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G = Dic9⋊C8order 288 = 25·32

The semidirect product of Dic9 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic9⋊C8
G = < a,b,c | a18=c8=1, b2=a9, bab-1=a-1, ac=ca, cbc-1=a9b >

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 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 4 4 6 6 6 8 8 8 8 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 18 18 18 18 2 2 2 2 2 2 2 18 18 18 18 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - + - image C1 C2 C2 C2 C4 C8 S3 D4 Q8 D6 M4(2) D9 Dic6 C3⋊D4 C4×S3 D18 S3×C8 C8⋊S3 Dic18 C9⋊D4 C4×D9 C8×D9 C8⋊D9 kernel Dic9⋊C8 C2×C9⋊C8 C4×Dic9 C2×C72 C2×Dic9 Dic9 C2×C24 C36 C36 C2×C12 C18 C2×C8 C12 C12 C2×C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 8 1 1 1 1 2 3 2 2 2 3 4 4 6 6 6 12 12

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

 1 0 0 0 31 28 0 45 3
,
 72 0 0 0 2 55 0 53 71
,
 51 0 0 0 70 67 0 6 3
G:=sub<GL(3,GF(73))| [1,0,0,0,31,45,0,28,3],[72,0,0,0,2,53,0,55,71],[51,0,0,0,70,6,0,67,3] >;

Dic9⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("Dic9:C8");
// GroupNames label

G:=SmallGroup(288,22);
// by ID

G=gap.SmallGroup(288,22);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,6725,292,9414]);
// Polycyclic

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

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