direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C42.C2, C36.11Q8, C42.4C18, C4⋊C4.4C18, C4.3(Q8×C9), C6.21(C6×Q8), C2.4(Q8×C18), (C4×C36).10C2, (C4×C12).19C6, C18.21(C2×Q8), C12.11(C3×Q8), C18.45(C4○D4), (C2×C36).65C22, (C2×C18).80C23, C22.15(C22×C18), C2.8(C9×C4○D4), (C3×C4⋊C4).15C6, (C9×C4⋊C4).11C2, C3.(C3×C42.C2), C6.45(C3×C4○D4), (C2×C4).21(C2×C18), (C2×C12).84(C2×C6), (C3×C42.C2).C3, (C2×C6).85(C22×C6), SmallGroup(288,175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C42.C2
G = < a,b,c,d | a9=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 102 in 84 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C12, C12, C2×C6, C42, C4⋊C4, C18, C18, C2×C12, C2×C12, C42.C2, C36, C36, C2×C18, C4×C12, C3×C4⋊C4, C2×C36, C2×C36, C3×C42.C2, C4×C36, C9×C4⋊C4, C9×C42.C2
Quotients: C1, C2, C3, C22, C6, Q8, C23, C9, C2×C6, C2×Q8, C4○D4, C18, C3×Q8, C22×C6, C42.C2, C2×C18, C6×Q8, C3×C4○D4, Q8×C9, C22×C18, C3×C42.C2, Q8×C18, C9×C4○D4, C9×C42.C2
►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)
(1 106 34 110)(2 107 35 111)(3 108 36 112)(4 100 28 113)(5 101 29 114)(6 102 30 115)(7 103 31 116)(8 104 32 117)(9 105 33 109)(10 206 282 232)(11 207 283 233)(12 199 284 234)(13 200 285 226)(14 201 286 227)(15 202 287 228)(16 203 288 229)(17 204 280 230)(18 205 281 231)(19 216 275 220)(20 208 276 221)(21 209 277 222)(22 210 278 223)(23 211 279 224)(24 212 271 225)(25 213 272 217)(26 214 273 218)(27 215 274 219)(37 121 49 95)(38 122 50 96)(39 123 51 97)(40 124 52 98)(41 125 53 99)(42 126 54 91)(43 118 46 92)(44 119 47 93)(45 120 48 94)(55 162 90 127)(56 154 82 128)(57 155 83 129)(58 156 84 130)(59 157 85 131)(60 158 86 132)(61 159 87 133)(62 160 88 134)(63 161 89 135)(64 149 77 136)(65 150 78 137)(66 151 79 138)(67 152 80 139)(68 153 81 140)(69 145 73 141)(70 146 74 142)(71 147 75 143)(72 148 76 144)(163 248 198 261)(164 249 190 253)(165 250 191 254)(166 251 192 255)(167 252 193 256)(168 244 194 257)(169 245 195 258)(170 246 196 259)(171 247 197 260)(172 240 185 266)(173 241 186 267)(174 242 187 268)(175 243 188 269)(176 235 189 270)(177 236 181 262)(178 237 182 263)(179 238 183 264)(180 239 184 265)
(1 57 44 74)(2 58 45 75)(3 59 37 76)(4 60 38 77)(5 61 39 78)(6 62 40 79)(7 63 41 80)(8 55 42 81)(9 56 43 73)(10 246 278 268)(11 247 279 269)(12 248 271 270)(13 249 272 262)(14 250 273 263)(15 251 274 264)(16 252 275 265)(17 244 276 266)(18 245 277 267)(19 239 288 256)(20 240 280 257)(21 241 281 258)(22 242 282 259)(23 243 283 260)(24 235 284 261)(25 236 285 253)(26 237 286 254)(27 238 287 255)(28 86 50 64)(29 87 51 65)(30 88 52 66)(31 89 53 67)(32 90 54 68)(33 82 46 69)(34 83 47 70)(35 84 48 71)(36 85 49 72)(91 153 117 127)(92 145 109 128)(93 146 110 129)(94 147 111 130)(95 148 112 131)(96 149 113 132)(97 150 114 133)(98 151 115 134)(99 152 116 135)(100 158 122 136)(101 159 123 137)(102 160 124 138)(103 161 125 139)(104 162 126 140)(105 154 118 141)(106 155 119 142)(107 156 120 143)(108 157 121 144)(163 212 189 234)(164 213 181 226)(165 214 182 227)(166 215 183 228)(167 216 184 229)(168 208 185 230)(169 209 186 231)(170 210 187 232)(171 211 188 233)(172 204 194 221)(173 205 195 222)(174 206 196 223)(175 207 197 224)(176 199 198 225)(177 200 190 217)(178 201 191 218)(179 202 192 219)(180 203 193 220)
(1 165 44 182)(2 166 45 183)(3 167 37 184)(4 168 38 185)(5 169 39 186)(6 170 40 187)(7 171 41 188)(8 163 42 189)(9 164 43 181)(10 151 278 134)(11 152 279 135)(12 153 271 127)(13 145 272 128)(14 146 273 129)(15 147 274 130)(16 148 275 131)(17 149 276 132)(18 150 277 133)(19 157 288 144)(20 158 280 136)(21 159 281 137)(22 160 282 138)(23 161 283 139)(24 162 284 140)(25 154 285 141)(26 155 286 142)(27 156 287 143)(28 194 50 172)(29 195 51 173)(30 196 52 174)(31 197 53 175)(32 198 54 176)(33 190 46 177)(34 191 47 178)(35 192 48 179)(36 193 49 180)(55 225 81 199)(56 217 73 200)(57 218 74 201)(58 219 75 202)(59 220 76 203)(60 221 77 204)(61 222 78 205)(62 223 79 206)(63 224 80 207)(64 230 86 208)(65 231 87 209)(66 232 88 210)(67 233 89 211)(68 234 90 212)(69 226 82 213)(70 227 83 214)(71 228 84 215)(72 229 85 216)(91 261 117 235)(92 253 109 236)(93 254 110 237)(94 255 111 238)(95 256 112 239)(96 257 113 240)(97 258 114 241)(98 259 115 242)(99 260 116 243)(100 266 122 244)(101 267 123 245)(102 268 124 246)(103 269 125 247)(104 270 126 248)(105 262 118 249)(106 263 119 250)(107 264 120 251)(108 265 121 252)
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), (1,106,34,110)(2,107,35,111)(3,108,36,112)(4,100,28,113)(5,101,29,114)(6,102,30,115)(7,103,31,116)(8,104,32,117)(9,105,33,109)(10,206,282,232)(11,207,283,233)(12,199,284,234)(13,200,285,226)(14,201,286,227)(15,202,287,228)(16,203,288,229)(17,204,280,230)(18,205,281,231)(19,216,275,220)(20,208,276,221)(21,209,277,222)(22,210,278,223)(23,211,279,224)(24,212,271,225)(25,213,272,217)(26,214,273,218)(27,215,274,219)(37,121,49,95)(38,122,50,96)(39,123,51,97)(40,124,52,98)(41,125,53,99)(42,126,54,91)(43,118,46,92)(44,119,47,93)(45,120,48,94)(55,162,90,127)(56,154,82,128)(57,155,83,129)(58,156,84,130)(59,157,85,131)(60,158,86,132)(61,159,87,133)(62,160,88,134)(63,161,89,135)(64,149,77,136)(65,150,78,137)(66,151,79,138)(67,152,80,139)(68,153,81,140)(69,145,73,141)(70,146,74,142)(71,147,75,143)(72,148,76,144)(163,248,198,261)(164,249,190,253)(165,250,191,254)(166,251,192,255)(167,252,193,256)(168,244,194,257)(169,245,195,258)(170,246,196,259)(171,247,197,260)(172,240,185,266)(173,241,186,267)(174,242,187,268)(175,243,188,269)(176,235,189,270)(177,236,181,262)(178,237,182,263)(179,238,183,264)(180,239,184,265), (1,57,44,74)(2,58,45,75)(3,59,37,76)(4,60,38,77)(5,61,39,78)(6,62,40,79)(7,63,41,80)(8,55,42,81)(9,56,43,73)(10,246,278,268)(11,247,279,269)(12,248,271,270)(13,249,272,262)(14,250,273,263)(15,251,274,264)(16,252,275,265)(17,244,276,266)(18,245,277,267)(19,239,288,256)(20,240,280,257)(21,241,281,258)(22,242,282,259)(23,243,283,260)(24,235,284,261)(25,236,285,253)(26,237,286,254)(27,238,287,255)(28,86,50,64)(29,87,51,65)(30,88,52,66)(31,89,53,67)(32,90,54,68)(33,82,46,69)(34,83,47,70)(35,84,48,71)(36,85,49,72)(91,153,117,127)(92,145,109,128)(93,146,110,129)(94,147,111,130)(95,148,112,131)(96,149,113,132)(97,150,114,133)(98,151,115,134)(99,152,116,135)(100,158,122,136)(101,159,123,137)(102,160,124,138)(103,161,125,139)(104,162,126,140)(105,154,118,141)(106,155,119,142)(107,156,120,143)(108,157,121,144)(163,212,189,234)(164,213,181,226)(165,214,182,227)(166,215,183,228)(167,216,184,229)(168,208,185,230)(169,209,186,231)(170,210,187,232)(171,211,188,233)(172,204,194,221)(173,205,195,222)(174,206,196,223)(175,207,197,224)(176,199,198,225)(177,200,190,217)(178,201,191,218)(179,202,192,219)(180,203,193,220), (1,165,44,182)(2,166,45,183)(3,167,37,184)(4,168,38,185)(5,169,39,186)(6,170,40,187)(7,171,41,188)(8,163,42,189)(9,164,43,181)(10,151,278,134)(11,152,279,135)(12,153,271,127)(13,145,272,128)(14,146,273,129)(15,147,274,130)(16,148,275,131)(17,149,276,132)(18,150,277,133)(19,157,288,144)(20,158,280,136)(21,159,281,137)(22,160,282,138)(23,161,283,139)(24,162,284,140)(25,154,285,141)(26,155,286,142)(27,156,287,143)(28,194,50,172)(29,195,51,173)(30,196,52,174)(31,197,53,175)(32,198,54,176)(33,190,46,177)(34,191,47,178)(35,192,48,179)(36,193,49,180)(55,225,81,199)(56,217,73,200)(57,218,74,201)(58,219,75,202)(59,220,76,203)(60,221,77,204)(61,222,78,205)(62,223,79,206)(63,224,80,207)(64,230,86,208)(65,231,87,209)(66,232,88,210)(67,233,89,211)(68,234,90,212)(69,226,82,213)(70,227,83,214)(71,228,84,215)(72,229,85,216)(91,261,117,235)(92,253,109,236)(93,254,110,237)(94,255,111,238)(95,256,112,239)(96,257,113,240)(97,258,114,241)(98,259,115,242)(99,260,116,243)(100,266,122,244)(101,267,123,245)(102,268,124,246)(103,269,125,247)(104,270,126,248)(105,262,118,249)(106,263,119,250)(107,264,120,251)(108,265,121,252)>;
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), (1,106,34,110)(2,107,35,111)(3,108,36,112)(4,100,28,113)(5,101,29,114)(6,102,30,115)(7,103,31,116)(8,104,32,117)(9,105,33,109)(10,206,282,232)(11,207,283,233)(12,199,284,234)(13,200,285,226)(14,201,286,227)(15,202,287,228)(16,203,288,229)(17,204,280,230)(18,205,281,231)(19,216,275,220)(20,208,276,221)(21,209,277,222)(22,210,278,223)(23,211,279,224)(24,212,271,225)(25,213,272,217)(26,214,273,218)(27,215,274,219)(37,121,49,95)(38,122,50,96)(39,123,51,97)(40,124,52,98)(41,125,53,99)(42,126,54,91)(43,118,46,92)(44,119,47,93)(45,120,48,94)(55,162,90,127)(56,154,82,128)(57,155,83,129)(58,156,84,130)(59,157,85,131)(60,158,86,132)(61,159,87,133)(62,160,88,134)(63,161,89,135)(64,149,77,136)(65,150,78,137)(66,151,79,138)(67,152,80,139)(68,153,81,140)(69,145,73,141)(70,146,74,142)(71,147,75,143)(72,148,76,144)(163,248,198,261)(164,249,190,253)(165,250,191,254)(166,251,192,255)(167,252,193,256)(168,244,194,257)(169,245,195,258)(170,246,196,259)(171,247,197,260)(172,240,185,266)(173,241,186,267)(174,242,187,268)(175,243,188,269)(176,235,189,270)(177,236,181,262)(178,237,182,263)(179,238,183,264)(180,239,184,265), (1,57,44,74)(2,58,45,75)(3,59,37,76)(4,60,38,77)(5,61,39,78)(6,62,40,79)(7,63,41,80)(8,55,42,81)(9,56,43,73)(10,246,278,268)(11,247,279,269)(12,248,271,270)(13,249,272,262)(14,250,273,263)(15,251,274,264)(16,252,275,265)(17,244,276,266)(18,245,277,267)(19,239,288,256)(20,240,280,257)(21,241,281,258)(22,242,282,259)(23,243,283,260)(24,235,284,261)(25,236,285,253)(26,237,286,254)(27,238,287,255)(28,86,50,64)(29,87,51,65)(30,88,52,66)(31,89,53,67)(32,90,54,68)(33,82,46,69)(34,83,47,70)(35,84,48,71)(36,85,49,72)(91,153,117,127)(92,145,109,128)(93,146,110,129)(94,147,111,130)(95,148,112,131)(96,149,113,132)(97,150,114,133)(98,151,115,134)(99,152,116,135)(100,158,122,136)(101,159,123,137)(102,160,124,138)(103,161,125,139)(104,162,126,140)(105,154,118,141)(106,155,119,142)(107,156,120,143)(108,157,121,144)(163,212,189,234)(164,213,181,226)(165,214,182,227)(166,215,183,228)(167,216,184,229)(168,208,185,230)(169,209,186,231)(170,210,187,232)(171,211,188,233)(172,204,194,221)(173,205,195,222)(174,206,196,223)(175,207,197,224)(176,199,198,225)(177,200,190,217)(178,201,191,218)(179,202,192,219)(180,203,193,220), (1,165,44,182)(2,166,45,183)(3,167,37,184)(4,168,38,185)(5,169,39,186)(6,170,40,187)(7,171,41,188)(8,163,42,189)(9,164,43,181)(10,151,278,134)(11,152,279,135)(12,153,271,127)(13,145,272,128)(14,146,273,129)(15,147,274,130)(16,148,275,131)(17,149,276,132)(18,150,277,133)(19,157,288,144)(20,158,280,136)(21,159,281,137)(22,160,282,138)(23,161,283,139)(24,162,284,140)(25,154,285,141)(26,155,286,142)(27,156,287,143)(28,194,50,172)(29,195,51,173)(30,196,52,174)(31,197,53,175)(32,198,54,176)(33,190,46,177)(34,191,47,178)(35,192,48,179)(36,193,49,180)(55,225,81,199)(56,217,73,200)(57,218,74,201)(58,219,75,202)(59,220,76,203)(60,221,77,204)(61,222,78,205)(62,223,79,206)(63,224,80,207)(64,230,86,208)(65,231,87,209)(66,232,88,210)(67,233,89,211)(68,234,90,212)(69,226,82,213)(70,227,83,214)(71,228,84,215)(72,229,85,216)(91,261,117,235)(92,253,109,236)(93,254,110,237)(94,255,111,238)(95,256,112,239)(96,257,113,240)(97,258,114,241)(98,259,115,242)(99,260,116,243)(100,266,122,244)(101,267,123,245)(102,268,124,246)(103,269,125,247)(104,270,126,248)(105,262,118,249)(106,263,119,250)(107,264,120,251)(108,265,121,252) );
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)], [(1,106,34,110),(2,107,35,111),(3,108,36,112),(4,100,28,113),(5,101,29,114),(6,102,30,115),(7,103,31,116),(8,104,32,117),(9,105,33,109),(10,206,282,232),(11,207,283,233),(12,199,284,234),(13,200,285,226),(14,201,286,227),(15,202,287,228),(16,203,288,229),(17,204,280,230),(18,205,281,231),(19,216,275,220),(20,208,276,221),(21,209,277,222),(22,210,278,223),(23,211,279,224),(24,212,271,225),(25,213,272,217),(26,214,273,218),(27,215,274,219),(37,121,49,95),(38,122,50,96),(39,123,51,97),(40,124,52,98),(41,125,53,99),(42,126,54,91),(43,118,46,92),(44,119,47,93),(45,120,48,94),(55,162,90,127),(56,154,82,128),(57,155,83,129),(58,156,84,130),(59,157,85,131),(60,158,86,132),(61,159,87,133),(62,160,88,134),(63,161,89,135),(64,149,77,136),(65,150,78,137),(66,151,79,138),(67,152,80,139),(68,153,81,140),(69,145,73,141),(70,146,74,142),(71,147,75,143),(72,148,76,144),(163,248,198,261),(164,249,190,253),(165,250,191,254),(166,251,192,255),(167,252,193,256),(168,244,194,257),(169,245,195,258),(170,246,196,259),(171,247,197,260),(172,240,185,266),(173,241,186,267),(174,242,187,268),(175,243,188,269),(176,235,189,270),(177,236,181,262),(178,237,182,263),(179,238,183,264),(180,239,184,265)], [(1,57,44,74),(2,58,45,75),(3,59,37,76),(4,60,38,77),(5,61,39,78),(6,62,40,79),(7,63,41,80),(8,55,42,81),(9,56,43,73),(10,246,278,268),(11,247,279,269),(12,248,271,270),(13,249,272,262),(14,250,273,263),(15,251,274,264),(16,252,275,265),(17,244,276,266),(18,245,277,267),(19,239,288,256),(20,240,280,257),(21,241,281,258),(22,242,282,259),(23,243,283,260),(24,235,284,261),(25,236,285,253),(26,237,286,254),(27,238,287,255),(28,86,50,64),(29,87,51,65),(30,88,52,66),(31,89,53,67),(32,90,54,68),(33,82,46,69),(34,83,47,70),(35,84,48,71),(36,85,49,72),(91,153,117,127),(92,145,109,128),(93,146,110,129),(94,147,111,130),(95,148,112,131),(96,149,113,132),(97,150,114,133),(98,151,115,134),(99,152,116,135),(100,158,122,136),(101,159,123,137),(102,160,124,138),(103,161,125,139),(104,162,126,140),(105,154,118,141),(106,155,119,142),(107,156,120,143),(108,157,121,144),(163,212,189,234),(164,213,181,226),(165,214,182,227),(166,215,183,228),(167,216,184,229),(168,208,185,230),(169,209,186,231),(170,210,187,232),(171,211,188,233),(172,204,194,221),(173,205,195,222),(174,206,196,223),(175,207,197,224),(176,199,198,225),(177,200,190,217),(178,201,191,218),(179,202,192,219),(180,203,193,220)], [(1,165,44,182),(2,166,45,183),(3,167,37,184),(4,168,38,185),(5,169,39,186),(6,170,40,187),(7,171,41,188),(8,163,42,189),(9,164,43,181),(10,151,278,134),(11,152,279,135),(12,153,271,127),(13,145,272,128),(14,146,273,129),(15,147,274,130),(16,148,275,131),(17,149,276,132),(18,150,277,133),(19,157,288,144),(20,158,280,136),(21,159,281,137),(22,160,282,138),(23,161,283,139),(24,162,284,140),(25,154,285,141),(26,155,286,142),(27,156,287,143),(28,194,50,172),(29,195,51,173),(30,196,52,174),(31,197,53,175),(32,198,54,176),(33,190,46,177),(34,191,47,178),(35,192,48,179),(36,193,49,180),(55,225,81,199),(56,217,73,200),(57,218,74,201),(58,219,75,202),(59,220,76,203),(60,221,77,204),(61,222,78,205),(62,223,79,206),(63,224,80,207),(64,230,86,208),(65,231,87,209),(66,232,88,210),(67,233,89,211),(68,234,90,212),(69,226,82,213),(70,227,83,214),(71,228,84,215),(72,229,85,216),(91,261,117,235),(92,253,109,236),(93,254,110,237),(94,255,111,238),(95,256,112,239),(96,257,113,240),(97,258,114,241),(98,259,115,242),(99,260,116,243),(100,266,122,244),(101,267,123,245),(102,268,124,246),(103,269,125,247),(104,270,126,248),(105,262,118,249),(106,263,119,250),(107,264,120,251),(108,265,121,252)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12L | 12M | ··· | 12T | 18A | ··· | 18R | 36A | ··· | 36AJ | 36AK | ··· | 36BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | Q8 | C4○D4 | C3×Q8 | C3×C4○D4 | Q8×C9 | C9×C4○D4 |
| kernel | C9×C42.C2 | C4×C36 | C9×C4⋊C4 | C3×C42.C2 | C4×C12 | C3×C4⋊C4 | C42.C2 | C42 | C4⋊C4 | C36 | C18 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 6 | 2 | 2 | 12 | 6 | 6 | 36 | 2 | 4 | 4 | 8 | 12 | 24 |
Matrix representation of C9×C42.C2 ►in GL4(𝔽37) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 12 | 22 | 0 | 0 |
| 12 | 25 | 0 | 0 |
| 0 | 0 | 5 | 23 |
| 0 | 0 | 23 | 32 |
G:=sub<GL(4,GF(37))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[6,0,0,0,0,6,0,0,0,0,0,36,0,0,1,0],[1,0,0,0,35,36,0,0,0,0,0,36,0,0,1,0],[12,12,0,0,22,25,0,0,0,0,5,23,0,0,23,32] >;
C9×C42.C2 in GAP, Magma, Sage, TeX
C_9\times C_4^2.C_2
% in TeX
G:=Group("C9xC4^2.C2"); // GroupNames label
G:=SmallGroup(288,175);
// by ID
G=gap.SmallGroup(288,175);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,365,344,1094,142,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations