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## G = C8×Dic9order 288 = 25·32

### Direct product of C8 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C8×Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×C36 — C4×Dic9 — C8×Dic9
 Lower central C9 — C8×Dic9
 Upper central C1 — C2×C8

Generators and relations for C8×Dic9
G = < a,b,c | a8=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 176 in 66 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], C9, Dic3 [×4], C12 [×2], C2×C6, C42, C2×C8, C2×C8, C18, C18 [×2], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×2], C2×C12, C4×C8, Dic9 [×4], C36 [×2], C2×C18, C2×C3⋊C8, C4×Dic3, C2×C24, C9⋊C8 [×2], C72 [×2], C2×Dic9 [×2], C2×C36, C8×Dic3, C2×C9⋊C8, C4×Dic9, C2×C72, C8×Dic9
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C8 [×4], C2×C4 [×3], Dic3 [×2], D6, C42, C2×C8 [×2], D9, C4×S3 [×2], C2×Dic3, C4×C8, Dic9 [×2], D18, S3×C8 [×2], C4×Dic3, C4×D9 [×2], C2×Dic9, C8×Dic3, C8×D9 [×2], C4×Dic9, C8×Dic9

Smallest permutation representation of C8×Dic9
Regular action on 288 points
Generators in S288
(1 27 287 42 93 110 184 259)(2 28 288 43 94 111 185 260)(3 29 271 44 95 112 186 261)(4 30 272 45 96 113 187 262)(5 31 273 46 97 114 188 263)(6 32 274 47 98 115 189 264)(7 33 275 48 99 116 190 265)(8 34 276 49 100 117 191 266)(9 35 277 50 101 118 192 267)(10 36 278 51 102 119 193 268)(11 19 279 52 103 120 194 269)(12 20 280 53 104 121 195 270)(13 21 281 54 105 122 196 253)(14 22 282 37 106 123 197 254)(15 23 283 38 107 124 198 255)(16 24 284 39 108 125 181 256)(17 25 285 40 91 126 182 257)(18 26 286 41 92 109 183 258)(55 74 146 215 138 164 221 250)(56 75 147 216 139 165 222 251)(57 76 148 199 140 166 223 252)(58 77 149 200 141 167 224 235)(59 78 150 201 142 168 225 236)(60 79 151 202 143 169 226 237)(61 80 152 203 144 170 227 238)(62 81 153 204 127 171 228 239)(63 82 154 205 128 172 229 240)(64 83 155 206 129 173 230 241)(65 84 156 207 130 174 231 242)(66 85 157 208 131 175 232 243)(67 86 158 209 132 176 233 244)(68 87 159 210 133 177 234 245)(69 88 160 211 134 178 217 246)(70 89 161 212 135 179 218 247)(71 90 162 213 136 180 219 248)(72 73 145 214 137 163 220 249)
(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)
(1 218 10 227)(2 217 11 226)(3 234 12 225)(4 233 13 224)(5 232 14 223)(6 231 15 222)(7 230 16 221)(8 229 17 220)(9 228 18 219)(19 237 28 246)(20 236 29 245)(21 235 30 244)(22 252 31 243)(23 251 32 242)(24 250 33 241)(25 249 34 240)(26 248 35 239)(27 247 36 238)(37 76 46 85)(38 75 47 84)(39 74 48 83)(40 73 49 82)(41 90 50 81)(42 89 51 80)(43 88 52 79)(44 87 53 78)(45 86 54 77)(55 275 64 284)(56 274 65 283)(57 273 66 282)(58 272 67 281)(59 271 68 280)(60 288 69 279)(61 287 70 278)(62 286 71 277)(63 285 72 276)(91 145 100 154)(92 162 101 153)(93 161 102 152)(94 160 103 151)(95 159 104 150)(96 158 105 149)(97 157 106 148)(98 156 107 147)(99 155 108 146)(109 213 118 204)(110 212 119 203)(111 211 120 202)(112 210 121 201)(113 209 122 200)(114 208 123 199)(115 207 124 216)(116 206 125 215)(117 205 126 214)(127 183 136 192)(128 182 137 191)(129 181 138 190)(130 198 139 189)(131 197 140 188)(132 196 141 187)(133 195 142 186)(134 194 143 185)(135 193 144 184)(163 266 172 257)(164 265 173 256)(165 264 174 255)(166 263 175 254)(167 262 176 253)(168 261 177 270)(169 260 178 269)(170 259 179 268)(171 258 180 267)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E ··· 4L 6A 6B 6C 8A ··· 8H 8I ··· 8P 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 24A ··· 24H 36A ··· 36L 72A ··· 72X order 1 2 2 2 3 4 4 4 4 4 ··· 4 6 6 6 8 ··· 8 8 ··· 8 9 9 9 12 12 12 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 2 1 1 1 1 9 ··· 9 2 2 2 1 ··· 1 9 ··· 9 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + image C1 C2 C2 C2 C4 C4 C4 C8 S3 Dic3 D6 D9 C4×S3 C4×S3 Dic9 D18 S3×C8 C4×D9 C4×D9 C8×D9 kernel C8×Dic9 C2×C9⋊C8 C4×Dic9 C2×C72 C9⋊C8 C72 C2×Dic9 Dic9 C2×C24 C24 C2×C12 C2×C8 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 1 1 1 4 4 4 16 1 2 1 3 2 2 6 3 8 6 6 24

Matrix representation of C8×Dic9 in GL3(𝔽73) generated by

 1 0 0 0 63 0 0 0 63
,
 72 0 0 0 70 42 0 31 28
,
 27 0 0 0 61 51 0 63 12
G:=sub<GL(3,GF(73))| [1,0,0,0,63,0,0,0,63],[72,0,0,0,70,31,0,42,28],[27,0,0,0,61,63,0,51,12] >;

C8×Dic9 in GAP, Magma, Sage, TeX

C_8\times {\rm Dic}_9
% in TeX

G:=Group("C8xDic9");
// GroupNames label

G:=SmallGroup(288,21);
// by ID

G=gap.SmallGroup(288,21);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,64,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^8=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽