direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C4⋊C8, C4⋊C72, C36⋊3C8, C12.4C24, C36.67D4, C36.12Q8, C42.2C18, C18.9M4(2), C4.4(Q8×C9), (C2×C36).8C4, C2.2(C2×C72), (C2×C24).4C6, (C4×C36).8C2, (C2×C4).4C36, (C2×C8).2C18, (C2×C72).4C2, C4.18(D4×C9), (C4×C12).15C6, C6.12(C2×C24), C18.12(C2×C8), C12.84(C3×D4), C18.11(C4⋊C4), C12.16(C3×Q8), (C2×C12).18C12, C2.3(C9×M4(2)), C6.9(C3×M4(2)), C22.10(C2×C36), (C2×C36).134C22, C3.(C3×C4⋊C8), (C3×C4⋊C8).C3, C2.2(C9×C4⋊C4), C6.11(C3×C4⋊C4), (C2×C4).33(C2×C18), (C2×C6).48(C2×C12), (C2×C18).39(C2×C4), (C2×C12).168(C2×C6), SmallGroup(288,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4⋊C8
G = < a,b,c | a9=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×C4⋊C8
►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)
(1 113 107 147)(2 114 108 148)(3 115 100 149)(4 116 101 150)(5 117 102 151)(6 109 103 152)(7 110 104 153)(8 111 105 145)(9 112 106 146)(10 98 75 179)(11 99 76 180)(12 91 77 172)(13 92 78 173)(14 93 79 174)(15 94 80 175)(16 95 81 176)(17 96 73 177)(18 97 74 178)(19 224 238 181)(20 225 239 182)(21 217 240 183)(22 218 241 184)(23 219 242 185)(24 220 243 186)(25 221 235 187)(26 222 236 188)(27 223 237 189)(28 251 257 170)(29 252 258 171)(30 244 259 163)(31 245 260 164)(32 246 261 165)(33 247 253 166)(34 248 254 167)(35 249 255 168)(36 250 256 169)(37 143 88 133)(38 144 89 134)(39 136 90 135)(40 137 82 127)(41 138 83 128)(42 139 84 129)(43 140 85 130)(44 141 86 131)(45 142 87 132)(46 204 265 214)(47 205 266 215)(48 206 267 216)(49 207 268 208)(50 199 269 209)(51 200 270 210)(52 201 262 211)(53 202 263 212)(54 203 264 213)(55 159 70 121)(56 160 71 122)(57 161 72 123)(58 162 64 124)(59 154 65 125)(60 155 66 126)(61 156 67 118)(62 157 68 119)(63 158 69 120)(190 273 228 283)(191 274 229 284)(192 275 230 285)(193 276 231 286)(194 277 232 287)(195 278 233 288)(196 279 234 280)(197 271 226 281)(198 272 227 282)
(1 190 42 219 75 201 61 170)(2 191 43 220 76 202 62 171)(3 192 44 221 77 203 63 163)(4 193 45 222 78 204 55 164)(5 194 37 223 79 205 56 165)(6 195 38 224 80 206 57 166)(7 196 39 225 81 207 58 167)(8 197 40 217 73 199 59 168)(9 198 41 218 74 200 60 169)(10 211 67 251 107 228 84 185)(11 212 68 252 108 229 85 186)(12 213 69 244 100 230 86 187)(13 214 70 245 101 231 87 188)(14 215 71 246 102 232 88 189)(15 216 72 247 103 233 89 181)(16 208 64 248 104 234 90 182)(17 209 65 249 105 226 82 183)(18 210 66 250 106 227 83 184)(19 175 48 161 253 109 288 144)(20 176 49 162 254 110 280 136)(21 177 50 154 255 111 281 137)(22 178 51 155 256 112 282 138)(23 179 52 156 257 113 283 139)(24 180 53 157 258 114 284 140)(25 172 54 158 259 115 285 141)(26 173 46 159 260 116 286 142)(27 174 47 160 261 117 287 143)(28 147 273 129 242 98 262 118)(29 148 274 130 243 99 263 119)(30 149 275 131 235 91 264 120)(31 150 276 132 236 92 265 121)(32 151 277 133 237 93 266 122)(33 152 278 134 238 94 267 123)(34 153 279 135 239 95 268 124)(35 145 271 127 240 96 269 125)(36 146 272 128 241 97 270 126)
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), (1,113,107,147)(2,114,108,148)(3,115,100,149)(4,116,101,150)(5,117,102,151)(6,109,103,152)(7,110,104,153)(8,111,105,145)(9,112,106,146)(10,98,75,179)(11,99,76,180)(12,91,77,172)(13,92,78,173)(14,93,79,174)(15,94,80,175)(16,95,81,176)(17,96,73,177)(18,97,74,178)(19,224,238,181)(20,225,239,182)(21,217,240,183)(22,218,241,184)(23,219,242,185)(24,220,243,186)(25,221,235,187)(26,222,236,188)(27,223,237,189)(28,251,257,170)(29,252,258,171)(30,244,259,163)(31,245,260,164)(32,246,261,165)(33,247,253,166)(34,248,254,167)(35,249,255,168)(36,250,256,169)(37,143,88,133)(38,144,89,134)(39,136,90,135)(40,137,82,127)(41,138,83,128)(42,139,84,129)(43,140,85,130)(44,141,86,131)(45,142,87,132)(46,204,265,214)(47,205,266,215)(48,206,267,216)(49,207,268,208)(50,199,269,209)(51,200,270,210)(52,201,262,211)(53,202,263,212)(54,203,264,213)(55,159,70,121)(56,160,71,122)(57,161,72,123)(58,162,64,124)(59,154,65,125)(60,155,66,126)(61,156,67,118)(62,157,68,119)(63,158,69,120)(190,273,228,283)(191,274,229,284)(192,275,230,285)(193,276,231,286)(194,277,232,287)(195,278,233,288)(196,279,234,280)(197,271,226,281)(198,272,227,282), (1,190,42,219,75,201,61,170)(2,191,43,220,76,202,62,171)(3,192,44,221,77,203,63,163)(4,193,45,222,78,204,55,164)(5,194,37,223,79,205,56,165)(6,195,38,224,80,206,57,166)(7,196,39,225,81,207,58,167)(8,197,40,217,73,199,59,168)(9,198,41,218,74,200,60,169)(10,211,67,251,107,228,84,185)(11,212,68,252,108,229,85,186)(12,213,69,244,100,230,86,187)(13,214,70,245,101,231,87,188)(14,215,71,246,102,232,88,189)(15,216,72,247,103,233,89,181)(16,208,64,248,104,234,90,182)(17,209,65,249,105,226,82,183)(18,210,66,250,106,227,83,184)(19,175,48,161,253,109,288,144)(20,176,49,162,254,110,280,136)(21,177,50,154,255,111,281,137)(22,178,51,155,256,112,282,138)(23,179,52,156,257,113,283,139)(24,180,53,157,258,114,284,140)(25,172,54,158,259,115,285,141)(26,173,46,159,260,116,286,142)(27,174,47,160,261,117,287,143)(28,147,273,129,242,98,262,118)(29,148,274,130,243,99,263,119)(30,149,275,131,235,91,264,120)(31,150,276,132,236,92,265,121)(32,151,277,133,237,93,266,122)(33,152,278,134,238,94,267,123)(34,153,279,135,239,95,268,124)(35,145,271,127,240,96,269,125)(36,146,272,128,241,97,270,126)>;
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), (1,113,107,147)(2,114,108,148)(3,115,100,149)(4,116,101,150)(5,117,102,151)(6,109,103,152)(7,110,104,153)(8,111,105,145)(9,112,106,146)(10,98,75,179)(11,99,76,180)(12,91,77,172)(13,92,78,173)(14,93,79,174)(15,94,80,175)(16,95,81,176)(17,96,73,177)(18,97,74,178)(19,224,238,181)(20,225,239,182)(21,217,240,183)(22,218,241,184)(23,219,242,185)(24,220,243,186)(25,221,235,187)(26,222,236,188)(27,223,237,189)(28,251,257,170)(29,252,258,171)(30,244,259,163)(31,245,260,164)(32,246,261,165)(33,247,253,166)(34,248,254,167)(35,249,255,168)(36,250,256,169)(37,143,88,133)(38,144,89,134)(39,136,90,135)(40,137,82,127)(41,138,83,128)(42,139,84,129)(43,140,85,130)(44,141,86,131)(45,142,87,132)(46,204,265,214)(47,205,266,215)(48,206,267,216)(49,207,268,208)(50,199,269,209)(51,200,270,210)(52,201,262,211)(53,202,263,212)(54,203,264,213)(55,159,70,121)(56,160,71,122)(57,161,72,123)(58,162,64,124)(59,154,65,125)(60,155,66,126)(61,156,67,118)(62,157,68,119)(63,158,69,120)(190,273,228,283)(191,274,229,284)(192,275,230,285)(193,276,231,286)(194,277,232,287)(195,278,233,288)(196,279,234,280)(197,271,226,281)(198,272,227,282), (1,190,42,219,75,201,61,170)(2,191,43,220,76,202,62,171)(3,192,44,221,77,203,63,163)(4,193,45,222,78,204,55,164)(5,194,37,223,79,205,56,165)(6,195,38,224,80,206,57,166)(7,196,39,225,81,207,58,167)(8,197,40,217,73,199,59,168)(9,198,41,218,74,200,60,169)(10,211,67,251,107,228,84,185)(11,212,68,252,108,229,85,186)(12,213,69,244,100,230,86,187)(13,214,70,245,101,231,87,188)(14,215,71,246,102,232,88,189)(15,216,72,247,103,233,89,181)(16,208,64,248,104,234,90,182)(17,209,65,249,105,226,82,183)(18,210,66,250,106,227,83,184)(19,175,48,161,253,109,288,144)(20,176,49,162,254,110,280,136)(21,177,50,154,255,111,281,137)(22,178,51,155,256,112,282,138)(23,179,52,156,257,113,283,139)(24,180,53,157,258,114,284,140)(25,172,54,158,259,115,285,141)(26,173,46,159,260,116,286,142)(27,174,47,160,261,117,287,143)(28,147,273,129,242,98,262,118)(29,148,274,130,243,99,263,119)(30,149,275,131,235,91,264,120)(31,150,276,132,236,92,265,121)(32,151,277,133,237,93,266,122)(33,152,278,134,238,94,267,123)(34,153,279,135,239,95,268,124)(35,145,271,127,240,96,269,125)(36,146,272,128,241,97,270,126) );
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)], [(1,113,107,147),(2,114,108,148),(3,115,100,149),(4,116,101,150),(5,117,102,151),(6,109,103,152),(7,110,104,153),(8,111,105,145),(9,112,106,146),(10,98,75,179),(11,99,76,180),(12,91,77,172),(13,92,78,173),(14,93,79,174),(15,94,80,175),(16,95,81,176),(17,96,73,177),(18,97,74,178),(19,224,238,181),(20,225,239,182),(21,217,240,183),(22,218,241,184),(23,219,242,185),(24,220,243,186),(25,221,235,187),(26,222,236,188),(27,223,237,189),(28,251,257,170),(29,252,258,171),(30,244,259,163),(31,245,260,164),(32,246,261,165),(33,247,253,166),(34,248,254,167),(35,249,255,168),(36,250,256,169),(37,143,88,133),(38,144,89,134),(39,136,90,135),(40,137,82,127),(41,138,83,128),(42,139,84,129),(43,140,85,130),(44,141,86,131),(45,142,87,132),(46,204,265,214),(47,205,266,215),(48,206,267,216),(49,207,268,208),(50,199,269,209),(51,200,270,210),(52,201,262,211),(53,202,263,212),(54,203,264,213),(55,159,70,121),(56,160,71,122),(57,161,72,123),(58,162,64,124),(59,154,65,125),(60,155,66,126),(61,156,67,118),(62,157,68,119),(63,158,69,120),(190,273,228,283),(191,274,229,284),(192,275,230,285),(193,276,231,286),(194,277,232,287),(195,278,233,288),(196,279,234,280),(197,271,226,281),(198,272,227,282)], [(1,190,42,219,75,201,61,170),(2,191,43,220,76,202,62,171),(3,192,44,221,77,203,63,163),(4,193,45,222,78,204,55,164),(5,194,37,223,79,205,56,165),(6,195,38,224,80,206,57,166),(7,196,39,225,81,207,58,167),(8,197,40,217,73,199,59,168),(9,198,41,218,74,200,60,169),(10,211,67,251,107,228,84,185),(11,212,68,252,108,229,85,186),(12,213,69,244,100,230,86,187),(13,214,70,245,101,231,87,188),(14,215,71,246,102,232,88,189),(15,216,72,247,103,233,89,181),(16,208,64,248,104,234,90,182),(17,209,65,249,105,226,82,183),(18,210,66,250,106,227,83,184),(19,175,48,161,253,109,288,144),(20,176,49,162,254,110,280,136),(21,177,50,154,255,111,281,137),(22,178,51,155,256,112,282,138),(23,179,52,156,257,113,283,139),(24,180,53,157,258,114,284,140),(25,172,54,158,259,115,285,141),(26,173,46,159,260,116,286,142),(27,174,47,160,261,117,287,143),(28,147,273,129,242,98,262,118),(29,148,274,130,243,99,263,119),(30,149,275,131,235,91,264,120),(31,150,276,132,236,92,265,121),(32,151,277,133,237,93,266,122),(33,152,278,134,238,94,267,123),(34,153,279,135,239,95,268,124),(35,145,271,127,240,96,269,125),(36,146,272,128,241,97,270,126)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8H | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12P | 18A | ··· | 18R | 24A | ··· | 24P | 36A | ··· | 36X | 36Y | ··· | 36AV | 72A | ··· | 72AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 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 | C8 | C9 | C12 | C18 | C18 | C24 | C36 | C72 | D4 | Q8 | M4(2) | C3×D4 | C3×Q8 | C3×M4(2) | D4×C9 | Q8×C9 | C9×M4(2) |
| kernel | C9×C4⋊C8 | C4×C36 | C2×C72 | C3×C4⋊C8 | C2×C36 | C4×C12 | C2×C24 | C36 | C4⋊C8 | C2×C12 | C42 | C2×C8 | C12 | C2×C4 | C4 | C36 | C36 | C18 | C12 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 6 | 8 | 6 | 12 | 16 | 24 | 48 | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 |
Matrix representation of C9×C4⋊C8 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 22 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 52 | 12 |
| 0 | 0 | 12 | 21 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[22,0,0,0,0,72,0,0,0,0,52,12,0,0,12,21] >;
C9×C4⋊C8 in GAP, Magma, Sage, TeX
C_9\times C_4\rtimes C_8
% in TeX
G:=Group("C9xC4:C8"); // GroupNames label
G:=SmallGroup(288,55);
// by ID
G=gap.SmallGroup(288,55);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,92,268,242]);
// Polycyclic
G:=Group<a,b,c|a^9=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export