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G = C9×C4⋊C8order 288 = 25·32

Direct product of C9 and C4⋊C8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C9×C4⋊C8, C4⋊C72, C363C8, C12.4C24, C36.67D4, C36.12Q8, C42.2C18, C18.9M4(2), C4.4(Q8×C9), (C2×C36).8C4, C2.2(C2×C72), (C2×C24).4C6, (C4×C36).8C2, (C2×C4).4C36, (C2×C8).2C18, (C2×C72).4C2, C4.18(D4×C9), (C4×C12).15C6, C6.12(C2×C24), C18.12(C2×C8), C12.84(C3×D4), C18.11(C4⋊C4), C12.16(C3×Q8), (C2×C12).18C12, C2.3(C9×M4(2)), C6.9(C3×M4(2)), C22.10(C2×C36), (C2×C36).134C22, C3.(C3×C4⋊C8), (C3×C4⋊C8).C3, C2.2(C9×C4⋊C4), C6.11(C3×C4⋊C4), (C2×C4).33(C2×C18), (C2×C6).48(C2×C12), (C2×C18).39(C2×C4), (C2×C12).168(C2×C6), SmallGroup(288,55)

Series: Derived Chief Lower central Upper central

C1C2 — C9×C4⋊C8
C1C2C6C12C2×C12C2×C36C2×C72 — C9×C4⋊C8
C1C2 — C9×C4⋊C8
C1C2×C36 — C9×C4⋊C8

Generators and relations for C9×C4⋊C8
 G = < a,b,c | a9=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C8
2C8
2C12
2C24
2C24
2C36
2C72
2C72

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

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

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

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

180 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H6A···6F8A···8H9A···9F12A···12H12I···12P18A···18R24A···24P36A···36X36Y···36AV72A···72AV
order122233444444446···68···89···912···1212···1218···1824···2436···3636···3672···72
size111111111122221···12···21···11···12···21···12···21···12···22···2

180 irreducible representations

dim111111111111111222222222
type++++-
imageC1C2C2C3C4C6C6C8C9C12C18C18C24C36C72D4Q8M4(2)C3×D4C3×Q8C3×M4(2)D4×C9Q8×C9C9×M4(2)
kernelC9×C4⋊C8C4×C36C2×C72C3×C4⋊C8C2×C36C4×C12C2×C24C36C4⋊C8C2×C12C42C2×C8C12C2×C4C4C36C36C18C12C12C6C4C4C2
# reps11224248686121624481122246612

Matrix representation of C9×C4⋊C8 in GL4(𝔽73) generated by

64000
03700
0010
0001
,
72000
07200
0001
00720
,
22000
07200
005212
001221
G:=sub<GL(4,GF(73))| [64,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[22,0,0,0,0,72,0,0,0,0,52,12,0,0,12,21] >;

C9×C4⋊C8 in GAP, Magma, Sage, TeX

C_9\times C_4\rtimes C_8
% in TeX

G:=Group("C9xC4:C8");
// GroupNames label

G:=SmallGroup(288,55);
// by ID

G=gap.SmallGroup(288,55);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,92,268,242]);
// Polycyclic

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

Export

Subgroup lattice of C9×C4⋊C8 in TeX

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