direct product, non-abelian, soluble
Aliases: C4×Q8⋊C9, Q8⋊C36, C12.SL2(𝔽3), (C4×Q8)⋊C9, C6.4(C4×A4), (Q8×C12).C3, (C2×C12).1A4, (C6×Q8).4C6, C6.2(C4.A4), (C2×Q8).1C18, (C3×Q8).2C12, C2.(Q8.C18), C3.(C4×SL2(𝔽3)), C6.2(C2×SL2(𝔽3)), C2.(C2×Q8⋊C9), C2.2(C4×C3.A4), (C2×Q8⋊C9).3C2, (C2×C6).22(C2×A4), (C2×C4).1(C3.A4), C22.6(C2×C3.A4), SmallGroup(288,72)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C4×Q8⋊C9 |
Generators and relations for C4×Q8⋊C9
G = < a,b,c,d | a4=b4=d9=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Smallest permutation representation of C4×Q8⋊C9
►Regular action on 288 points
Generators in S288
(1 55 139 181)(2 56 140 182)(3 57 141 183)(4 58 142 184)(5 59 143 185)(6 60 144 186)(7 61 136 187)(8 62 137 188)(9 63 138 189)(10 134 193 164)(11 135 194 165)(12 127 195 166)(13 128 196 167)(14 129 197 168)(15 130 198 169)(16 131 190 170)(17 132 191 171)(18 133 192 163)(19 284 156 265)(20 285 157 266)(21 286 158 267)(22 287 159 268)(23 288 160 269)(24 280 161 270)(25 281 162 262)(26 282 154 263)(27 283 155 264)(28 52 258 178)(29 53 259 179)(30 54 260 180)(31 46 261 172)(32 47 253 173)(33 48 254 174)(34 49 255 175)(35 50 256 176)(36 51 257 177)(37 93 204 275)(38 94 205 276)(39 95 206 277)(40 96 207 278)(41 97 199 279)(42 98 200 271)(43 99 201 272)(44 91 202 273)(45 92 203 274)(64 81 229 151)(65 73 230 152)(66 74 231 153)(67 75 232 145)(68 76 233 146)(69 77 234 147)(70 78 226 148)(71 79 227 149)(72 80 228 150)(82 111 208 237)(83 112 209 238)(84 113 210 239)(85 114 211 240)(86 115 212 241)(87 116 213 242)(88 117 214 243)(89 109 215 235)(90 110 216 236)(100 246 217 125)(101 247 218 126)(102 248 219 118)(103 249 220 119)(104 250 221 120)(105 251 222 121)(106 252 223 122)(107 244 224 123)(108 245 225 124)
(1 83 27 30)(2 248 19 202)(3 166 20 74)(4 86 21 33)(5 251 22 205)(6 169 23 77)(7 89 24 36)(8 245 25 199)(9 163 26 80)(10 218 229 272)(11 46 230 113)(12 285 231 57)(13 221 232 275)(14 49 233 116)(15 288 234 60)(16 224 226 278)(17 52 227 110)(18 282 228 63)(28 79 90 171)(29 246 82 200)(31 73 84 165)(32 249 85 203)(34 76 87 168)(35 252 88 206)(37 128 120 145)(38 143 121 159)(39 256 122 214)(40 131 123 148)(41 137 124 162)(42 259 125 208)(43 134 126 151)(44 140 118 156)(45 253 119 211)(47 220 114 274)(48 58 115 286)(50 223 117 277)(51 61 109 280)(53 217 111 271)(54 55 112 283)(56 219 284 273)(59 222 287 276)(62 225 281 279)(64 99 193 101)(65 239 194 172)(66 183 195 266)(67 93 196 104)(68 242 197 175)(69 186 198 269)(70 96 190 107)(71 236 191 178)(72 189 192 263)(75 204 167 250)(78 207 170 244)(81 201 164 247)(91 182 102 265)(92 173 103 240)(94 185 105 268)(95 176 106 243)(97 188 108 262)(98 179 100 237)(127 157 153 141)(129 255 146 213)(130 160 147 144)(132 258 149 216)(133 154 150 138)(135 261 152 210)(136 215 161 257)(139 209 155 260)(142 212 158 254)(174 184 241 267)(177 187 235 270)(180 181 238 264)
(1 247 27 201)(2 165 19 73)(3 85 20 32)(4 250 21 204)(5 168 22 76)(6 88 23 35)(7 244 24 207)(8 171 25 79)(9 82 26 29)(10 54 229 112)(11 284 230 56)(12 220 231 274)(13 48 232 115)(14 287 233 59)(15 223 234 277)(16 51 226 109)(17 281 227 62)(18 217 228 271)(28 245 90 199)(30 81 83 164)(31 248 84 202)(33 75 86 167)(34 251 87 205)(36 78 89 170)(37 142 120 158)(38 255 121 213)(39 130 122 147)(40 136 123 161)(41 258 124 216)(42 133 125 150)(43 139 126 155)(44 261 118 210)(45 127 119 153)(46 219 113 273)(47 57 114 285)(49 222 116 276)(50 60 117 288)(52 225 110 279)(53 63 111 282)(55 218 283 272)(58 221 286 275)(61 224 280 278)(64 238 193 180)(65 182 194 265)(66 92 195 103)(67 241 196 174)(68 185 197 268)(69 95 198 106)(70 235 190 177)(71 188 191 262)(72 98 192 100)(74 203 166 249)(77 206 169 252)(80 200 163 246)(91 172 102 239)(93 184 104 267)(94 175 105 242)(96 187 107 270)(97 178 108 236)(99 181 101 264)(128 254 145 212)(129 159 146 143)(131 257 148 215)(132 162 149 137)(134 260 151 209)(135 156 152 140)(138 208 154 259)(141 211 157 253)(144 214 160 256)(173 183 240 266)(176 186 243 269)(179 189 237 263)
(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)| (1,55,139,181)(2,56,140,182)(3,57,141,183)(4,58,142,184)(5,59,143,185)(6,60,144,186)(7,61,136,187)(8,62,137,188)(9,63,138,189)(10,134,193,164)(11,135,194,165)(12,127,195,166)(13,128,196,167)(14,129,197,168)(15,130,198,169)(16,131,190,170)(17,132,191,171)(18,133,192,163)(19,284,156,265)(20,285,157,266)(21,286,158,267)(22,287,159,268)(23,288,160,269)(24,280,161,270)(25,281,162,262)(26,282,154,263)(27,283,155,264)(28,52,258,178)(29,53,259,179)(30,54,260,180)(31,46,261,172)(32,47,253,173)(33,48,254,174)(34,49,255,175)(35,50,256,176)(36,51,257,177)(37,93,204,275)(38,94,205,276)(39,95,206,277)(40,96,207,278)(41,97,199,279)(42,98,200,271)(43,99,201,272)(44,91,202,273)(45,92,203,274)(64,81,229,151)(65,73,230,152)(66,74,231,153)(67,75,232,145)(68,76,233,146)(69,77,234,147)(70,78,226,148)(71,79,227,149)(72,80,228,150)(82,111,208,237)(83,112,209,238)(84,113,210,239)(85,114,211,240)(86,115,212,241)(87,116,213,242)(88,117,214,243)(89,109,215,235)(90,110,216,236)(100,246,217,125)(101,247,218,126)(102,248,219,118)(103,249,220,119)(104,250,221,120)(105,251,222,121)(106,252,223,122)(107,244,224,123)(108,245,225,124), (1,83,27,30)(2,248,19,202)(3,166,20,74)(4,86,21,33)(5,251,22,205)(6,169,23,77)(7,89,24,36)(8,245,25,199)(9,163,26,80)(10,218,229,272)(11,46,230,113)(12,285,231,57)(13,221,232,275)(14,49,233,116)(15,288,234,60)(16,224,226,278)(17,52,227,110)(18,282,228,63)(28,79,90,171)(29,246,82,200)(31,73,84,165)(32,249,85,203)(34,76,87,168)(35,252,88,206)(37,128,120,145)(38,143,121,159)(39,256,122,214)(40,131,123,148)(41,137,124,162)(42,259,125,208)(43,134,126,151)(44,140,118,156)(45,253,119,211)(47,220,114,274)(48,58,115,286)(50,223,117,277)(51,61,109,280)(53,217,111,271)(54,55,112,283)(56,219,284,273)(59,222,287,276)(62,225,281,279)(64,99,193,101)(65,239,194,172)(66,183,195,266)(67,93,196,104)(68,242,197,175)(69,186,198,269)(70,96,190,107)(71,236,191,178)(72,189,192,263)(75,204,167,250)(78,207,170,244)(81,201,164,247)(91,182,102,265)(92,173,103,240)(94,185,105,268)(95,176,106,243)(97,188,108,262)(98,179,100,237)(127,157,153,141)(129,255,146,213)(130,160,147,144)(132,258,149,216)(133,154,150,138)(135,261,152,210)(136,215,161,257)(139,209,155,260)(142,212,158,254)(174,184,241,267)(177,187,235,270)(180,181,238,264), (1,247,27,201)(2,165,19,73)(3,85,20,32)(4,250,21,204)(5,168,22,76)(6,88,23,35)(7,244,24,207)(8,171,25,79)(9,82,26,29)(10,54,229,112)(11,284,230,56)(12,220,231,274)(13,48,232,115)(14,287,233,59)(15,223,234,277)(16,51,226,109)(17,281,227,62)(18,217,228,271)(28,245,90,199)(30,81,83,164)(31,248,84,202)(33,75,86,167)(34,251,87,205)(36,78,89,170)(37,142,120,158)(38,255,121,213)(39,130,122,147)(40,136,123,161)(41,258,124,216)(42,133,125,150)(43,139,126,155)(44,261,118,210)(45,127,119,153)(46,219,113,273)(47,57,114,285)(49,222,116,276)(50,60,117,288)(52,225,110,279)(53,63,111,282)(55,218,283,272)(58,221,286,275)(61,224,280,278)(64,238,193,180)(65,182,194,265)(66,92,195,103)(67,241,196,174)(68,185,197,268)(69,95,198,106)(70,235,190,177)(71,188,191,262)(72,98,192,100)(74,203,166,249)(77,206,169,252)(80,200,163,246)(91,172,102,239)(93,184,104,267)(94,175,105,242)(96,187,107,270)(97,178,108,236)(99,181,101,264)(128,254,145,212)(129,159,146,143)(131,257,148,215)(132,162,149,137)(134,260,151,209)(135,156,152,140)(138,208,154,259)(141,211,157,253)(144,214,160,256)(173,183,240,266)(176,186,243,269)(179,189,237,263), (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( (1,55,139,181)(2,56,140,182)(3,57,141,183)(4,58,142,184)(5,59,143,185)(6,60,144,186)(7,61,136,187)(8,62,137,188)(9,63,138,189)(10,134,193,164)(11,135,194,165)(12,127,195,166)(13,128,196,167)(14,129,197,168)(15,130,198,169)(16,131,190,170)(17,132,191,171)(18,133,192,163)(19,284,156,265)(20,285,157,266)(21,286,158,267)(22,287,159,268)(23,288,160,269)(24,280,161,270)(25,281,162,262)(26,282,154,263)(27,283,155,264)(28,52,258,178)(29,53,259,179)(30,54,260,180)(31,46,261,172)(32,47,253,173)(33,48,254,174)(34,49,255,175)(35,50,256,176)(36,51,257,177)(37,93,204,275)(38,94,205,276)(39,95,206,277)(40,96,207,278)(41,97,199,279)(42,98,200,271)(43,99,201,272)(44,91,202,273)(45,92,203,274)(64,81,229,151)(65,73,230,152)(66,74,231,153)(67,75,232,145)(68,76,233,146)(69,77,234,147)(70,78,226,148)(71,79,227,149)(72,80,228,150)(82,111,208,237)(83,112,209,238)(84,113,210,239)(85,114,211,240)(86,115,212,241)(87,116,213,242)(88,117,214,243)(89,109,215,235)(90,110,216,236)(100,246,217,125)(101,247,218,126)(102,248,219,118)(103,249,220,119)(104,250,221,120)(105,251,222,121)(106,252,223,122)(107,244,224,123)(108,245,225,124), (1,83,27,30)(2,248,19,202)(3,166,20,74)(4,86,21,33)(5,251,22,205)(6,169,23,77)(7,89,24,36)(8,245,25,199)(9,163,26,80)(10,218,229,272)(11,46,230,113)(12,285,231,57)(13,221,232,275)(14,49,233,116)(15,288,234,60)(16,224,226,278)(17,52,227,110)(18,282,228,63)(28,79,90,171)(29,246,82,200)(31,73,84,165)(32,249,85,203)(34,76,87,168)(35,252,88,206)(37,128,120,145)(38,143,121,159)(39,256,122,214)(40,131,123,148)(41,137,124,162)(42,259,125,208)(43,134,126,151)(44,140,118,156)(45,253,119,211)(47,220,114,274)(48,58,115,286)(50,223,117,277)(51,61,109,280)(53,217,111,271)(54,55,112,283)(56,219,284,273)(59,222,287,276)(62,225,281,279)(64,99,193,101)(65,239,194,172)(66,183,195,266)(67,93,196,104)(68,242,197,175)(69,186,198,269)(70,96,190,107)(71,236,191,178)(72,189,192,263)(75,204,167,250)(78,207,170,244)(81,201,164,247)(91,182,102,265)(92,173,103,240)(94,185,105,268)(95,176,106,243)(97,188,108,262)(98,179,100,237)(127,157,153,141)(129,255,146,213)(130,160,147,144)(132,258,149,216)(133,154,150,138)(135,261,152,210)(136,215,161,257)(139,209,155,260)(142,212,158,254)(174,184,241,267)(177,187,235,270)(180,181,238,264), (1,247,27,201)(2,165,19,73)(3,85,20,32)(4,250,21,204)(5,168,22,76)(6,88,23,35)(7,244,24,207)(8,171,25,79)(9,82,26,29)(10,54,229,112)(11,284,230,56)(12,220,231,274)(13,48,232,115)(14,287,233,59)(15,223,234,277)(16,51,226,109)(17,281,227,62)(18,217,228,271)(28,245,90,199)(30,81,83,164)(31,248,84,202)(33,75,86,167)(34,251,87,205)(36,78,89,170)(37,142,120,158)(38,255,121,213)(39,130,122,147)(40,136,123,161)(41,258,124,216)(42,133,125,150)(43,139,126,155)(44,261,118,210)(45,127,119,153)(46,219,113,273)(47,57,114,285)(49,222,116,276)(50,60,117,288)(52,225,110,279)(53,63,111,282)(55,218,283,272)(58,221,286,275)(61,224,280,278)(64,238,193,180)(65,182,194,265)(66,92,195,103)(67,241,196,174)(68,185,197,268)(69,95,198,106)(70,235,190,177)(71,188,191,262)(72,98,192,100)(74,203,166,249)(77,206,169,252)(80,200,163,246)(91,172,102,239)(93,184,104,267)(94,175,105,242)(96,187,107,270)(97,178,108,236)(99,181,101,264)(128,254,145,212)(129,159,146,143)(131,257,148,215)(132,162,149,137)(134,260,151,209)(135,156,152,140)(138,208,154,259)(141,211,157,253)(144,214,160,256)(173,183,240,266)(176,186,243,269)(179,189,237,263), (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([[(1,55,139,181),(2,56,140,182),(3,57,141,183),(4,58,142,184),(5,59,143,185),(6,60,144,186),(7,61,136,187),(8,62,137,188),(9,63,138,189),(10,134,193,164),(11,135,194,165),(12,127,195,166),(13,128,196,167),(14,129,197,168),(15,130,198,169),(16,131,190,170),(17,132,191,171),(18,133,192,163),(19,284,156,265),(20,285,157,266),(21,286,158,267),(22,287,159,268),(23,288,160,269),(24,280,161,270),(25,281,162,262),(26,282,154,263),(27,283,155,264),(28,52,258,178),(29,53,259,179),(30,54,260,180),(31,46,261,172),(32,47,253,173),(33,48,254,174),(34,49,255,175),(35,50,256,176),(36,51,257,177),(37,93,204,275),(38,94,205,276),(39,95,206,277),(40,96,207,278),(41,97,199,279),(42,98,200,271),(43,99,201,272),(44,91,202,273),(45,92,203,274),(64,81,229,151),(65,73,230,152),(66,74,231,153),(67,75,232,145),(68,76,233,146),(69,77,234,147),(70,78,226,148),(71,79,227,149),(72,80,228,150),(82,111,208,237),(83,112,209,238),(84,113,210,239),(85,114,211,240),(86,115,212,241),(87,116,213,242),(88,117,214,243),(89,109,215,235),(90,110,216,236),(100,246,217,125),(101,247,218,126),(102,248,219,118),(103,249,220,119),(104,250,221,120),(105,251,222,121),(106,252,223,122),(107,244,224,123),(108,245,225,124)], [(1,83,27,30),(2,248,19,202),(3,166,20,74),(4,86,21,33),(5,251,22,205),(6,169,23,77),(7,89,24,36),(8,245,25,199),(9,163,26,80),(10,218,229,272),(11,46,230,113),(12,285,231,57),(13,221,232,275),(14,49,233,116),(15,288,234,60),(16,224,226,278),(17,52,227,110),(18,282,228,63),(28,79,90,171),(29,246,82,200),(31,73,84,165),(32,249,85,203),(34,76,87,168),(35,252,88,206),(37,128,120,145),(38,143,121,159),(39,256,122,214),(40,131,123,148),(41,137,124,162),(42,259,125,208),(43,134,126,151),(44,140,118,156),(45,253,119,211),(47,220,114,274),(48,58,115,286),(50,223,117,277),(51,61,109,280),(53,217,111,271),(54,55,112,283),(56,219,284,273),(59,222,287,276),(62,225,281,279),(64,99,193,101),(65,239,194,172),(66,183,195,266),(67,93,196,104),(68,242,197,175),(69,186,198,269),(70,96,190,107),(71,236,191,178),(72,189,192,263),(75,204,167,250),(78,207,170,244),(81,201,164,247),(91,182,102,265),(92,173,103,240),(94,185,105,268),(95,176,106,243),(97,188,108,262),(98,179,100,237),(127,157,153,141),(129,255,146,213),(130,160,147,144),(132,258,149,216),(133,154,150,138),(135,261,152,210),(136,215,161,257),(139,209,155,260),(142,212,158,254),(174,184,241,267),(177,187,235,270),(180,181,238,264)], [(1,247,27,201),(2,165,19,73),(3,85,20,32),(4,250,21,204),(5,168,22,76),(6,88,23,35),(7,244,24,207),(8,171,25,79),(9,82,26,29),(10,54,229,112),(11,284,230,56),(12,220,231,274),(13,48,232,115),(14,287,233,59),(15,223,234,277),(16,51,226,109),(17,281,227,62),(18,217,228,271),(28,245,90,199),(30,81,83,164),(31,248,84,202),(33,75,86,167),(34,251,87,205),(36,78,89,170),(37,142,120,158),(38,255,121,213),(39,130,122,147),(40,136,123,161),(41,258,124,216),(42,133,125,150),(43,139,126,155),(44,261,118,210),(45,127,119,153),(46,219,113,273),(47,57,114,285),(49,222,116,276),(50,60,117,288),(52,225,110,279),(53,63,111,282),(55,218,283,272),(58,221,286,275),(61,224,280,278),(64,238,193,180),(65,182,194,265),(66,92,195,103),(67,241,196,174),(68,185,197,268),(69,95,198,106),(70,235,190,177),(71,188,191,262),(72,98,192,100),(74,203,166,249),(77,206,169,252),(80,200,163,246),(91,172,102,239),(93,184,104,267),(94,175,105,242),(96,187,107,270),(97,178,108,236),(99,181,101,264),(128,254,145,212),(129,159,146,143),(131,257,148,215),(132,162,149,137),(134,260,151,209),(135,156,152,140),(138,208,154,259),(141,211,157,253),(144,214,160,256),(173,183,240,266),(176,186,243,269),(179,189,237,263)], [(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 | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12P | 18A | ··· | 18R | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | - | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | SL2(𝔽3) | SL2(𝔽3) | C4.A4 | Q8⋊C9 | Q8.C18 | A4 | C2×A4 | C3.A4 | C4×A4 | C2×C3.A4 | C4×C3.A4 |
| kernel | C4×Q8⋊C9 | C2×Q8⋊C9 | Q8×C12 | Q8⋊C9 | C6×Q8 | C4×Q8 | C3×Q8 | C2×Q8 | Q8 | C12 | C12 | C6 | C4 | C2 | C2×C12 | C2×C6 | C2×C4 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 2 | 4 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C4×Q8⋊C9 ►in GL3(𝔽37) generated by
| 31 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 29 | 3 |
| 0 | 3 | 8 |
| 1 | 0 | 0 |
| 0 | 0 | 36 |
| 0 | 1 | 0 |
| 26 | 0 | 0 |
| 0 | 35 | 21 |
| 0 | 14 | 32 |
G:=sub<GL(3,GF(37))| [31,0,0,0,1,0,0,0,1],[1,0,0,0,29,3,0,3,8],[1,0,0,0,0,1,0,36,0],[26,0,0,0,35,14,0,21,32] >;
C4×Q8⋊C9 in GAP, Magma, Sage, TeX
C_4\times Q_8\rtimes C_9
% in TeX
G:=Group("C4xQ8:C9"); // GroupNames label
G:=SmallGroup(288,72);
// by ID
G=gap.SmallGroup(288,72);
# by ID
G:=PCGroup([7,-2,-3,-2,-3,-2,2,-2,42,92,1271,172,2280,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^9=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations
Export