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G = C8⋊Dic9order 288 = 25·32

2nd semidirect product of C8 and Dic9 acting via Dic9/C18=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C722C4, C82Dic9, C36.4Q8, C4.4Dic18, C24.4Dic3, C18.2SD16, C22.8D36, C12.19Dic6, (C2×C8).6D9, C92(C4.Q8), (C2×C72).8C2, C18.6(C4⋊C4), C3.(C8⋊Dic3), C36.36(C2×C4), (C2×C24).17S3, (C2×C4).74D18, (C2×C18).13D4, (C2×C6).21D12, C4⋊Dic9.2C2, C4.7(C2×Dic9), C6.2(C24⋊C2), (C2×C12).363D6, C2.2(C72⋊C2), C2.4(C4⋊Dic9), C6.8(C4⋊Dic3), (C2×C36).82C22, C12.42(C2×Dic3), SmallGroup(288,25)

Series: Derived Chief Lower central Upper central

C1C36 — C8⋊Dic9
C1C3C9C18C2×C18C2×C36C4⋊Dic9 — C8⋊Dic9
C9C18C36 — C8⋊Dic9
C1C22C2×C4C2×C8

Generators and relations for C8⋊Dic9
 G = < a,b,c | a8=b18=1, c2=b9, ab=ba, cac-1=a3, cbc-1=b-1 >

36C4
36C4
18C2×C4
18C2×C4
12Dic3
12Dic3
9C4⋊C4
9C4⋊C4
6C2×Dic3
6C2×Dic3
4Dic9
4Dic9
9C4.Q8
3C4⋊Dic3
3C4⋊Dic3
2C2×Dic9
2C2×Dic9
3C8⋊Dic3

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

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

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

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

78 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222344444466688889991212121218···1824···2436···3672···72
size111122236363636222222222222222···22···22···22···2

78 irreducible representations

dim1111222222222222222
type++++-+-++-+-+-+
imageC1C2C2C4S3Q8D4Dic3D6SD16D9Dic6D12Dic9D18C24⋊C2Dic18D36C72⋊C2
kernelC8⋊Dic9C4⋊Dic9C2×C72C72C2×C24C36C2×C18C24C2×C12C18C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps12141112143226386624

Matrix representation of C8⋊Dic9 in GL4(𝔽73) generated by

481100
623700
004862
001137
,
07200
1100
007045
002842
,
516300
122200
002368
001850
G:=sub<GL(4,GF(73))| [48,62,0,0,11,37,0,0,0,0,48,11,0,0,62,37],[0,1,0,0,72,1,0,0,0,0,70,28,0,0,45,42],[51,12,0,0,63,22,0,0,0,0,23,18,0,0,68,50] >;

C8⋊Dic9 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,25);
// by ID

G=gap.SmallGroup(288,25);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^8=b^18=1,c^2=b^9,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊Dic9 in TeX

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