38,237)(39,238)(40,239)(41,240)(42,241)(43,242)(44,243)(45,235)(46,245)(47,246)(48,247)(49,248)(50,249)(51,250)(52,251)(53,252)(54,244)(55,254)(56,255)(57,256)(58,257)(59,258)(60,259)(61,260)(62,261)(63,253)(64,263)(65,264)(66,265)(67,266)(68,267)(69,268)(70,269)(71,270)(72,262)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,197)(126,198)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(208,280)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288), (1,129,109,83)(2,79,110,99)(3,102,111,121)(4,132,112,86)(5,73,113,93)(6,105,114,124)(7,135,115,89)(8,76,116,96)(9,108,117,118)(10,46,29,65)(11,45,30,25)(12,57,31,285)(13,49,32,68)(14,39,33,19)(15,60,34,288)(16,52,35,71)(17,42,36,22)(18,63,28,282)(20,70,40,51)(21,61,41,280)(23,64,43,54)(24,55,44,283)(26,67,37,48)(27,58,38,286)(47,56,66,284)(50,59,69,287)(53,62,72,281)(74,134,94,88)(75,125,95,106)(77,128,97,82)(78,119,98,100)(80,131,91,85)(81,122,92,103)(84,120,130,101)(87,123,133,104)(90,126,127,107)(136,148,188,168)(137,180,189,190)(138,201,181,155)(139,151,182,171)(140,174,183,193)(141,204,184,158)(142,145,185,165)(143,177,186,196)(144,207,187,161)(146,206,166,160)(147,197,167,178)(149,200,169,154)(150,191,170,172)(152,203,163,157)(153,194,164,175)(156,192,202,173)(159,195,205,176)(162,198,199,179)(208,220,260,240)(209,252,261,262)(210,273,253,227)(211,223,254,243)(212,246,255,265)(213,276,256,230)(214,217,257,237)(215,249,258,268)(216,279,259,233)(218,278,238,232)(219,269,239,250)(221,272,241,226)(222,263,242,244)(224,275,235,229)(225,266,236,247)(228,264,274,245)(231,267,277,248)(234,270,271,251), (1,78,109,98)(2,101,110,120)(3,131,111,85)(4,81,112,92)(5,104,113,123)(6,134,114,88)(7,75,115,95)(8,107,116,126)(9,128,117,82)(10,44,29,24)(11,56,30,284)(12,48,31,67)(13,38,32,27)(14,59,33,287)(15,51,34,70)(16,41,35,21)(17,62,36,281)(18,54,28,64)(19,69,39,50)(20,60,40,288)(22,72,42,53)(23,63,43,282)(25,66,45,47)(26,57,37,285)(46,55,65,283)(49,58,68,286)(52,61,71,280)(73,133,93,87)(74,124,94,105)(76,127,96,90)(77,118,97,108)(79,130,99,84)(80,121,91,102)(83,119,129,100)(86,122,132,103)(89,125,135,106)(136,179,188,198)(137,200,189,154)(138,150,181,170)(139,173,182,192)(140,203,183,157)(141,153,184,164)(142,176,185,195)(143,206,186,160)(144,147,187,167)(145,205,165,159)(146,196,166,177)(148,199,168,162)(149,190,169,180)(151,202,171,156)(152,193,163,174)(155,191,201,172)(158,194,204,175)(161,197,207,178)(208,251,260,270)(209,272,261,226)(210,222,253,242)(211,245,254,264)(212,275,255,229)(213,225,256,236)(214,248,257,267)(215,278,258,232)(216,219,259,239)(217,277,237,231)(218,268,238,249)(220,271,240,234)(221,262,241,252)(223,274,243,228)(224,265,235,246)(227,263,273,244)(230,266,276,247)(233,269,279,250), (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)>;
G:=Group( (1,253)(2,254)(3,255)(4,256)(5,257)(6,258)(7,259)(8,260)(9,261)(10,156)(11,157)(12,158)(13,159)(14,160)(15,161)(16,162)(17,154)(18,155)(19,166)(20,167)(21,168)(22,169)(23,170)(24,171)(25,163)(26,164)(27,165)(28,201)(29,202)(30,203)(31,204)(32,205)(33,206)(34,207)(35,199)(36,200)(37,153)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,192)(47,193)(48,194)(49,195)(50,196)(51,197)(52,198)(53,190)(54,191)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,136)(62,137)(63,138)(64,172)(65,173)(66,174)(67,175)(68,176)(69,177)(70,178)(71,179)(72,180)(73,237)(74,238)(75,239)(76,240)(77,241)(78,242)(79,243)(80,235)(81,236)(82,272)(83,273)(84,274)(85,275)(86,276)(87,277)(88,278)(89,279)(90,271)(91,224)(92,225)(93,217)(94,218)(95,219)(96,220)(97,221)(98,222)(99,223)(100,263)(101,264)(102,265)(103,266)(104,267)(105,268)(106,269)(107,270)(108,262)(109,210)(110,211)(111,212)(112,213)(113,214)(114,215)(115,216)(116,208)(117,209)(118,252)(119,244)(120,245)(121,246)(122,247)(123,248)(124,249)(125,250)(126,251)(127,234)(128,226)(129,227)(130,228)(131,229)(132,230)(133,231)(134,232)(135,233)(181,282)(182,283)(183,284)(184,285)(185,286)(186,287)(187,288)(188,280)(189,281), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,136)(9,137)(10,274)(11,275)(12,276)(13,277)(14,278)(15,279)(16,271)(17,272)(18,273)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,224)(26,225)(27,217)(28,227)(29,228)(30,229)(31,230)(32,231)(33,232)(34,233)(35,234)(36,226)(37,236)(38,237)(39,238)(40,239)(41,240)(42,241)(43,242)(44,243)(45,235)(46,245)(47,246)(48,247)(49,248)(50,249)(51,250)(52,251)(53,252)(54,244)(55,254)(56,255)(57,256)(58,257)(59,258)(60,259)(61,260)(62,261)(63,253)(64,263)(65,264)(66,265)(67,266)(68,267)(69,268)(70,269)(71,270)(72,262)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,197)(126,198)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(208,280)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288), (1,129,109,83)(2,79,110,99)(3,102,111,121)(4,132,112,86)(5,73,113,93)(6,105,114,124)(7,135,115,89)(8,76,116,96)(9,108,117,118)(10,46,29,65)(11,45,30,25)(12,57,31,285)(13,49,32,68)(14,39,33,19)(15,60,34,288)(16,52,35,71)(17,42,36,22)(18,63,28,282)(20,70,40,51)(21,61,41,280)(23,64,43,54)(24,55,44,283)(26,67,37,48)(27,58,38,286)(47,56,66,284)(50,59,69,287)(53,62,72,281)(74,134,94,88)(75,125,95,106)(77,128,97,82)(78,119,98,100)(80,131,91,85)(81,122,92,103)(84,120,130,101)(87,123,133,104)(90,126,127,107)(136,148,188,168)(137,180,189,190)(138,201,181,155)(139,151,182,171)(140,174,183,193)(141,204,184,158)(142,145,185,165)(143,177,186,196)(144,207,187,161)(146,206,166,160)(147,197,167,178)(149,200,169,154)(150,191,170,172)(152,203,163,157)(153,194,164,175)(156,192,202,173)(159,195,205,176)(162,198,199,179)(208,220,260,240)(209,252,261,262)(210,273,253,227)(211,223,254,243)(212,246,255,265)(213,276,256,230)(214,217,257,237)(215,249,258,268)(216,279,259,233)(218,278,238,232)(219,269,239,250)(221,272,241,226)(222,263,242,244)(224,275,235,229)(225,266,236,247)(228,264,274,245)(231,267,277,248)(234,270,271,251), (1,78,109,98)(2,101,110,120)(3,131,111,85)(4,81,112,92)(5,104,113,123)(6,134,114,88)(7,75,115,95)(8,107,116,126)(9,128,117,82)(10,44,29,24)(11,56,30,284)(12,48,31,67)(13,38,32,27)(14,59,33,287)(15,51,34,70)(16,41,35,21)(17,62,36,281)(18,54,28,64)(19,69,39,50)(20,60,40,288)(22,72,42,53)(23,63,43,282)(25,66,45,47)(26,57,37,285)(46,55,65,283)(49,58,68,286)(52,61,71,280)(73,133,93,87)(74,124,94,105)(76,127,96,90)(77,118,97,108)(79,130,99,84)(80,121,91,102)(83,119,129,100)(86,122,132,103)(89,125,135,106)(136,179,188,198)(137,200,189,154)(138,150,181,170)(139,173,182,192)(140,203,183,157)(141,153,184,164)(142,176,185,195)(143,206,186,160)(144,147,187,167)(145,205,165,159)(146,196,166,177)(148,199,168,162)(149,190,169,180)(151,202,171,156)(152,193,163,174)(155,191,201,172)(158,194,204,175)(161,197,207,178)(208,251,260,270)(209,272,261,226)(210,222,253,242)(211,245,254,264)(212,275,255,229)(213,225,256,236)(214,248,257,267)(215,278,258,232)(216,219,259,239)(217,277,237,231)(218,268,238,249)(220,271,240,234)(221,262,241,252)(223,274,243,228)(224,265,235,246)(227,263,273,244)(230,266,276,247)(233,269,279,250), (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) );
G=PermutationGroup([[(1,253),(2,254),(3,255),(4,256),(5,257),(6,258),(7,259),(8,260),(9,261),(10,156),(11,157),(12,158),(13,159),(14,160),(15,161),(16,162),(17,154),(18,155),(19,166),(20,167),(21,168),(22,169),(23,170),(24,171),(25,163),(26,164),(27,165),(28,201),(29,202),(30,203),(31,204),(32,205),(33,206),(34,207),(35,199),(36,200),(37,153),(38,145),(39,146),(40,147),(41,148),(42,149),(43,150),(44,151),(45,152),(46,192),(47,193),(48,194),(49,195),(50,196),(51,197),(52,198),(53,190),(54,191),(55,139),(56,140),(57,141),(58,142),(59,143),(60,144),(61,136),(62,137),(63,138),(64,172),(65,173),(66,174),(67,175),(68,176),(69,177),(70,178),(71,179),(72,180),(73,237),(74,238),(75,239),(76,240),(77,241),(78,242),(79,243),(80,235),(81,236),(82,272),(83,273),(84,274),(85,275),(86,276),(87,277),(88,278),(89,279),(90,271),(91,224),(92,225),(93,217),(94,218),(95,219),(96,220),(97,221),(98,222),(99,223),(100,263),(101,264),(102,265),(103,266),(104,267),(105,268),(106,269),(107,270),(108,262),(109,210),(110,211),(111,212),(112,213),(113,214),(114,215),(115,216),(116,208),(117,209),(118,252),(119,244),(120,245),(121,246),(122,247),(123,248),(124,249),(125,250),(126,251),(127,234),(128,226),(129,227),(130,228),(131,229),(132,230),(133,231),(134,232),(135,233),(181,282),(182,283),(183,284),(184,285),(185,286),(186,287),(187,288),(188,280),(189,281)], [(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,136),(9,137),(10,274),(11,275),(12,276),(13,277),(14,278),(15,279),(16,271),(17,272),(18,273),(19,218),(20,219),(21,220),(22,221),(23,222),(24,223),(25,224),(26,225),(27,217),(28,227),(29,228),(30,229),(31,230),(32,231),(33,232),(34,233),(35,234),(36,226),(37,236),(38,237),(39,238),(40,239),(41,240),(42,241),(43,242),(44,243),(45,235),(46,245),(47,246),(48,247),(49,248),(50,249),(51,250),(52,251),(53,252),(54,244),(55,254),(56,255),(57,256),(58,257),(59,258),(60,259),(61,260),(62,261),(63,253),(64,263),(65,264),(66,265),(67,266),(68,267),(69,268),(70,269),(71,270),(72,262),(73,145),(74,146),(75,147),(76,148),(77,149),(78,150),(79,151),(80,152),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,161),(90,162),(91,163),(92,164),(93,165),(94,166),(95,167),(96,168),(97,169),(98,170),(99,171),(100,172),(101,173),(102,174),(103,175),(104,176),(105,177),(106,178),(107,179),(108,180),(109,181),(110,182),(111,183),(112,184),(113,185),(114,186),(115,187),(116,188),(117,189),(118,190),(119,191),(120,192),(121,193),(122,194),(123,195),(124,196),(125,197),(126,198),(127,199),(128,200),(129,201),(130,202),(131,203),(132,204),(133,205),(134,206),(135,207),(208,280),(209,281),(210,282),(211,283),(212,284),(213,285),(214,286),(215,287),(216,288)], [(1,129,109,83),(2,79,110,99),(3,102,111,121),(4,132,112,86),(5,73,113,93),(6,105,114,124),(7,135,115,89),(8,76,116,96),(9,108,117,118),(10,46,29,65),(11,45,30,25),(12,57,31,285),(13,49,32,68),(14,39,33,19),(15,60,34,288),(16,52,35,71),(17,42,36,22),(18,63,28,282),(20,70,40,51),(21,61,41,280),(23,64,43,54),(24,55,44,283),(26,67,37,48),(27,58,38,286),(47,56,66,284),(50,59,69,287),(53,62,72,281),(74,134,94,88),(75,125,95,106),(77,128,97,82),(78,119,98,100),(80,131,91,85),(81,122,92,103),(84,120,130,101),(87,123,133,104),(90,126,127,107),(136,148,188,168),(137,180,189,190),(138,201,181,155),(139,151,182,171),(140,174,183,193),(141,204,184,158),(142,145,185,165),(143,177,186,196),(144,207,187,161),(146,206,166,160),(147,197,167,178),(149,200,169,154),(150,191,170,172),(152,203,163,157),(153,194,164,175),(156,192,202,173),(159,195,205,176),(162,198,199,179),(208,220,260,240),(209,252,261,262),(210,273,253,227),(211,223,254,243),(212,246,255,265),(213,276,256,230),(214,217,257,237),(215,249,258,268),(216,279,259,233),(218,278,238,232),(219,269,239,250),(221,272,241,226),(222,263,242,244),(224,275,235,229),(225,266,236,247),(228,264,274,245),(231,267,277,248),(234,270,271,251)], [(1,78,109,98),(2,101,110,120),(3,131,111,85),(4,81,112,92),(5,104,113,123),(6,134,114,88),(7,75,115,95),(8,107,116,126),(9,128,117,82),(10,44,29,24),(11,56,30,284),(12,48,31,67),(13,38,32,27),(14,59,33,287),(15,51,34,70),(16,41,35,21),(17,62,36,281),(18,54,28,64),(19,69,39,50),(20,60,40,288),(22,72,42,53),(23,63,43,282),(25,66,45,47),(26,57,37,285),(46,55,65,283),(49,58,68,286),(52,61,71,280),(73,133,93,87),(74,124,94,105),(76,127,96,90),(77,118,97,108),(79,130,99,84),(80,121,91,102),(83,119,129,100),(86,122,132,103),(89,125,135,106),(136,179,188,198),(137,200,189,154),(138,150,181,170),(139,173,182,192),(140,203,183,157),(141,153,184,164),(142,176,185,195),(143,206,186,160),(144,147,187,167),(145,205,165,159),(146,196,166,177),(148,199,168,162),(149,190,169,180),(151,202,171,156),(152,193,163,174),(155,191,201,172),(158,194,204,175),(161,197,207,178),(208,251,260,270),(209,272,261,226),(210,222,253,242),(211,245,254,264),(212,275,255,229),(213,225,256,236),(214,248,257,267),(215,278,258,232),(216,219,259,239),(217,277,237,231),(218,268,238,249),(220,271,240,234),(221,262,241,252),(223,274,243,228),(224,265,235,246),(227,263,273,244),(230,266,276,247),(233,269,279,250)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6N | 9A | ··· | 9F | 12A | ··· | 12H | 18A | ··· | 18AP |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | - | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | SL2(𝔽3) | SL2(𝔽3) | Q8⋊C9 | A4 | C2×A4 | C3.A4 | C2×C3.A4 |
| kernel | C22×Q8⋊C9 | C2×Q8⋊C9 | Q8×C2×C6 | C6×Q8 | C22×Q8 | C2×Q8 | C2×C6 | C2×C6 | C22 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 4 | 8 | 24 | 1 | 3 | 2 | 6 |
Matrix representation of C22×Q8⋊C9 ►in GL5(𝔽37)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 31 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 0 | 2 | 35 |
| 0 | 0 | 0 | 25 | 25 |
G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,36,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,31],[12,0,0,0,0,0,10,0,0,0,0,0,26,0,0,0,0,0,2,25,0,0,0,35,25] >;
C22×Q8⋊C9 in GAP, Magma, Sage, TeX
C_2^2\times Q_8\rtimes C_9
% in TeX
G:=Group("C2^2xQ8:C9"); // GroupNames label
G:=SmallGroup(288,345);
// by ID
G=gap.SmallGroup(288,345);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,79,648,172,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^9=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=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations