metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.1D9, C18.2C42, C18.4M4(2), C9⋊C8⋊3C4, C9⋊1(C8⋊C4), (C4×C36).7C2, (C2×C36).3C4, C4.19(C4×D9), C12.68(C4×S3), C36.24(C2×C4), (C4×C12).17S3, (C2×C4).88D18, (C2×C4).2Dic9, C2.3(C4×Dic9), C6.7(C4×Dic3), (C2×C12).402D6, (C2×C12).5Dic3, C3.(C42.S3), C6.4(C4.Dic3), C2.1(C4.Dic9), C22.7(C2×Dic9), (C2×C36).100C22, (C2×C9⋊C8).7C2, (C2×C18).25(C2×C4), (C2×C6).29(C2×Dic3), SmallGroup(288,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.D9
G = < a,b,c,d | a4=b4=c9=1, d2=b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of C42.D9
►Regular action on 288 points
Generators in S288
(1 95 23 77)(2 96 24 78)(3 97 25 79)(4 98 26 80)(5 99 27 81)(6 91 19 73)(7 92 20 74)(8 93 21 75)(9 94 22 76)(10 100 28 82)(11 101 29 83)(12 102 30 84)(13 103 31 85)(14 104 32 86)(15 105 33 87)(16 106 34 88)(17 107 35 89)(18 108 36 90)(37 127 55 109)(38 128 56 110)(39 129 57 111)(40 130 58 112)(41 131 59 113)(42 132 60 114)(43 133 61 115)(44 134 62 116)(45 135 63 117)(46 136 64 118)(47 137 65 119)(48 138 66 120)(49 139 67 121)(50 140 68 122)(51 141 69 123)(52 142 70 124)(53 143 71 125)(54 144 72 126)(145 244 163 226)(146 245 164 227)(147 246 165 228)(148 247 166 229)(149 248 167 230)(150 249 168 231)(151 250 169 232)(152 251 170 233)(153 252 171 234)(154 235 172 217)(155 236 173 218)(156 237 174 219)(157 238 175 220)(158 239 176 221)(159 240 177 222)(160 241 178 223)(161 242 179 224)(162 243 180 225)(181 280 199 262)(182 281 200 263)(183 282 201 264)(184 283 202 265)(185 284 203 266)(186 285 204 267)(187 286 205 268)(188 287 206 269)(189 288 207 270)(190 271 208 253)(191 272 209 254)(192 273 210 255)(193 274 211 256)(194 275 212 257)(195 276 213 258)(196 277 214 259)(197 278 215 260)(198 279 216 261)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)(91 136 100 127)(92 137 101 128)(93 138 102 129)(94 139 103 130)(95 140 104 131)(96 141 105 132)(97 142 106 133)(98 143 107 134)(99 144 108 135)(145 190 154 181)(146 191 155 182)(147 192 156 183)(148 193 157 184)(149 194 158 185)(150 195 159 186)(151 196 160 187)(152 197 161 188)(153 198 162 189)(163 208 172 199)(164 209 173 200)(165 210 174 201)(166 211 175 202)(167 212 176 203)(168 213 177 204)(169 214 178 205)(170 215 179 206)(171 216 180 207)(217 262 226 253)(218 263 227 254)(219 264 228 255)(220 265 229 256)(221 266 230 257)(222 267 231 258)(223 268 232 259)(224 269 233 260)(225 270 234 261)(235 280 244 271)(236 281 245 272)(237 282 246 273)(238 283 247 274)(239 284 248 275)(240 285 249 276)(241 286 250 277)(242 287 251 278)(243 288 252 279)
(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)
(1 192 50 156 14 183 41 147)(2 191 51 155 15 182 42 146)(3 190 52 154 16 181 43 145)(4 198 53 162 17 189 44 153)(5 197 54 161 18 188 45 152)(6 196 46 160 10 187 37 151)(7 195 47 159 11 186 38 150)(8 194 48 158 12 185 39 149)(9 193 49 157 13 184 40 148)(19 214 64 178 28 205 55 169)(20 213 65 177 29 204 56 168)(21 212 66 176 30 203 57 167)(22 211 67 175 31 202 58 166)(23 210 68 174 32 201 59 165)(24 209 69 173 33 200 60 164)(25 208 70 172 34 199 61 163)(26 216 71 180 35 207 62 171)(27 215 72 179 36 206 63 170)(73 268 118 232 82 259 109 223)(74 267 119 231 83 258 110 222)(75 266 120 230 84 257 111 221)(76 265 121 229 85 256 112 220)(77 264 122 228 86 255 113 219)(78 263 123 227 87 254 114 218)(79 262 124 226 88 253 115 217)(80 270 125 234 89 261 116 225)(81 269 126 233 90 260 117 224)(91 286 136 250 100 277 127 241)(92 285 137 249 101 276 128 240)(93 284 138 248 102 275 129 239)(94 283 139 247 103 274 130 238)(95 282 140 246 104 273 131 237)(96 281 141 245 105 272 132 236)(97 280 142 244 106 271 133 235)(98 288 143 252 107 279 134 243)(99 287 144 251 108 278 135 242)
G:=sub<Sym(288)| (1,95,23,77)(2,96,24,78)(3,97,25,79)(4,98,26,80)(5,99,27,81)(6,91,19,73)(7,92,20,74)(8,93,21,75)(9,94,22,76)(10,100,28,82)(11,101,29,83)(12,102,30,84)(13,103,31,85)(14,104,32,86)(15,105,33,87)(16,106,34,88)(17,107,35,89)(18,108,36,90)(37,127,55,109)(38,128,56,110)(39,129,57,111)(40,130,58,112)(41,131,59,113)(42,132,60,114)(43,133,61,115)(44,134,62,116)(45,135,63,117)(46,136,64,118)(47,137,65,119)(48,138,66,120)(49,139,67,121)(50,140,68,122)(51,141,69,123)(52,142,70,124)(53,143,71,125)(54,144,72,126)(145,244,163,226)(146,245,164,227)(147,246,165,228)(148,247,166,229)(149,248,167,230)(150,249,168,231)(151,250,169,232)(152,251,170,233)(153,252,171,234)(154,235,172,217)(155,236,173,218)(156,237,174,219)(157,238,175,220)(158,239,176,221)(159,240,177,222)(160,241,178,223)(161,242,179,224)(162,243,180,225)(181,280,199,262)(182,281,200,263)(183,282,201,264)(184,283,202,265)(185,284,203,266)(186,285,204,267)(187,286,205,268)(188,287,206,269)(189,288,207,270)(190,271,208,253)(191,272,209,254)(192,273,210,255)(193,274,211,256)(194,275,212,257)(195,276,213,258)(196,277,214,259)(197,278,215,260)(198,279,216,261), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135)(145,190,154,181)(146,191,155,182)(147,192,156,183)(148,193,157,184)(149,194,158,185)(150,195,159,186)(151,196,160,187)(152,197,161,188)(153,198,162,189)(163,208,172,199)(164,209,173,200)(165,210,174,201)(166,211,175,202)(167,212,176,203)(168,213,177,204)(169,214,178,205)(170,215,179,206)(171,216,180,207)(217,262,226,253)(218,263,227,254)(219,264,228,255)(220,265,229,256)(221,266,230,257)(222,267,231,258)(223,268,232,259)(224,269,233,260)(225,270,234,261)(235,280,244,271)(236,281,245,272)(237,282,246,273)(238,283,247,274)(239,284,248,275)(240,285,249,276)(241,286,250,277)(242,287,251,278)(243,288,252,279), (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), (1,192,50,156,14,183,41,147)(2,191,51,155,15,182,42,146)(3,190,52,154,16,181,43,145)(4,198,53,162,17,189,44,153)(5,197,54,161,18,188,45,152)(6,196,46,160,10,187,37,151)(7,195,47,159,11,186,38,150)(8,194,48,158,12,185,39,149)(9,193,49,157,13,184,40,148)(19,214,64,178,28,205,55,169)(20,213,65,177,29,204,56,168)(21,212,66,176,30,203,57,167)(22,211,67,175,31,202,58,166)(23,210,68,174,32,201,59,165)(24,209,69,173,33,200,60,164)(25,208,70,172,34,199,61,163)(26,216,71,180,35,207,62,171)(27,215,72,179,36,206,63,170)(73,268,118,232,82,259,109,223)(74,267,119,231,83,258,110,222)(75,266,120,230,84,257,111,221)(76,265,121,229,85,256,112,220)(77,264,122,228,86,255,113,219)(78,263,123,227,87,254,114,218)(79,262,124,226,88,253,115,217)(80,270,125,234,89,261,116,225)(81,269,126,233,90,260,117,224)(91,286,136,250,100,277,127,241)(92,285,137,249,101,276,128,240)(93,284,138,248,102,275,129,239)(94,283,139,247,103,274,130,238)(95,282,140,246,104,273,131,237)(96,281,141,245,105,272,132,236)(97,280,142,244,106,271,133,235)(98,288,143,252,107,279,134,243)(99,287,144,251,108,278,135,242)>;
G:=Group( (1,95,23,77)(2,96,24,78)(3,97,25,79)(4,98,26,80)(5,99,27,81)(6,91,19,73)(7,92,20,74)(8,93,21,75)(9,94,22,76)(10,100,28,82)(11,101,29,83)(12,102,30,84)(13,103,31,85)(14,104,32,86)(15,105,33,87)(16,106,34,88)(17,107,35,89)(18,108,36,90)(37,127,55,109)(38,128,56,110)(39,129,57,111)(40,130,58,112)(41,131,59,113)(42,132,60,114)(43,133,61,115)(44,134,62,116)(45,135,63,117)(46,136,64,118)(47,137,65,119)(48,138,66,120)(49,139,67,121)(50,140,68,122)(51,141,69,123)(52,142,70,124)(53,143,71,125)(54,144,72,126)(145,244,163,226)(146,245,164,227)(147,246,165,228)(148,247,166,229)(149,248,167,230)(150,249,168,231)(151,250,169,232)(152,251,170,233)(153,252,171,234)(154,235,172,217)(155,236,173,218)(156,237,174,219)(157,238,175,220)(158,239,176,221)(159,240,177,222)(160,241,178,223)(161,242,179,224)(162,243,180,225)(181,280,199,262)(182,281,200,263)(183,282,201,264)(184,283,202,265)(185,284,203,266)(186,285,204,267)(187,286,205,268)(188,287,206,269)(189,288,207,270)(190,271,208,253)(191,272,209,254)(192,273,210,255)(193,274,211,256)(194,275,212,257)(195,276,213,258)(196,277,214,259)(197,278,215,260)(198,279,216,261), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135)(145,190,154,181)(146,191,155,182)(147,192,156,183)(148,193,157,184)(149,194,158,185)(150,195,159,186)(151,196,160,187)(152,197,161,188)(153,198,162,189)(163,208,172,199)(164,209,173,200)(165,210,174,201)(166,211,175,202)(167,212,176,203)(168,213,177,204)(169,214,178,205)(170,215,179,206)(171,216,180,207)(217,262,226,253)(218,263,227,254)(219,264,228,255)(220,265,229,256)(221,266,230,257)(222,267,231,258)(223,268,232,259)(224,269,233,260)(225,270,234,261)(235,280,244,271)(236,281,245,272)(237,282,246,273)(238,283,247,274)(239,284,248,275)(240,285,249,276)(241,286,250,277)(242,287,251,278)(243,288,252,279), (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), (1,192,50,156,14,183,41,147)(2,191,51,155,15,182,42,146)(3,190,52,154,16,181,43,145)(4,198,53,162,17,189,44,153)(5,197,54,161,18,188,45,152)(6,196,46,160,10,187,37,151)(7,195,47,159,11,186,38,150)(8,194,48,158,12,185,39,149)(9,193,49,157,13,184,40,148)(19,214,64,178,28,205,55,169)(20,213,65,177,29,204,56,168)(21,212,66,176,30,203,57,167)(22,211,67,175,31,202,58,166)(23,210,68,174,32,201,59,165)(24,209,69,173,33,200,60,164)(25,208,70,172,34,199,61,163)(26,216,71,180,35,207,62,171)(27,215,72,179,36,206,63,170)(73,268,118,232,82,259,109,223)(74,267,119,231,83,258,110,222)(75,266,120,230,84,257,111,221)(76,265,121,229,85,256,112,220)(77,264,122,228,86,255,113,219)(78,263,123,227,87,254,114,218)(79,262,124,226,88,253,115,217)(80,270,125,234,89,261,116,225)(81,269,126,233,90,260,117,224)(91,286,136,250,100,277,127,241)(92,285,137,249,101,276,128,240)(93,284,138,248,102,275,129,239)(94,283,139,247,103,274,130,238)(95,282,140,246,104,273,131,237)(96,281,141,245,105,272,132,236)(97,280,142,244,106,271,133,235)(98,288,143,252,107,279,134,243)(99,287,144,251,108,278,135,242) );
G=PermutationGroup([[(1,95,23,77),(2,96,24,78),(3,97,25,79),(4,98,26,80),(5,99,27,81),(6,91,19,73),(7,92,20,74),(8,93,21,75),(9,94,22,76),(10,100,28,82),(11,101,29,83),(12,102,30,84),(13,103,31,85),(14,104,32,86),(15,105,33,87),(16,106,34,88),(17,107,35,89),(18,108,36,90),(37,127,55,109),(38,128,56,110),(39,129,57,111),(40,130,58,112),(41,131,59,113),(42,132,60,114),(43,133,61,115),(44,134,62,116),(45,135,63,117),(46,136,64,118),(47,137,65,119),(48,138,66,120),(49,139,67,121),(50,140,68,122),(51,141,69,123),(52,142,70,124),(53,143,71,125),(54,144,72,126),(145,244,163,226),(146,245,164,227),(147,246,165,228),(148,247,166,229),(149,248,167,230),(150,249,168,231),(151,250,169,232),(152,251,170,233),(153,252,171,234),(154,235,172,217),(155,236,173,218),(156,237,174,219),(157,238,175,220),(158,239,176,221),(159,240,177,222),(160,241,178,223),(161,242,179,224),(162,243,180,225),(181,280,199,262),(182,281,200,263),(183,282,201,264),(184,283,202,265),(185,284,203,266),(186,285,204,267),(187,286,205,268),(188,287,206,269),(189,288,207,270),(190,271,208,253),(191,272,209,254),(192,273,210,255),(193,274,211,256),(194,275,212,257),(195,276,213,258),(196,277,214,259),(197,278,215,260),(198,279,216,261)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63),(73,118,82,109),(74,119,83,110),(75,120,84,111),(76,121,85,112),(77,122,86,113),(78,123,87,114),(79,124,88,115),(80,125,89,116),(81,126,90,117),(91,136,100,127),(92,137,101,128),(93,138,102,129),(94,139,103,130),(95,140,104,131),(96,141,105,132),(97,142,106,133),(98,143,107,134),(99,144,108,135),(145,190,154,181),(146,191,155,182),(147,192,156,183),(148,193,157,184),(149,194,158,185),(150,195,159,186),(151,196,160,187),(152,197,161,188),(153,198,162,189),(163,208,172,199),(164,209,173,200),(165,210,174,201),(166,211,175,202),(167,212,176,203),(168,213,177,204),(169,214,178,205),(170,215,179,206),(171,216,180,207),(217,262,226,253),(218,263,227,254),(219,264,228,255),(220,265,229,256),(221,266,230,257),(222,267,231,258),(223,268,232,259),(224,269,233,260),(225,270,234,261),(235,280,244,271),(236,281,245,272),(237,282,246,273),(238,283,247,274),(239,284,248,275),(240,285,249,276),(241,286,250,277),(242,287,251,278),(243,288,252,279)], [(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)], [(1,192,50,156,14,183,41,147),(2,191,51,155,15,182,42,146),(3,190,52,154,16,181,43,145),(4,198,53,162,17,189,44,153),(5,197,54,161,18,188,45,152),(6,196,46,160,10,187,37,151),(7,195,47,159,11,186,38,150),(8,194,48,158,12,185,39,149),(9,193,49,157,13,184,40,148),(19,214,64,178,28,205,55,169),(20,213,65,177,29,204,56,168),(21,212,66,176,30,203,57,167),(22,211,67,175,31,202,58,166),(23,210,68,174,32,201,59,165),(24,209,69,173,33,200,60,164),(25,208,70,172,34,199,61,163),(26,216,71,180,35,207,62,171),(27,215,72,179,36,206,63,170),(73,268,118,232,82,259,109,223),(74,267,119,231,83,258,110,222),(75,266,120,230,84,257,111,221),(76,265,121,229,85,256,112,220),(77,264,122,228,86,255,113,219),(78,263,123,227,87,254,114,218),(79,262,124,226,88,253,115,217),(80,270,125,234,89,261,116,225),(81,269,126,233,90,260,117,224),(91,286,136,250,100,277,127,241),(92,285,137,249,101,276,128,240),(93,284,138,248,102,275,129,239),(94,283,139,247,103,274,130,238),(95,282,140,246,104,273,131,237),(96,281,141,245,105,272,132,236),(97,280,142,244,106,271,133,235),(98,288,143,252,107,279,134,243),(99,287,144,251,108,278,135,242)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | M4(2) | D9 | C4×S3 | Dic9 | D18 | C4.Dic3 | C4×D9 | C4.Dic9 |
| kernel | C42.D9 | C2×C9⋊C8 | C4×C36 | C9⋊C8 | C2×C36 | C4×C12 | C2×C12 | C2×C12 | C18 | C42 | C12 | C2×C4 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 4 | 3 | 4 | 6 | 3 | 8 | 12 | 24 |
Matrix representation of C42.D9 ►in GL3(𝔽73) generated by
| 46 | 0 | 0 |
| 0 | 7 | 59 |
| 0 | 14 | 66 |
| 72 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 45 | 31 |
| 0 | 42 | 3 |
| 46 | 0 | 0 |
| 0 | 71 | 31 |
| 0 | 29 | 2 |
G:=sub<GL(3,GF(73))| [46,0,0,0,7,14,0,59,66],[72,0,0,0,27,0,0,0,27],[1,0,0,0,45,42,0,31,3],[46,0,0,0,71,29,0,31,2] >;
C42.D9 in GAP, Magma, Sage, TeX
C_4^2.D_9
% in TeX
G:=Group("C4^2.D9"); // GroupNames label
G:=SmallGroup(288,10);
// by ID
G=gap.SmallGroup(288,10);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=1,d^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export