direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×C4.Q8, C8⋊2C36, C72⋊6C4, C36.9Q8, C24.7C12, C18.11SD16, C4⋊C4.2C18, C4.1(Q8×C9), (C2×C8).6C18, C4.6(C2×C36), C12.9(C3×Q8), (C2×C72).16C2, (C2×C24).25C6, C36.43(C2×C4), (C2×C18).48D4, C18.12(C4⋊C4), C2.3(C9×SD16), C12.49(C2×C12), C6.11(C3×SD16), C22.10(D4×C9), (C2×C36).116C22, C2.3(C9×C4⋊C4), C3.(C3×C4.Q8), (C9×C4⋊C4).9C2, (C3×C4.Q8).C3, C6.12(C3×C4⋊C4), (C3×C4⋊C4).10C6, (C2×C6).57(C3×D4), (C2×C4).15(C2×C18), (C2×C12).136(C2×C6), SmallGroup(288,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4.Q8
G = < a,b,c,d | a9=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C9×C4.Q8
►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 86 77 67)(2 87 78 68)(3 88 79 69)(4 89 80 70)(5 90 81 71)(6 82 73 72)(7 83 74 64)(8 84 75 65)(9 85 76 66)(10 40 105 57)(11 41 106 58)(12 42 107 59)(13 43 108 60)(14 44 100 61)(15 45 101 62)(16 37 102 63)(17 38 103 55)(18 39 104 56)(19 54 237 285)(20 46 238 286)(21 47 239 287)(22 48 240 288)(23 49 241 280)(24 50 242 281)(25 51 243 282)(26 52 235 283)(27 53 236 284)(28 265 260 276)(29 266 261 277)(30 267 253 278)(31 268 254 279)(32 269 255 271)(33 270 256 272)(34 262 257 273)(35 263 258 274)(36 264 259 275)(91 158 149 139)(92 159 150 140)(93 160 151 141)(94 161 152 142)(95 162 153 143)(96 154 145 144)(97 155 146 136)(98 156 147 137)(99 157 148 138)(109 123 175 134)(110 124 176 135)(111 125 177 127)(112 126 178 128)(113 118 179 129)(114 119 180 130)(115 120 172 131)(116 121 173 132)(117 122 174 133)(163 211 221 230)(164 212 222 231)(165 213 223 232)(166 214 224 233)(167 215 225 234)(168 216 217 226)(169 208 218 227)(170 209 219 228)(171 210 220 229)(181 206 247 195)(182 207 248 196)(183 199 249 197)(184 200 250 198)(185 201 251 190)(186 202 252 191)(187 203 244 192)(188 204 245 193)(189 205 246 194)
(1 113 12 149 77 179 107 91)(2 114 13 150 78 180 108 92)(3 115 14 151 79 172 100 93)(4 116 15 152 80 173 101 94)(5 117 16 153 81 174 102 95)(6 109 17 145 73 175 103 96)(7 110 18 146 74 176 104 97)(8 111 10 147 75 177 105 98)(9 112 11 148 76 178 106 99)(19 244 259 223 237 187 36 165)(20 245 260 224 238 188 28 166)(21 246 261 225 239 189 29 167)(22 247 253 217 240 181 30 168)(23 248 254 218 241 182 31 169)(24 249 255 219 242 183 32 170)(25 250 256 220 243 184 33 171)(26 251 257 221 235 185 34 163)(27 252 258 222 236 186 35 164)(37 143 71 133 63 162 90 122)(38 144 72 134 55 154 82 123)(39 136 64 135 56 155 83 124)(40 137 65 127 57 156 84 125)(41 138 66 128 58 157 85 126)(42 139 67 129 59 158 86 118)(43 140 68 130 60 159 87 119)(44 141 69 131 61 160 88 120)(45 142 70 132 62 161 89 121)(46 193 276 233 286 204 265 214)(47 194 277 234 287 205 266 215)(48 195 278 226 288 206 267 216)(49 196 279 227 280 207 268 208)(50 197 271 228 281 199 269 209)(51 198 272 229 282 200 270 210)(52 190 273 230 283 201 262 211)(53 191 274 231 284 202 263 212)(54 192 275 232 285 203 264 213)
(1 201 59 163)(2 202 60 164)(3 203 61 165)(4 204 62 166)(5 205 63 167)(6 206 55 168)(7 207 56 169)(8 199 57 170)(9 200 58 171)(10 228 84 183)(11 229 85 184)(12 230 86 185)(13 231 87 186)(14 232 88 187)(15 233 89 188)(16 234 90 189)(17 226 82 181)(18 227 83 182)(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)(28 173 286 142)(29 174 287 143)(30 175 288 144)(31 176 280 136)(32 177 281 137)(33 178 282 138)(34 179 283 139)(35 180 284 140)(36 172 285 141)(37 225 81 194)(38 217 73 195)(39 218 74 196)(40 219 75 197)(41 220 76 198)(42 221 77 190)(43 222 78 191)(44 223 79 192)(45 224 80 193)(46 161 260 116)(47 162 261 117)(48 154 253 109)(49 155 254 110)(50 156 255 111)(51 157 256 112)(52 158 257 113)(53 159 258 114)(54 160 259 115)(64 248 104 208)(65 249 105 209)(66 250 106 210)(67 251 107 211)(68 252 108 212)(69 244 100 213)(70 245 101 214)(71 246 102 215)(72 247 103 216)(91 273 129 235)(92 274 130 236)(93 275 131 237)(94 276 132 238)(95 277 133 239)(96 278 134 240)(97 279 135 241)(98 271 127 242)(99 272 128 243)
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), (1,86,77,67)(2,87,78,68)(3,88,79,69)(4,89,80,70)(5,90,81,71)(6,82,73,72)(7,83,74,64)(8,84,75,65)(9,85,76,66)(10,40,105,57)(11,41,106,58)(12,42,107,59)(13,43,108,60)(14,44,100,61)(15,45,101,62)(16,37,102,63)(17,38,103,55)(18,39,104,56)(19,54,237,285)(20,46,238,286)(21,47,239,287)(22,48,240,288)(23,49,241,280)(24,50,242,281)(25,51,243,282)(26,52,235,283)(27,53,236,284)(28,265,260,276)(29,266,261,277)(30,267,253,278)(31,268,254,279)(32,269,255,271)(33,270,256,272)(34,262,257,273)(35,263,258,274)(36,264,259,275)(91,158,149,139)(92,159,150,140)(93,160,151,141)(94,161,152,142)(95,162,153,143)(96,154,145,144)(97,155,146,136)(98,156,147,137)(99,157,148,138)(109,123,175,134)(110,124,176,135)(111,125,177,127)(112,126,178,128)(113,118,179,129)(114,119,180,130)(115,120,172,131)(116,121,173,132)(117,122,174,133)(163,211,221,230)(164,212,222,231)(165,213,223,232)(166,214,224,233)(167,215,225,234)(168,216,217,226)(169,208,218,227)(170,209,219,228)(171,210,220,229)(181,206,247,195)(182,207,248,196)(183,199,249,197)(184,200,250,198)(185,201,251,190)(186,202,252,191)(187,203,244,192)(188,204,245,193)(189,205,246,194), (1,113,12,149,77,179,107,91)(2,114,13,150,78,180,108,92)(3,115,14,151,79,172,100,93)(4,116,15,152,80,173,101,94)(5,117,16,153,81,174,102,95)(6,109,17,145,73,175,103,96)(7,110,18,146,74,176,104,97)(8,111,10,147,75,177,105,98)(9,112,11,148,76,178,106,99)(19,244,259,223,237,187,36,165)(20,245,260,224,238,188,28,166)(21,246,261,225,239,189,29,167)(22,247,253,217,240,181,30,168)(23,248,254,218,241,182,31,169)(24,249,255,219,242,183,32,170)(25,250,256,220,243,184,33,171)(26,251,257,221,235,185,34,163)(27,252,258,222,236,186,35,164)(37,143,71,133,63,162,90,122)(38,144,72,134,55,154,82,123)(39,136,64,135,56,155,83,124)(40,137,65,127,57,156,84,125)(41,138,66,128,58,157,85,126)(42,139,67,129,59,158,86,118)(43,140,68,130,60,159,87,119)(44,141,69,131,61,160,88,120)(45,142,70,132,62,161,89,121)(46,193,276,233,286,204,265,214)(47,194,277,234,287,205,266,215)(48,195,278,226,288,206,267,216)(49,196,279,227,280,207,268,208)(50,197,271,228,281,199,269,209)(51,198,272,229,282,200,270,210)(52,190,273,230,283,201,262,211)(53,191,274,231,284,202,263,212)(54,192,275,232,285,203,264,213), (1,201,59,163)(2,202,60,164)(3,203,61,165)(4,204,62,166)(5,205,63,167)(6,206,55,168)(7,207,56,169)(8,199,57,170)(9,200,58,171)(10,228,84,183)(11,229,85,184)(12,230,86,185)(13,231,87,186)(14,232,88,187)(15,233,89,188)(16,234,90,189)(17,226,82,181)(18,227,83,182)(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)(28,173,286,142)(29,174,287,143)(30,175,288,144)(31,176,280,136)(32,177,281,137)(33,178,282,138)(34,179,283,139)(35,180,284,140)(36,172,285,141)(37,225,81,194)(38,217,73,195)(39,218,74,196)(40,219,75,197)(41,220,76,198)(42,221,77,190)(43,222,78,191)(44,223,79,192)(45,224,80,193)(46,161,260,116)(47,162,261,117)(48,154,253,109)(49,155,254,110)(50,156,255,111)(51,157,256,112)(52,158,257,113)(53,159,258,114)(54,160,259,115)(64,248,104,208)(65,249,105,209)(66,250,106,210)(67,251,107,211)(68,252,108,212)(69,244,100,213)(70,245,101,214)(71,246,102,215)(72,247,103,216)(91,273,129,235)(92,274,130,236)(93,275,131,237)(94,276,132,238)(95,277,133,239)(96,278,134,240)(97,279,135,241)(98,271,127,242)(99,272,128,243)>;
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), (1,86,77,67)(2,87,78,68)(3,88,79,69)(4,89,80,70)(5,90,81,71)(6,82,73,72)(7,83,74,64)(8,84,75,65)(9,85,76,66)(10,40,105,57)(11,41,106,58)(12,42,107,59)(13,43,108,60)(14,44,100,61)(15,45,101,62)(16,37,102,63)(17,38,103,55)(18,39,104,56)(19,54,237,285)(20,46,238,286)(21,47,239,287)(22,48,240,288)(23,49,241,280)(24,50,242,281)(25,51,243,282)(26,52,235,283)(27,53,236,284)(28,265,260,276)(29,266,261,277)(30,267,253,278)(31,268,254,279)(32,269,255,271)(33,270,256,272)(34,262,257,273)(35,263,258,274)(36,264,259,275)(91,158,149,139)(92,159,150,140)(93,160,151,141)(94,161,152,142)(95,162,153,143)(96,154,145,144)(97,155,146,136)(98,156,147,137)(99,157,148,138)(109,123,175,134)(110,124,176,135)(111,125,177,127)(112,126,178,128)(113,118,179,129)(114,119,180,130)(115,120,172,131)(116,121,173,132)(117,122,174,133)(163,211,221,230)(164,212,222,231)(165,213,223,232)(166,214,224,233)(167,215,225,234)(168,216,217,226)(169,208,218,227)(170,209,219,228)(171,210,220,229)(181,206,247,195)(182,207,248,196)(183,199,249,197)(184,200,250,198)(185,201,251,190)(186,202,252,191)(187,203,244,192)(188,204,245,193)(189,205,246,194), (1,113,12,149,77,179,107,91)(2,114,13,150,78,180,108,92)(3,115,14,151,79,172,100,93)(4,116,15,152,80,173,101,94)(5,117,16,153,81,174,102,95)(6,109,17,145,73,175,103,96)(7,110,18,146,74,176,104,97)(8,111,10,147,75,177,105,98)(9,112,11,148,76,178,106,99)(19,244,259,223,237,187,36,165)(20,245,260,224,238,188,28,166)(21,246,261,225,239,189,29,167)(22,247,253,217,240,181,30,168)(23,248,254,218,241,182,31,169)(24,249,255,219,242,183,32,170)(25,250,256,220,243,184,33,171)(26,251,257,221,235,185,34,163)(27,252,258,222,236,186,35,164)(37,143,71,133,63,162,90,122)(38,144,72,134,55,154,82,123)(39,136,64,135,56,155,83,124)(40,137,65,127,57,156,84,125)(41,138,66,128,58,157,85,126)(42,139,67,129,59,158,86,118)(43,140,68,130,60,159,87,119)(44,141,69,131,61,160,88,120)(45,142,70,132,62,161,89,121)(46,193,276,233,286,204,265,214)(47,194,277,234,287,205,266,215)(48,195,278,226,288,206,267,216)(49,196,279,227,280,207,268,208)(50,197,271,228,281,199,269,209)(51,198,272,229,282,200,270,210)(52,190,273,230,283,201,262,211)(53,191,274,231,284,202,263,212)(54,192,275,232,285,203,264,213), (1,201,59,163)(2,202,60,164)(3,203,61,165)(4,204,62,166)(5,205,63,167)(6,206,55,168)(7,207,56,169)(8,199,57,170)(9,200,58,171)(10,228,84,183)(11,229,85,184)(12,230,86,185)(13,231,87,186)(14,232,88,187)(15,233,89,188)(16,234,90,189)(17,226,82,181)(18,227,83,182)(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)(28,173,286,142)(29,174,287,143)(30,175,288,144)(31,176,280,136)(32,177,281,137)(33,178,282,138)(34,179,283,139)(35,180,284,140)(36,172,285,141)(37,225,81,194)(38,217,73,195)(39,218,74,196)(40,219,75,197)(41,220,76,198)(42,221,77,190)(43,222,78,191)(44,223,79,192)(45,224,80,193)(46,161,260,116)(47,162,261,117)(48,154,253,109)(49,155,254,110)(50,156,255,111)(51,157,256,112)(52,158,257,113)(53,159,258,114)(54,160,259,115)(64,248,104,208)(65,249,105,209)(66,250,106,210)(67,251,107,211)(68,252,108,212)(69,244,100,213)(70,245,101,214)(71,246,102,215)(72,247,103,216)(91,273,129,235)(92,274,130,236)(93,275,131,237)(94,276,132,238)(95,277,133,239)(96,278,134,240)(97,279,135,241)(98,271,127,242)(99,272,128,243) );
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)], [(1,86,77,67),(2,87,78,68),(3,88,79,69),(4,89,80,70),(5,90,81,71),(6,82,73,72),(7,83,74,64),(8,84,75,65),(9,85,76,66),(10,40,105,57),(11,41,106,58),(12,42,107,59),(13,43,108,60),(14,44,100,61),(15,45,101,62),(16,37,102,63),(17,38,103,55),(18,39,104,56),(19,54,237,285),(20,46,238,286),(21,47,239,287),(22,48,240,288),(23,49,241,280),(24,50,242,281),(25,51,243,282),(26,52,235,283),(27,53,236,284),(28,265,260,276),(29,266,261,277),(30,267,253,278),(31,268,254,279),(32,269,255,271),(33,270,256,272),(34,262,257,273),(35,263,258,274),(36,264,259,275),(91,158,149,139),(92,159,150,140),(93,160,151,141),(94,161,152,142),(95,162,153,143),(96,154,145,144),(97,155,146,136),(98,156,147,137),(99,157,148,138),(109,123,175,134),(110,124,176,135),(111,125,177,127),(112,126,178,128),(113,118,179,129),(114,119,180,130),(115,120,172,131),(116,121,173,132),(117,122,174,133),(163,211,221,230),(164,212,222,231),(165,213,223,232),(166,214,224,233),(167,215,225,234),(168,216,217,226),(169,208,218,227),(170,209,219,228),(171,210,220,229),(181,206,247,195),(182,207,248,196),(183,199,249,197),(184,200,250,198),(185,201,251,190),(186,202,252,191),(187,203,244,192),(188,204,245,193),(189,205,246,194)], [(1,113,12,149,77,179,107,91),(2,114,13,150,78,180,108,92),(3,115,14,151,79,172,100,93),(4,116,15,152,80,173,101,94),(5,117,16,153,81,174,102,95),(6,109,17,145,73,175,103,96),(7,110,18,146,74,176,104,97),(8,111,10,147,75,177,105,98),(9,112,11,148,76,178,106,99),(19,244,259,223,237,187,36,165),(20,245,260,224,238,188,28,166),(21,246,261,225,239,189,29,167),(22,247,253,217,240,181,30,168),(23,248,254,218,241,182,31,169),(24,249,255,219,242,183,32,170),(25,250,256,220,243,184,33,171),(26,251,257,221,235,185,34,163),(27,252,258,222,236,186,35,164),(37,143,71,133,63,162,90,122),(38,144,72,134,55,154,82,123),(39,136,64,135,56,155,83,124),(40,137,65,127,57,156,84,125),(41,138,66,128,58,157,85,126),(42,139,67,129,59,158,86,118),(43,140,68,130,60,159,87,119),(44,141,69,131,61,160,88,120),(45,142,70,132,62,161,89,121),(46,193,276,233,286,204,265,214),(47,194,277,234,287,205,266,215),(48,195,278,226,288,206,267,216),(49,196,279,227,280,207,268,208),(50,197,271,228,281,199,269,209),(51,198,272,229,282,200,270,210),(52,190,273,230,283,201,262,211),(53,191,274,231,284,202,263,212),(54,192,275,232,285,203,264,213)], [(1,201,59,163),(2,202,60,164),(3,203,61,165),(4,204,62,166),(5,205,63,167),(6,206,55,168),(7,207,56,169),(8,199,57,170),(9,200,58,171),(10,228,84,183),(11,229,85,184),(12,230,86,185),(13,231,87,186),(14,232,88,187),(15,233,89,188),(16,234,90,189),(17,226,82,181),(18,227,83,182),(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),(28,173,286,142),(29,174,287,143),(30,175,288,144),(31,176,280,136),(32,177,281,137),(33,178,282,138),(34,179,283,139),(35,180,284,140),(36,172,285,141),(37,225,81,194),(38,217,73,195),(39,218,74,196),(40,219,75,197),(41,220,76,198),(42,221,77,190),(43,222,78,191),(44,223,79,192),(45,224,80,193),(46,161,260,116),(47,162,261,117),(48,154,253,109),(49,155,254,110),(50,156,255,111),(51,157,256,112),(52,158,257,113),(53,159,258,114),(54,160,259,115),(64,248,104,208),(65,249,105,209),(66,250,106,210),(67,251,107,211),(68,252,108,212),(69,244,100,213),(70,245,101,214),(71,246,102,215),(72,247,103,216),(91,273,129,235),(92,274,130,236),(93,275,131,237),(94,276,132,238),(95,277,133,239),(96,278,134,240),(97,279,135,241),(98,271,127,242),(99,272,128,243)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 18A | ··· | 18R | 24A | ··· | 24H | 36A | ··· | 36L | 36M | ··· | 36AJ | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | Q8 | D4 | SD16 | C3×Q8 | C3×D4 | C3×SD16 | Q8×C9 | D4×C9 | C9×SD16 |
| kernel | C9×C4.Q8 | C9×C4⋊C4 | C2×C72 | C3×C4.Q8 | C72 | C3×C4⋊C4 | C2×C24 | C4.Q8 | C24 | C4⋊C4 | C2×C8 | C8 | C36 | C2×C18 | C18 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 24 | 1 | 1 | 4 | 2 | 2 | 8 | 6 | 6 | 24 |
Matrix representation of C9×C4.Q8 ►in GL4(𝔽73) generated by
| 16 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 67 | 6 |
| 0 | 0 | 67 | 67 |
| 1 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 32 |
| 0 | 0 | 32 | 27 |
G:=sub<GL(4,GF(73))| [16,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,67,67,0,0,6,67],[1,0,0,0,0,27,0,0,0,0,46,32,0,0,32,27] >;
C9×C4.Q8 in GAP, Magma, Sage, TeX
C_9\times C_4.Q_8
% in TeX
G:=Group("C9xC4.Q8"); // GroupNames label
G:=SmallGroup(288,56);
// by ID
G=gap.SmallGroup(288,56);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,92,268,4371,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
Export