direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C2.C42, C18.6C42, (C2×C36)⋊4C4, (C2×C4)⋊2C36, C2.1(C4×C36), C6.6(C4×C12), (C2×C18).7Q8, (C2×C12).5C12, (C2×C18).45D4, C18.10(C4⋊C4), C22.7(D4×C9), C22.2(Q8×C9), (C22×C36).2C2, (C22×C12).9C6, C22.7(C2×C36), (C22×C4).3C18, C23.15(C2×C18), C18.19(C22⋊C4), (C22×C18).48C22, C2.1(C9×C4⋊C4), C6.10(C3×C4⋊C4), (C2×C6).54(C3×D4), C2.1(C9×C22⋊C4), (C2×C6).10(C3×Q8), (C2×C18).36(C2×C4), (C2×C6).45(C2×C12), C6.19(C3×C22⋊C4), C3.(C3×C2.C42), (C22×C6).73(C2×C6), (C3×C2.C42).2C3, SmallGroup(288,45)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C2.C42
G = < a,b,c,d | a9=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 150 in 114 conjugacy classes, 78 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C9, C12, C2×C6, C2×C6, C22×C4, C18, C18, C2×C12, C2×C12, C22×C6, C2.C42, C36, C2×C18, C2×C18, C22×C12, C2×C36, C2×C36, C22×C18, C3×C2.C42, C22×C36, C9×C2.C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C9, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C18, C2×C12, C3×D4, C3×Q8, C2.C42, C36, C2×C18, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C36, D4×C9, Q8×C9, C3×C2.C42, C4×C36, C9×C22⋊C4, C9×C4⋊C4, C9×C2.C42
►Regular action on 288 points
Generators in S288
(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 77)(2 78)(3 79)(4 80)(5 81)(6 73)(7 74)(8 75)(9 76)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 279)(20 271)(21 272)(22 273)(23 274)(24 275)(25 276)(26 277)(27 278)(28 61)(29 62)(30 63)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 100)(46 243)(47 235)(48 236)(49 237)(50 238)(51 239)(52 240)(53 241)(54 242)(82 125)(83 126)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 179)(92 180)(93 172)(94 173)(95 174)(96 175)(97 176)(98 177)(99 178)(109 144)(110 136)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(127 152)(128 153)(129 145)(130 146)(131 147)(132 148)(133 149)(134 150)(135 151)(154 197)(155 198)(156 190)(157 191)(158 192)(159 193)(160 194)(161 195)(162 196)(163 251)(164 252)(165 244)(166 245)(167 246)(168 247)(169 248)(170 249)(171 250)(181 216)(182 208)(183 209)(184 210)(185 211)(186 212)(187 213)(188 214)(189 215)(199 224)(200 225)(201 217)(202 218)(203 219)(204 220)(205 221)(206 222)(207 223)(226 269)(227 270)(228 262)(229 263)(230 264)(231 265)(232 266)(233 267)(234 268)(253 288)(254 280)(255 281)(256 282)(257 283)(258 284)(259 285)(260 286)(261 287)
(1 156 28 205)(2 157 29 206)(3 158 30 207)(4 159 31 199)(5 160 32 200)(6 161 33 201)(7 162 34 202)(8 154 35 203)(9 155 36 204)(10 247 103 182)(11 248 104 183)(12 249 105 184)(13 250 106 185)(14 251 107 186)(15 252 108 187)(16 244 100 188)(17 245 101 189)(18 246 102 181)(19 151 264 120)(20 152 265 121)(21 153 266 122)(22 145 267 123)(23 146 268 124)(24 147 269 125)(25 148 270 126)(26 149 262 118)(27 150 263 119)(37 215 71 166)(38 216 72 167)(39 208 64 168)(40 209 65 169)(41 210 66 170)(42 211 67 171)(43 212 68 163)(44 213 69 164)(45 214 70 165)(46 178 257 113)(47 179 258 114)(48 180 259 115)(49 172 260 116)(50 173 261 117)(51 174 253 109)(52 175 254 110)(53 176 255 111)(54 177 256 112)(55 224 80 193)(56 225 81 194)(57 217 73 195)(58 218 74 196)(59 219 75 197)(60 220 76 198)(61 221 77 190)(62 222 78 191)(63 223 79 192)(82 275 131 226)(83 276 132 227)(84 277 133 228)(85 278 134 229)(86 279 135 230)(87 271 127 231)(88 272 128 232)(89 273 129 233)(90 274 130 234)(91 284 140 235)(92 285 141 236)(93 286 142 237)(94 287 143 238)(95 288 144 239)(96 280 136 240)(97 281 137 241)(98 282 138 242)(99 283 139 243)
(1 143 71 133)(2 144 72 134)(3 136 64 135)(4 137 65 127)(5 138 66 128)(6 139 67 129)(7 140 68 130)(8 141 69 131)(9 142 70 132)(10 151 79 110)(11 152 80 111)(12 153 81 112)(13 145 73 113)(14 146 74 114)(15 147 75 115)(16 148 76 116)(17 149 77 117)(18 150 78 109)(19 207 254 208)(20 199 255 209)(21 200 256 210)(22 201 257 211)(23 202 258 212)(24 203 259 213)(25 204 260 214)(26 205 261 215)(27 206 253 216)(28 94 37 84)(29 95 38 85)(30 96 39 86)(31 97 40 87)(32 98 41 88)(33 99 42 89)(34 91 43 90)(35 92 44 82)(36 93 45 83)(46 171 267 161)(47 163 268 162)(48 164 269 154)(49 165 270 155)(50 166 262 156)(51 167 263 157)(52 168 264 158)(53 169 265 159)(54 170 266 160)(55 176 104 121)(56 177 105 122)(57 178 106 123)(58 179 107 124)(59 180 108 125)(60 172 100 126)(61 173 101 118)(62 174 102 119)(63 175 103 120)(181 278 222 288)(182 279 223 280)(183 271 224 281)(184 272 225 282)(185 273 217 283)(186 274 218 284)(187 275 219 285)(188 276 220 286)(189 277 221 287)(190 238 245 228)(191 239 246 229)(192 240 247 230)(193 241 248 231)(194 242 249 232)(195 243 250 233)(196 235 251 234)(197 236 252 226)(198 237 244 227)
G:=sub<Sym(288)| (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), (1,77)(2,78)(3,79)(4,80)(5,81)(6,73)(7,74)(8,75)(9,76)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,279)(20,271)(21,272)(22,273)(23,274)(24,275)(25,276)(26,277)(27,278)(28,61)(29,62)(30,63)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,100)(46,243)(47,235)(48,236)(49,237)(50,238)(51,239)(52,240)(53,241)(54,242)(82,125)(83,126)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,179)(92,180)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(109,144)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(127,152)(128,153)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(154,197)(155,198)(156,190)(157,191)(158,192)(159,193)(160,194)(161,195)(162,196)(163,251)(164,252)(165,244)(166,245)(167,246)(168,247)(169,248)(170,249)(171,250)(181,216)(182,208)(183,209)(184,210)(185,211)(186,212)(187,213)(188,214)(189,215)(199,224)(200,225)(201,217)(202,218)(203,219)(204,220)(205,221)(206,222)(207,223)(226,269)(227,270)(228,262)(229,263)(230,264)(231,265)(232,266)(233,267)(234,268)(253,288)(254,280)(255,281)(256,282)(257,283)(258,284)(259,285)(260,286)(261,287), (1,156,28,205)(2,157,29,206)(3,158,30,207)(4,159,31,199)(5,160,32,200)(6,161,33,201)(7,162,34,202)(8,154,35,203)(9,155,36,204)(10,247,103,182)(11,248,104,183)(12,249,105,184)(13,250,106,185)(14,251,107,186)(15,252,108,187)(16,244,100,188)(17,245,101,189)(18,246,102,181)(19,151,264,120)(20,152,265,121)(21,153,266,122)(22,145,267,123)(23,146,268,124)(24,147,269,125)(25,148,270,126)(26,149,262,118)(27,150,263,119)(37,215,71,166)(38,216,72,167)(39,208,64,168)(40,209,65,169)(41,210,66,170)(42,211,67,171)(43,212,68,163)(44,213,69,164)(45,214,70,165)(46,178,257,113)(47,179,258,114)(48,180,259,115)(49,172,260,116)(50,173,261,117)(51,174,253,109)(52,175,254,110)(53,176,255,111)(54,177,256,112)(55,224,80,193)(56,225,81,194)(57,217,73,195)(58,218,74,196)(59,219,75,197)(60,220,76,198)(61,221,77,190)(62,222,78,191)(63,223,79,192)(82,275,131,226)(83,276,132,227)(84,277,133,228)(85,278,134,229)(86,279,135,230)(87,271,127,231)(88,272,128,232)(89,273,129,233)(90,274,130,234)(91,284,140,235)(92,285,141,236)(93,286,142,237)(94,287,143,238)(95,288,144,239)(96,280,136,240)(97,281,137,241)(98,282,138,242)(99,283,139,243), (1,143,71,133)(2,144,72,134)(3,136,64,135)(4,137,65,127)(5,138,66,128)(6,139,67,129)(7,140,68,130)(8,141,69,131)(9,142,70,132)(10,151,79,110)(11,152,80,111)(12,153,81,112)(13,145,73,113)(14,146,74,114)(15,147,75,115)(16,148,76,116)(17,149,77,117)(18,150,78,109)(19,207,254,208)(20,199,255,209)(21,200,256,210)(22,201,257,211)(23,202,258,212)(24,203,259,213)(25,204,260,214)(26,205,261,215)(27,206,253,216)(28,94,37,84)(29,95,38,85)(30,96,39,86)(31,97,40,87)(32,98,41,88)(33,99,42,89)(34,91,43,90)(35,92,44,82)(36,93,45,83)(46,171,267,161)(47,163,268,162)(48,164,269,154)(49,165,270,155)(50,166,262,156)(51,167,263,157)(52,168,264,158)(53,169,265,159)(54,170,266,160)(55,176,104,121)(56,177,105,122)(57,178,106,123)(58,179,107,124)(59,180,108,125)(60,172,100,126)(61,173,101,118)(62,174,102,119)(63,175,103,120)(181,278,222,288)(182,279,223,280)(183,271,224,281)(184,272,225,282)(185,273,217,283)(186,274,218,284)(187,275,219,285)(188,276,220,286)(189,277,221,287)(190,238,245,228)(191,239,246,229)(192,240,247,230)(193,241,248,231)(194,242,249,232)(195,243,250,233)(196,235,251,234)(197,236,252,226)(198,237,244,227)>;
G:=Group( (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), (1,77)(2,78)(3,79)(4,80)(5,81)(6,73)(7,74)(8,75)(9,76)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,279)(20,271)(21,272)(22,273)(23,274)(24,275)(25,276)(26,277)(27,278)(28,61)(29,62)(30,63)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,100)(46,243)(47,235)(48,236)(49,237)(50,238)(51,239)(52,240)(53,241)(54,242)(82,125)(83,126)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,179)(92,180)(93,172)(94,173)(95,174)(96,175)(97,176)(98,177)(99,178)(109,144)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(127,152)(128,153)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(154,197)(155,198)(156,190)(157,191)(158,192)(159,193)(160,194)(161,195)(162,196)(163,251)(164,252)(165,244)(166,245)(167,246)(168,247)(169,248)(170,249)(171,250)(181,216)(182,208)(183,209)(184,210)(185,211)(186,212)(187,213)(188,214)(189,215)(199,224)(200,225)(201,217)(202,218)(203,219)(204,220)(205,221)(206,222)(207,223)(226,269)(227,270)(228,262)(229,263)(230,264)(231,265)(232,266)(233,267)(234,268)(253,288)(254,280)(255,281)(256,282)(257,283)(258,284)(259,285)(260,286)(261,287), (1,156,28,205)(2,157,29,206)(3,158,30,207)(4,159,31,199)(5,160,32,200)(6,161,33,201)(7,162,34,202)(8,154,35,203)(9,155,36,204)(10,247,103,182)(11,248,104,183)(12,249,105,184)(13,250,106,185)(14,251,107,186)(15,252,108,187)(16,244,100,188)(17,245,101,189)(18,246,102,181)(19,151,264,120)(20,152,265,121)(21,153,266,122)(22,145,267,123)(23,146,268,124)(24,147,269,125)(25,148,270,126)(26,149,262,118)(27,150,263,119)(37,215,71,166)(38,216,72,167)(39,208,64,168)(40,209,65,169)(41,210,66,170)(42,211,67,171)(43,212,68,163)(44,213,69,164)(45,214,70,165)(46,178,257,113)(47,179,258,114)(48,180,259,115)(49,172,260,116)(50,173,261,117)(51,174,253,109)(52,175,254,110)(53,176,255,111)(54,177,256,112)(55,224,80,193)(56,225,81,194)(57,217,73,195)(58,218,74,196)(59,219,75,197)(60,220,76,198)(61,221,77,190)(62,222,78,191)(63,223,79,192)(82,275,131,226)(83,276,132,227)(84,277,133,228)(85,278,134,229)(86,279,135,230)(87,271,127,231)(88,272,128,232)(89,273,129,233)(90,274,130,234)(91,284,140,235)(92,285,141,236)(93,286,142,237)(94,287,143,238)(95,288,144,239)(96,280,136,240)(97,281,137,241)(98,282,138,242)(99,283,139,243), (1,143,71,133)(2,144,72,134)(3,136,64,135)(4,137,65,127)(5,138,66,128)(6,139,67,129)(7,140,68,130)(8,141,69,131)(9,142,70,132)(10,151,79,110)(11,152,80,111)(12,153,81,112)(13,145,73,113)(14,146,74,114)(15,147,75,115)(16,148,76,116)(17,149,77,117)(18,150,78,109)(19,207,254,208)(20,199,255,209)(21,200,256,210)(22,201,257,211)(23,202,258,212)(24,203,259,213)(25,204,260,214)(26,205,261,215)(27,206,253,216)(28,94,37,84)(29,95,38,85)(30,96,39,86)(31,97,40,87)(32,98,41,88)(33,99,42,89)(34,91,43,90)(35,92,44,82)(36,93,45,83)(46,171,267,161)(47,163,268,162)(48,164,269,154)(49,165,270,155)(50,166,262,156)(51,167,263,157)(52,168,264,158)(53,169,265,159)(54,170,266,160)(55,176,104,121)(56,177,105,122)(57,178,106,123)(58,179,107,124)(59,180,108,125)(60,172,100,126)(61,173,101,118)(62,174,102,119)(63,175,103,120)(181,278,222,288)(182,279,223,280)(183,271,224,281)(184,272,225,282)(185,273,217,283)(186,274,218,284)(187,275,219,285)(188,276,220,286)(189,277,221,287)(190,238,245,228)(191,239,246,229)(192,240,247,230)(193,241,248,231)(194,242,249,232)(195,243,250,233)(196,235,251,234)(197,236,252,226)(198,237,244,227) );
G=PermutationGroup([[(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)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,73),(7,74),(8,75),(9,76),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,279),(20,271),(21,272),(22,273),(23,274),(24,275),(25,276),(26,277),(27,278),(28,61),(29,62),(30,63),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,100),(46,243),(47,235),(48,236),(49,237),(50,238),(51,239),(52,240),(53,241),(54,242),(82,125),(83,126),(84,118),(85,119),(86,120),(87,121),(88,122),(89,123),(90,124),(91,179),(92,180),(93,172),(94,173),(95,174),(96,175),(97,176),(98,177),(99,178),(109,144),(110,136),(111,137),(112,138),(113,139),(114,140),(115,141),(116,142),(117,143),(127,152),(128,153),(129,145),(130,146),(131,147),(132,148),(133,149),(134,150),(135,151),(154,197),(155,198),(156,190),(157,191),(158,192),(159,193),(160,194),(161,195),(162,196),(163,251),(164,252),(165,244),(166,245),(167,246),(168,247),(169,248),(170,249),(171,250),(181,216),(182,208),(183,209),(184,210),(185,211),(186,212),(187,213),(188,214),(189,215),(199,224),(200,225),(201,217),(202,218),(203,219),(204,220),(205,221),(206,222),(207,223),(226,269),(227,270),(228,262),(229,263),(230,264),(231,265),(232,266),(233,267),(234,268),(253,288),(254,280),(255,281),(256,282),(257,283),(258,284),(259,285),(260,286),(261,287)], [(1,156,28,205),(2,157,29,206),(3,158,30,207),(4,159,31,199),(5,160,32,200),(6,161,33,201),(7,162,34,202),(8,154,35,203),(9,155,36,204),(10,247,103,182),(11,248,104,183),(12,249,105,184),(13,250,106,185),(14,251,107,186),(15,252,108,187),(16,244,100,188),(17,245,101,189),(18,246,102,181),(19,151,264,120),(20,152,265,121),(21,153,266,122),(22,145,267,123),(23,146,268,124),(24,147,269,125),(25,148,270,126),(26,149,262,118),(27,150,263,119),(37,215,71,166),(38,216,72,167),(39,208,64,168),(40,209,65,169),(41,210,66,170),(42,211,67,171),(43,212,68,163),(44,213,69,164),(45,214,70,165),(46,178,257,113),(47,179,258,114),(48,180,259,115),(49,172,260,116),(50,173,261,117),(51,174,253,109),(52,175,254,110),(53,176,255,111),(54,177,256,112),(55,224,80,193),(56,225,81,194),(57,217,73,195),(58,218,74,196),(59,219,75,197),(60,220,76,198),(61,221,77,190),(62,222,78,191),(63,223,79,192),(82,275,131,226),(83,276,132,227),(84,277,133,228),(85,278,134,229),(86,279,135,230),(87,271,127,231),(88,272,128,232),(89,273,129,233),(90,274,130,234),(91,284,140,235),(92,285,141,236),(93,286,142,237),(94,287,143,238),(95,288,144,239),(96,280,136,240),(97,281,137,241),(98,282,138,242),(99,283,139,243)], [(1,143,71,133),(2,144,72,134),(3,136,64,135),(4,137,65,127),(5,138,66,128),(6,139,67,129),(7,140,68,130),(8,141,69,131),(9,142,70,132),(10,151,79,110),(11,152,80,111),(12,153,81,112),(13,145,73,113),(14,146,74,114),(15,147,75,115),(16,148,76,116),(17,149,77,117),(18,150,78,109),(19,207,254,208),(20,199,255,209),(21,200,256,210),(22,201,257,211),(23,202,258,212),(24,203,259,213),(25,204,260,214),(26,205,261,215),(27,206,253,216),(28,94,37,84),(29,95,38,85),(30,96,39,86),(31,97,40,87),(32,98,41,88),(33,99,42,89),(34,91,43,90),(35,92,44,82),(36,93,45,83),(46,171,267,161),(47,163,268,162),(48,164,269,154),(49,165,270,155),(50,166,262,156),(51,167,263,157),(52,168,264,158),(53,169,265,159),(54,170,266,160),(55,176,104,121),(56,177,105,122),(57,178,106,123),(58,179,107,124),(59,180,108,125),(60,172,100,126),(61,173,101,118),(62,174,102,119),(63,175,103,120),(181,278,222,288),(182,279,223,280),(183,271,224,281),(184,272,225,282),(185,273,217,283),(186,274,218,284),(187,275,219,285),(188,276,220,286),(189,277,221,287),(190,238,245,228),(191,239,246,229),(192,240,247,230),(193,241,248,231),(194,242,249,232),(195,243,250,233),(196,235,251,234),(197,236,252,226),(198,237,244,227)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6N | 9A | ··· | 9F | 12A | ··· | 12X | 18A | ··· | 18AP | 36A | ··· | 36BT |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | D4 | Q8 | C3×D4 | C3×Q8 | D4×C9 | Q8×C9 |
| kernel | C9×C2.C42 | C22×C36 | C3×C2.C42 | C2×C36 | C22×C12 | C2.C42 | C2×C12 | C22×C4 | C2×C4 | C2×C18 | C2×C18 | C2×C6 | C2×C6 | C22 | C22 |
| # reps | 1 | 3 | 2 | 12 | 6 | 6 | 24 | 18 | 72 | 3 | 1 | 6 | 2 | 18 | 6 |
Matrix representation of C9×C2.C42 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 31 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 26 | 19 |
| 0 | 0 | 19 | 11 |
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[31,0,0,0,0,6,0,0,0,0,26,19,0,0,19,11],[6,0,0,0,0,1,0,0,0,0,0,36,0,0,1,0] >;
C9×C2.C42 in GAP, Magma, Sage, TeX
C_9\times C_2.C_4^2
% in TeX
G:=Group("C9xC2.C4^2"); // GroupNames label
G:=SmallGroup(288,45);
// by ID
G=gap.SmallGroup(288,45);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,197,344,520]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations