direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C8⋊C4, C72⋊7C4, C8⋊3C36, C24.9C12, C42.1C18, C18.7C42, C18.7M4(2), (C2×C36).6C4, C6.7(C4×C12), (C2×C4).2C36, (C2×C8).7C18, (C4×C36).1C2, C2.2(C4×C36), (C4×C12).4C6, (C2×C12).6C12, (C2×C24).28C6, C36.48(C2×C4), (C2×C72).17C2, C4.10(C2×C36), C12.59(C2×C12), C22.8(C2×C36), C6.7(C3×M4(2)), C2.1(C9×M4(2)), (C2×C36).132C22, C3.(C3×C8⋊C4), (C3×C8⋊C4).C3, (C2×C18).37(C2×C4), (C2×C4).31(C2×C18), (C2×C6).46(C2×C12), (C2×C12).166(C2×C6), SmallGroup(288,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C8⋊C4
G = < a,b,c | a9=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C9×C8⋊C4
►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 34 67 262 77 257 86 273)(2 35 68 263 78 258 87 274)(3 36 69 264 79 259 88 275)(4 28 70 265 80 260 89 276)(5 29 71 266 81 261 90 277)(6 30 72 267 73 253 82 278)(7 31 64 268 74 254 83 279)(8 32 65 269 75 255 84 271)(9 33 66 270 76 256 85 272)(10 242 56 281 105 24 40 50)(11 243 57 282 106 25 41 51)(12 235 58 283 107 26 42 52)(13 236 59 284 108 27 43 53)(14 237 60 285 100 19 44 54)(15 238 61 286 101 20 45 46)(16 239 62 287 102 21 37 47)(17 240 63 288 103 22 38 48)(18 241 55 280 104 23 39 49)(91 211 139 221 149 230 158 163)(92 212 140 222 150 231 159 164)(93 213 141 223 151 232 160 165)(94 214 142 224 152 233 161 166)(95 215 143 225 153 234 162 167)(96 216 144 217 145 226 154 168)(97 208 136 218 146 227 155 169)(98 209 137 219 147 228 156 170)(99 210 138 220 148 229 157 171)(109 206 134 247 175 195 123 181)(110 207 135 248 176 196 124 182)(111 199 127 249 177 197 125 183)(112 200 128 250 178 198 126 184)(113 201 129 251 179 190 118 185)(114 202 130 252 180 191 119 186)(115 203 131 244 172 192 120 187)(116 204 132 245 173 193 121 188)(117 205 133 246 174 194 122 189)
(1 129 58 91)(2 130 59 92)(3 131 60 93)(4 132 61 94)(5 133 62 95)(6 134 63 96)(7 135 55 97)(8 127 56 98)(9 128 57 99)(10 156 84 111)(11 157 85 112)(12 158 86 113)(13 159 87 114)(14 160 88 115)(15 161 89 116)(16 162 90 117)(17 154 82 109)(18 155 83 110)(19 165 264 203)(20 166 265 204)(21 167 266 205)(22 168 267 206)(23 169 268 207)(24 170 269 199)(25 171 270 200)(26 163 262 201)(27 164 263 202)(28 188 286 233)(29 189 287 234)(30 181 288 226)(31 182 280 227)(32 183 281 228)(33 184 282 229)(34 185 283 230)(35 186 284 231)(36 187 285 232)(37 153 81 122)(38 145 73 123)(39 146 74 124)(40 147 75 125)(41 148 76 126)(42 149 77 118)(43 150 78 119)(44 151 79 120)(45 152 80 121)(46 214 260 245)(47 215 261 246)(48 216 253 247)(49 208 254 248)(50 209 255 249)(51 210 256 250)(52 211 257 251)(53 212 258 252)(54 213 259 244)(64 176 104 136)(65 177 105 137)(66 178 106 138)(67 179 107 139)(68 180 108 140)(69 172 100 141)(70 173 101 142)(71 174 102 143)(72 175 103 144)(190 235 221 273)(191 236 222 274)(192 237 223 275)(193 238 224 276)(194 239 225 277)(195 240 217 278)(196 241 218 279)(197 242 219 271)(198 243 220 272)
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), (1,34,67,262,77,257,86,273)(2,35,68,263,78,258,87,274)(3,36,69,264,79,259,88,275)(4,28,70,265,80,260,89,276)(5,29,71,266,81,261,90,277)(6,30,72,267,73,253,82,278)(7,31,64,268,74,254,83,279)(8,32,65,269,75,255,84,271)(9,33,66,270,76,256,85,272)(10,242,56,281,105,24,40,50)(11,243,57,282,106,25,41,51)(12,235,58,283,107,26,42,52)(13,236,59,284,108,27,43,53)(14,237,60,285,100,19,44,54)(15,238,61,286,101,20,45,46)(16,239,62,287,102,21,37,47)(17,240,63,288,103,22,38,48)(18,241,55,280,104,23,39,49)(91,211,139,221,149,230,158,163)(92,212,140,222,150,231,159,164)(93,213,141,223,151,232,160,165)(94,214,142,224,152,233,161,166)(95,215,143,225,153,234,162,167)(96,216,144,217,145,226,154,168)(97,208,136,218,146,227,155,169)(98,209,137,219,147,228,156,170)(99,210,138,220,148,229,157,171)(109,206,134,247,175,195,123,181)(110,207,135,248,176,196,124,182)(111,199,127,249,177,197,125,183)(112,200,128,250,178,198,126,184)(113,201,129,251,179,190,118,185)(114,202,130,252,180,191,119,186)(115,203,131,244,172,192,120,187)(116,204,132,245,173,193,121,188)(117,205,133,246,174,194,122,189), (1,129,58,91)(2,130,59,92)(3,131,60,93)(4,132,61,94)(5,133,62,95)(6,134,63,96)(7,135,55,97)(8,127,56,98)(9,128,57,99)(10,156,84,111)(11,157,85,112)(12,158,86,113)(13,159,87,114)(14,160,88,115)(15,161,89,116)(16,162,90,117)(17,154,82,109)(18,155,83,110)(19,165,264,203)(20,166,265,204)(21,167,266,205)(22,168,267,206)(23,169,268,207)(24,170,269,199)(25,171,270,200)(26,163,262,201)(27,164,263,202)(28,188,286,233)(29,189,287,234)(30,181,288,226)(31,182,280,227)(32,183,281,228)(33,184,282,229)(34,185,283,230)(35,186,284,231)(36,187,285,232)(37,153,81,122)(38,145,73,123)(39,146,74,124)(40,147,75,125)(41,148,76,126)(42,149,77,118)(43,150,78,119)(44,151,79,120)(45,152,80,121)(46,214,260,245)(47,215,261,246)(48,216,253,247)(49,208,254,248)(50,209,255,249)(51,210,256,250)(52,211,257,251)(53,212,258,252)(54,213,259,244)(64,176,104,136)(65,177,105,137)(66,178,106,138)(67,179,107,139)(68,180,108,140)(69,172,100,141)(70,173,101,142)(71,174,102,143)(72,175,103,144)(190,235,221,273)(191,236,222,274)(192,237,223,275)(193,238,224,276)(194,239,225,277)(195,240,217,278)(196,241,218,279)(197,242,219,271)(198,243,220,272)>;
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), (1,34,67,262,77,257,86,273)(2,35,68,263,78,258,87,274)(3,36,69,264,79,259,88,275)(4,28,70,265,80,260,89,276)(5,29,71,266,81,261,90,277)(6,30,72,267,73,253,82,278)(7,31,64,268,74,254,83,279)(8,32,65,269,75,255,84,271)(9,33,66,270,76,256,85,272)(10,242,56,281,105,24,40,50)(11,243,57,282,106,25,41,51)(12,235,58,283,107,26,42,52)(13,236,59,284,108,27,43,53)(14,237,60,285,100,19,44,54)(15,238,61,286,101,20,45,46)(16,239,62,287,102,21,37,47)(17,240,63,288,103,22,38,48)(18,241,55,280,104,23,39,49)(91,211,139,221,149,230,158,163)(92,212,140,222,150,231,159,164)(93,213,141,223,151,232,160,165)(94,214,142,224,152,233,161,166)(95,215,143,225,153,234,162,167)(96,216,144,217,145,226,154,168)(97,208,136,218,146,227,155,169)(98,209,137,219,147,228,156,170)(99,210,138,220,148,229,157,171)(109,206,134,247,175,195,123,181)(110,207,135,248,176,196,124,182)(111,199,127,249,177,197,125,183)(112,200,128,250,178,198,126,184)(113,201,129,251,179,190,118,185)(114,202,130,252,180,191,119,186)(115,203,131,244,172,192,120,187)(116,204,132,245,173,193,121,188)(117,205,133,246,174,194,122,189), (1,129,58,91)(2,130,59,92)(3,131,60,93)(4,132,61,94)(5,133,62,95)(6,134,63,96)(7,135,55,97)(8,127,56,98)(9,128,57,99)(10,156,84,111)(11,157,85,112)(12,158,86,113)(13,159,87,114)(14,160,88,115)(15,161,89,116)(16,162,90,117)(17,154,82,109)(18,155,83,110)(19,165,264,203)(20,166,265,204)(21,167,266,205)(22,168,267,206)(23,169,268,207)(24,170,269,199)(25,171,270,200)(26,163,262,201)(27,164,263,202)(28,188,286,233)(29,189,287,234)(30,181,288,226)(31,182,280,227)(32,183,281,228)(33,184,282,229)(34,185,283,230)(35,186,284,231)(36,187,285,232)(37,153,81,122)(38,145,73,123)(39,146,74,124)(40,147,75,125)(41,148,76,126)(42,149,77,118)(43,150,78,119)(44,151,79,120)(45,152,80,121)(46,214,260,245)(47,215,261,246)(48,216,253,247)(49,208,254,248)(50,209,255,249)(51,210,256,250)(52,211,257,251)(53,212,258,252)(54,213,259,244)(64,176,104,136)(65,177,105,137)(66,178,106,138)(67,179,107,139)(68,180,108,140)(69,172,100,141)(70,173,101,142)(71,174,102,143)(72,175,103,144)(190,235,221,273)(191,236,222,274)(192,237,223,275)(193,238,224,276)(194,239,225,277)(195,240,217,278)(196,241,218,279)(197,242,219,271)(198,243,220,272) );
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)], [(1,34,67,262,77,257,86,273),(2,35,68,263,78,258,87,274),(3,36,69,264,79,259,88,275),(4,28,70,265,80,260,89,276),(5,29,71,266,81,261,90,277),(6,30,72,267,73,253,82,278),(7,31,64,268,74,254,83,279),(8,32,65,269,75,255,84,271),(9,33,66,270,76,256,85,272),(10,242,56,281,105,24,40,50),(11,243,57,282,106,25,41,51),(12,235,58,283,107,26,42,52),(13,236,59,284,108,27,43,53),(14,237,60,285,100,19,44,54),(15,238,61,286,101,20,45,46),(16,239,62,287,102,21,37,47),(17,240,63,288,103,22,38,48),(18,241,55,280,104,23,39,49),(91,211,139,221,149,230,158,163),(92,212,140,222,150,231,159,164),(93,213,141,223,151,232,160,165),(94,214,142,224,152,233,161,166),(95,215,143,225,153,234,162,167),(96,216,144,217,145,226,154,168),(97,208,136,218,146,227,155,169),(98,209,137,219,147,228,156,170),(99,210,138,220,148,229,157,171),(109,206,134,247,175,195,123,181),(110,207,135,248,176,196,124,182),(111,199,127,249,177,197,125,183),(112,200,128,250,178,198,126,184),(113,201,129,251,179,190,118,185),(114,202,130,252,180,191,119,186),(115,203,131,244,172,192,120,187),(116,204,132,245,173,193,121,188),(117,205,133,246,174,194,122,189)], [(1,129,58,91),(2,130,59,92),(3,131,60,93),(4,132,61,94),(5,133,62,95),(6,134,63,96),(7,135,55,97),(8,127,56,98),(9,128,57,99),(10,156,84,111),(11,157,85,112),(12,158,86,113),(13,159,87,114),(14,160,88,115),(15,161,89,116),(16,162,90,117),(17,154,82,109),(18,155,83,110),(19,165,264,203),(20,166,265,204),(21,167,266,205),(22,168,267,206),(23,169,268,207),(24,170,269,199),(25,171,270,200),(26,163,262,201),(27,164,263,202),(28,188,286,233),(29,189,287,234),(30,181,288,226),(31,182,280,227),(32,183,281,228),(33,184,282,229),(34,185,283,230),(35,186,284,231),(36,187,285,232),(37,153,81,122),(38,145,73,123),(39,146,74,124),(40,147,75,125),(41,148,76,126),(42,149,77,118),(43,150,78,119),(44,151,79,120),(45,152,80,121),(46,214,260,245),(47,215,261,246),(48,216,253,247),(49,208,254,248),(50,209,255,249),(51,210,256,250),(52,211,257,251),(53,212,258,252),(54,213,259,244),(64,176,104,136),(65,177,105,137),(66,178,106,138),(67,179,107,139),(68,180,108,140),(69,172,100,141),(70,173,101,142),(71,174,102,143),(72,175,103,144),(190,235,221,273),(191,236,222,274),(192,237,223,275),(193,238,224,276),(194,239,225,277),(195,240,217,278),(196,241,218,279),(197,242,219,271),(198,243,220,272)]])
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 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C36 | C36 | M4(2) | C3×M4(2) | C9×M4(2) |
| kernel | C9×C8⋊C4 | C4×C36 | C2×C72 | C3×C8⋊C4 | C72 | C2×C36 | C4×C12 | C2×C24 | C8⋊C4 | C24 | C2×C12 | C42 | C2×C8 | C8 | C2×C4 | C18 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 6 | 16 | 8 | 6 | 12 | 48 | 24 | 4 | 8 | 24 |
Matrix representation of C9×C8⋊C4 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 72 | 0 | 0 |
| 0 | 56 | 63 |
| 0 | 17 | 17 |
| 46 | 0 | 0 |
| 0 | 72 | 71 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(73))| [1,0,0,0,32,0,0,0,32],[72,0,0,0,56,17,0,63,17],[46,0,0,0,72,0,0,71,1] >;
C9×C8⋊C4 in GAP, Magma, Sage, TeX
C_9\times C_8\rtimes C_4
% in TeX
G:=Group("C9xC8:C4"); // GroupNames label
G:=SmallGroup(288,47);
// by ID
G=gap.SmallGroup(288,47);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,84,2045,176,268,360]);
// Polycyclic
G:=Group<a,b,c|a^9=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export