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G = C9⋊C32order 288 = 25·32

The semidirect product of C9 and C32 acting via C32/C16=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C9⋊C32, C18.C16, C72.2C4, C36.2C8, C48.6S3, C16.2D9, C144.2C2, C8.3Dic9, C24.5Dic3, C3.(C3⋊C32), C2.(C9⋊C16), C4.2(C9⋊C8), C6.1(C3⋊C16), C12.4(C3⋊C8), SmallGroup(288,1)

Series: Derived Chief Lower central Upper central

C1C9 — C9⋊C32
C1C3C9C18C36C72C144 — C9⋊C32
C9 — C9⋊C32
C1C16

Generators and relations for C9⋊C32
 G = < a,b | a9=b32=1, bab-1=a-1 >

9C32
3C3⋊C32

Smallest permutation representation of C9⋊C32
Regular action on 288 points
Generators in S288
(1 183 141 63 263 241 107 74 216)(2 217 75 108 242 264 64 142 184)(3 185 143 33 265 243 109 76 218)(4 219 77 110 244 266 34 144 186)(5 187 145 35 267 245 111 78 220)(6 221 79 112 246 268 36 146 188)(7 189 147 37 269 247 113 80 222)(8 223 81 114 248 270 38 148 190)(9 191 149 39 271 249 115 82 224)(10 193 83 116 250 272 40 150 192)(11 161 151 41 273 251 117 84 194)(12 195 85 118 252 274 42 152 162)(13 163 153 43 275 253 119 86 196)(14 197 87 120 254 276 44 154 164)(15 165 155 45 277 255 121 88 198)(16 199 89 122 256 278 46 156 166)(17 167 157 47 279 225 123 90 200)(18 201 91 124 226 280 48 158 168)(19 169 159 49 281 227 125 92 202)(20 203 93 126 228 282 50 160 170)(21 171 129 51 283 229 127 94 204)(22 205 95 128 230 284 52 130 172)(23 173 131 53 285 231 97 96 206)(24 207 65 98 232 286 54 132 174)(25 175 133 55 287 233 99 66 208)(26 209 67 100 234 288 56 134 176)(27 177 135 57 257 235 101 68 210)(28 211 69 102 236 258 58 136 178)(29 179 137 59 259 237 103 70 212)(30 213 71 104 238 260 60 138 180)(31 181 139 61 261 239 105 72 214)(32 215 73 106 240 262 62 140 182)
(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)

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

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

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

96 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D9A9B9C12A12B16A···16H18A18B18C24A24B24C24D32A···32P36A···36F48A···48H72A···72L144A···144X
order1234468888999121216···161818182424242432···3236···3648···4872···72144···144
size1121121111222221···122222229···92···22···22···22···2

96 irreducible representations

dim1111112222222222
type+++-+-
imageC1C2C4C8C16C32S3Dic3D9C3⋊C8Dic9C3⋊C16C9⋊C8C3⋊C32C9⋊C16C9⋊C32
kernelC9⋊C32C144C72C36C18C9C48C24C16C12C8C6C4C3C2C1
# reps1124816113234681224

Matrix representation of C9⋊C32 in GL2(𝔽17) generated by

145
410
,
03
10
G:=sub<GL(2,GF(17))| [14,4,5,10],[0,1,3,0] >;

C9⋊C32 in GAP, Magma, Sage, TeX

C_9\rtimes C_{32}
% in TeX

G:=Group("C9:C32");
// GroupNames label

G:=SmallGroup(288,1);
// by ID

G=gap.SmallGroup(288,1);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,14,36,58,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b|a^9=b^32=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C9⋊C32 in TeX

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