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## G = C4×C9⋊C8order 288 = 25·32

### Direct product of C4 and C9⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C4×C9⋊C8
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — C2×C9⋊C8 — C4×C9⋊C8
 Lower central C9 — C4×C9⋊C8
 Upper central C1 — C42

Generators and relations for C4×C9⋊C8
G = < a,b,c | a4=b9=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 136 in 66 conjugacy classes, 52 normal (16 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C22, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C9, C12 [×6], C2×C6, C42, C2×C8 [×2], C18, C18 [×2], C3⋊C8 [×4], C2×C12, C2×C12 [×2], C4×C8, C36 [×6], C2×C18, C2×C3⋊C8 [×2], C4×C12, C9⋊C8 [×4], C2×C36, C2×C36 [×2], C4×C3⋊C8, C2×C9⋊C8 [×2], C4×C36, C4×C9⋊C8
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C8 [×4], C2×C4 [×3], Dic3 [×2], D6, C42, C2×C8 [×2], D9, C3⋊C8 [×4], C4×S3 [×2], C2×Dic3, C4×C8, Dic9 [×2], D18, C2×C3⋊C8 [×2], C4×Dic3, C9⋊C8 [×4], C4×D9 [×2], C2×Dic9, C4×C3⋊C8, C2×C9⋊C8 [×2], C4×Dic9, C4×C9⋊C8

Smallest permutation representation of C4×C9⋊C8
Regular action on 288 points
Generators in S288
(1 103 22 85)(2 104 23 86)(3 105 24 87)(4 106 25 88)(5 107 26 89)(6 108 27 90)(7 100 19 82)(8 101 20 83)(9 102 21 84)(10 91 28 73)(11 92 29 74)(12 93 30 75)(13 94 31 76)(14 95 32 77)(15 96 33 78)(16 97 34 79)(17 98 35 80)(18 99 36 81)(37 136 55 118)(38 137 56 119)(39 138 57 120)(40 139 58 121)(41 140 59 122)(42 141 60 123)(43 142 61 124)(44 143 62 125)(45 144 63 126)(46 127 64 109)(47 128 65 110)(48 129 66 111)(49 130 67 112)(50 131 68 113)(51 132 69 114)(52 133 70 115)(53 134 71 116)(54 135 72 117)(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 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)
(1 194 49 158 13 185 40 149)(2 193 50 157 14 184 41 148)(3 192 51 156 15 183 42 147)(4 191 52 155 16 182 43 146)(5 190 53 154 17 181 44 145)(6 198 54 162 18 189 45 153)(7 197 46 161 10 188 37 152)(8 196 47 160 11 187 38 151)(9 195 48 159 12 186 39 150)(19 215 64 179 28 206 55 170)(20 214 65 178 29 205 56 169)(21 213 66 177 30 204 57 168)(22 212 67 176 31 203 58 167)(23 211 68 175 32 202 59 166)(24 210 69 174 33 201 60 165)(25 209 70 173 34 200 61 164)(26 208 71 172 35 199 62 163)(27 216 72 180 36 207 63 171)(73 269 118 233 82 260 109 224)(74 268 119 232 83 259 110 223)(75 267 120 231 84 258 111 222)(76 266 121 230 85 257 112 221)(77 265 122 229 86 256 113 220)(78 264 123 228 87 255 114 219)(79 263 124 227 88 254 115 218)(80 262 125 226 89 253 116 217)(81 270 126 234 90 261 117 225)(91 287 136 251 100 278 127 242)(92 286 137 250 101 277 128 241)(93 285 138 249 102 276 129 240)(94 284 139 248 103 275 130 239)(95 283 140 247 104 274 131 238)(96 282 141 246 105 273 132 237)(97 281 142 245 106 272 133 236)(98 280 143 244 107 271 134 235)(99 288 144 252 108 279 135 243)

G:=sub<Sym(288)| (1,103,22,85)(2,104,23,86)(3,105,24,87)(4,106,25,88)(5,107,26,89)(6,108,27,90)(7,100,19,82)(8,101,20,83)(9,102,21,84)(10,91,28,73)(11,92,29,74)(12,93,30,75)(13,94,31,76)(14,95,32,77)(15,96,33,78)(16,97,34,79)(17,98,35,80)(18,99,36,81)(37,136,55,118)(38,137,56,119)(39,138,57,120)(40,139,58,121)(41,140,59,122)(42,141,60,123)(43,142,61,124)(44,143,62,125)(45,144,63,126)(46,127,64,109)(47,128,65,110)(48,129,66,111)(49,130,67,112)(50,131,68,113)(51,132,69,114)(52,133,70,115)(53,134,71,116)(54,135,72,117)(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), (1,2,3,4,5,6,7,8,9)(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), (1,194,49,158,13,185,40,149)(2,193,50,157,14,184,41,148)(3,192,51,156,15,183,42,147)(4,191,52,155,16,182,43,146)(5,190,53,154,17,181,44,145)(6,198,54,162,18,189,45,153)(7,197,46,161,10,188,37,152)(8,196,47,160,11,187,38,151)(9,195,48,159,12,186,39,150)(19,215,64,179,28,206,55,170)(20,214,65,178,29,205,56,169)(21,213,66,177,30,204,57,168)(22,212,67,176,31,203,58,167)(23,211,68,175,32,202,59,166)(24,210,69,174,33,201,60,165)(25,209,70,173,34,200,61,164)(26,208,71,172,35,199,62,163)(27,216,72,180,36,207,63,171)(73,269,118,233,82,260,109,224)(74,268,119,232,83,259,110,223)(75,267,120,231,84,258,111,222)(76,266,121,230,85,257,112,221)(77,265,122,229,86,256,113,220)(78,264,123,228,87,255,114,219)(79,263,124,227,88,254,115,218)(80,262,125,226,89,253,116,217)(81,270,126,234,90,261,117,225)(91,287,136,251,100,278,127,242)(92,286,137,250,101,277,128,241)(93,285,138,249,102,276,129,240)(94,284,139,248,103,275,130,239)(95,283,140,247,104,274,131,238)(96,282,141,246,105,273,132,237)(97,281,142,245,106,272,133,236)(98,280,143,244,107,271,134,235)(99,288,144,252,108,279,135,243)>;

G:=Group( (1,103,22,85)(2,104,23,86)(3,105,24,87)(4,106,25,88)(5,107,26,89)(6,108,27,90)(7,100,19,82)(8,101,20,83)(9,102,21,84)(10,91,28,73)(11,92,29,74)(12,93,30,75)(13,94,31,76)(14,95,32,77)(15,96,33,78)(16,97,34,79)(17,98,35,80)(18,99,36,81)(37,136,55,118)(38,137,56,119)(39,138,57,120)(40,139,58,121)(41,140,59,122)(42,141,60,123)(43,142,61,124)(44,143,62,125)(45,144,63,126)(46,127,64,109)(47,128,65,110)(48,129,66,111)(49,130,67,112)(50,131,68,113)(51,132,69,114)(52,133,70,115)(53,134,71,116)(54,135,72,117)(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), (1,2,3,4,5,6,7,8,9)(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), (1,194,49,158,13,185,40,149)(2,193,50,157,14,184,41,148)(3,192,51,156,15,183,42,147)(4,191,52,155,16,182,43,146)(5,190,53,154,17,181,44,145)(6,198,54,162,18,189,45,153)(7,197,46,161,10,188,37,152)(8,196,47,160,11,187,38,151)(9,195,48,159,12,186,39,150)(19,215,64,179,28,206,55,170)(20,214,65,178,29,205,56,169)(21,213,66,177,30,204,57,168)(22,212,67,176,31,203,58,167)(23,211,68,175,32,202,59,166)(24,210,69,174,33,201,60,165)(25,209,70,173,34,200,61,164)(26,208,71,172,35,199,62,163)(27,216,72,180,36,207,63,171)(73,269,118,233,82,260,109,224)(74,268,119,232,83,259,110,223)(75,267,120,231,84,258,111,222)(76,266,121,230,85,257,112,221)(77,265,122,229,86,256,113,220)(78,264,123,228,87,255,114,219)(79,263,124,227,88,254,115,218)(80,262,125,226,89,253,116,217)(81,270,126,234,90,261,117,225)(91,287,136,251,100,278,127,242)(92,286,137,250,101,277,128,241)(93,285,138,249,102,276,129,240)(94,284,139,248,103,275,130,239)(95,283,140,247,104,274,131,238)(96,282,141,246,105,273,132,237)(97,281,142,245,106,272,133,236)(98,280,143,244,107,271,134,235)(99,288,144,252,108,279,135,243) );

G=PermutationGroup([(1,103,22,85),(2,104,23,86),(3,105,24,87),(4,106,25,88),(5,107,26,89),(6,108,27,90),(7,100,19,82),(8,101,20,83),(9,102,21,84),(10,91,28,73),(11,92,29,74),(12,93,30,75),(13,94,31,76),(14,95,32,77),(15,96,33,78),(16,97,34,79),(17,98,35,80),(18,99,36,81),(37,136,55,118),(38,137,56,119),(39,138,57,120),(40,139,58,121),(41,140,59,122),(42,141,60,123),(43,142,61,124),(44,143,62,125),(45,144,63,126),(46,127,64,109),(47,128,65,110),(48,129,66,111),(49,130,67,112),(50,131,68,113),(51,132,69,114),(52,133,70,115),(53,134,71,116),(54,135,72,117),(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)], [(1,2,3,4,5,6,7,8,9),(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)], [(1,194,49,158,13,185,40,149),(2,193,50,157,14,184,41,148),(3,192,51,156,15,183,42,147),(4,191,52,155,16,182,43,146),(5,190,53,154,17,181,44,145),(6,198,54,162,18,189,45,153),(7,197,46,161,10,188,37,152),(8,196,47,160,11,187,38,151),(9,195,48,159,12,186,39,150),(19,215,64,179,28,206,55,170),(20,214,65,178,29,205,56,169),(21,213,66,177,30,204,57,168),(22,212,67,176,31,203,58,167),(23,211,68,175,32,202,59,166),(24,210,69,174,33,201,60,165),(25,209,70,173,34,200,61,164),(26,208,71,172,35,199,62,163),(27,216,72,180,36,207,63,171),(73,269,118,233,82,260,109,224),(74,268,119,232,83,259,110,223),(75,267,120,231,84,258,111,222),(76,266,121,230,85,257,112,221),(77,265,122,229,86,256,113,220),(78,264,123,228,87,255,114,219),(79,263,124,227,88,254,115,218),(80,262,125,226,89,253,116,217),(81,270,126,234,90,261,117,225),(91,287,136,251,100,278,127,242),(92,286,137,250,101,277,128,241),(93,285,138,249,102,276,129,240),(94,284,139,248,103,275,130,239),(95,283,140,247,104,274,131,238),(96,282,141,246,105,273,132,237),(97,281,142,245,106,272,133,236),(98,280,143,244,107,271,134,235),(99,288,144,252,108,279,135,243)])

96 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4L 6A 6B 6C 8A ··· 8P 9A 9B 9C 12A ··· 12L 18A ··· 18I 36A ··· 36AJ order 1 2 2 2 3 4 ··· 4 6 6 6 8 ··· 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 1 ··· 1 2 2 2 9 ··· 9 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + + - + image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 D9 C3⋊C8 C4×S3 Dic9 D18 C9⋊C8 C4×D9 kernel C4×C9⋊C8 C2×C9⋊C8 C4×C36 C9⋊C8 C2×C36 C36 C4×C12 C2×C12 C2×C12 C42 C12 C12 C2×C4 C2×C4 C4 C4 # reps 1 2 1 8 4 16 1 2 1 3 8 4 6 3 24 12

Matrix representation of C4×C9⋊C8 in GL3(𝔽73) generated by

 27 0 0 0 46 0 0 0 46
,
 1 0 0 0 31 3 0 70 28
,
 46 0 0 0 3 6 0 3 70
G:=sub<GL(3,GF(73))| [27,0,0,0,46,0,0,0,46],[1,0,0,0,31,70,0,3,28],[46,0,0,0,3,3,0,6,70] >;

C4×C9⋊C8 in GAP, Magma, Sage, TeX

C_4\times C_9\rtimes C_8
% in TeX

G:=Group("C4xC9:C8");
// GroupNames label

G:=SmallGroup(288,9);
// by ID

G=gap.SmallGroup(288,9);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,64,100,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^4=b^9=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽