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G = C9×Q32order 288 = 25·32

Direct product of C9 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C9×Q32, C16.C18, C48.3C6, Q16.C18, C144.3C2, C36.39D4, C18.17D8, C72.26C22, C3.(C3×Q32), C4.3(D4×C9), C2.5(C9×D8), (C3×Q32).C3, C8.4(C2×C18), C6.17(C3×D8), C24.23(C2×C6), C12.38(C3×D4), (C3×Q16).3C6, (C9×Q16).2C2, SmallGroup(288,63)

Series: Derived Chief Lower central Upper central

C1C8 — C9×Q32
C1C2C4C12C24C72C9×Q16 — C9×Q32
C1C2C4C8 — C9×Q32
C1C18C36C72 — C9×Q32

Generators and relations for C9×Q32
 G = < a,b,c | a9=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

4C4
4C4
2Q8
2Q8
4C12
4C12
2C3×Q8
2C3×Q8
4C36
4C36
2Q8×C9
2Q8×C9

Smallest permutation representation of C9×Q32
Regular action on 288 points
Generators in S288
(1 85 19 124 220 229 38 176 137)(2 86 20 125 221 230 39 161 138)(3 87 21 126 222 231 40 162 139)(4 88 22 127 223 232 41 163 140)(5 89 23 128 224 233 42 164 141)(6 90 24 113 209 234 43 165 142)(7 91 25 114 210 235 44 166 143)(8 92 26 115 211 236 45 167 144)(9 93 27 116 212 237 46 168 129)(10 94 28 117 213 238 47 169 130)(11 95 29 118 214 239 48 170 131)(12 96 30 119 215 240 33 171 132)(13 81 31 120 216 225 34 172 133)(14 82 32 121 217 226 35 173 134)(15 83 17 122 218 227 36 174 135)(16 84 18 123 219 228 37 175 136)(49 65 244 262 157 181 205 108 288)(50 66 245 263 158 182 206 109 273)(51 67 246 264 159 183 207 110 274)(52 68 247 265 160 184 208 111 275)(53 69 248 266 145 185 193 112 276)(54 70 249 267 146 186 194 97 277)(55 71 250 268 147 187 195 98 278)(56 72 251 269 148 188 196 99 279)(57 73 252 270 149 189 197 100 280)(58 74 253 271 150 190 198 101 281)(59 75 254 272 151 191 199 102 282)(60 76 255 257 152 192 200 103 283)(61 77 256 258 153 177 201 104 284)(62 78 241 259 154 178 202 105 285)(63 79 242 260 155 179 203 106 286)(64 80 243 261 156 180 204 107 287)
(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)
(1 267 9 259)(2 266 10 258)(3 265 11 257)(4 264 12 272)(5 263 13 271)(6 262 14 270)(7 261 15 269)(8 260 16 268)(17 188 25 180)(18 187 26 179)(19 186 27 178)(20 185 28 177)(21 184 29 192)(22 183 30 191)(23 182 31 190)(24 181 32 189)(33 59 41 51)(34 58 42 50)(35 57 43 49)(36 56 44 64)(37 55 45 63)(38 54 46 62)(39 53 47 61)(40 52 48 60)(65 173 73 165)(66 172 74 164)(67 171 75 163)(68 170 76 162)(69 169 77 161)(70 168 78 176)(71 167 79 175)(72 166 80 174)(81 150 89 158)(82 149 90 157)(83 148 91 156)(84 147 92 155)(85 146 93 154)(86 145 94 153)(87 160 95 152)(88 159 96 151)(97 212 105 220)(98 211 106 219)(99 210 107 218)(100 209 108 217)(101 224 109 216)(102 223 110 215)(103 222 111 214)(104 221 112 213)(113 205 121 197)(114 204 122 196)(115 203 123 195)(116 202 124 194)(117 201 125 193)(118 200 126 208)(119 199 127 207)(120 198 128 206)(129 241 137 249)(130 256 138 248)(131 255 139 247)(132 254 140 246)(133 253 141 245)(134 252 142 244)(135 251 143 243)(136 250 144 242)(225 281 233 273)(226 280 234 288)(227 279 235 287)(228 278 236 286)(229 277 237 285)(230 276 238 284)(231 275 239 283)(232 274 240 282)

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

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

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

99 conjugacy classes

class 1  2 3A3B4A4B4C6A6B8A8B9A···9F12A12B12C12D12E12F16A16B16C16D18A···18F24A24B24C24D36A···36F36G···36R48A···48H72A···72L144A···144X
order123344466889···91212121212121616161618···182424242436···3636···3648···4872···72144···144
size111128811221···122888822221···122222···28···82···22···22···2

99 irreducible representations

dim111111111222222222
type+++++-
imageC1C2C2C3C6C6C9C18C18D4D8C3×D4Q32C3×D8D4×C9C3×Q32C9×D8C9×Q32
kernelC9×Q32C144C9×Q16C3×Q32C48C3×Q16Q32C16Q16C36C18C12C9C6C4C3C2C1
# reps112224661212244681224

Matrix representation of C9×Q32 in GL2(𝔽433) generated by

1530
0153
,
13282
392214
,
315423
310118
G:=sub<GL(2,GF(433))| [153,0,0,153],[132,392,82,214],[315,310,423,118] >;

C9×Q32 in GAP, Magma, Sage, TeX

C_9\times Q_{32}
% in TeX

G:=Group("C9xQ32");
// GroupNames label

G:=SmallGroup(288,63);
// by ID

G=gap.SmallGroup(288,63);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,1008,197,1016,142,2355,1186,528,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c|a^9=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×Q32 in TeX

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