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G = C9⋊Q32order 288 = 25·32

The semidirect product of C9 and Q32 acting via Q32/Q16=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C92Q32, Q16.D9, C36.6D4, C24.9D6, C8.7D18, C18.11D8, C72.5C22, Dic36.2C2, C9⋊C16.C2, C3.(C3⋊Q32), C2.7(D4⋊D9), C4.4(C9⋊D4), (C9×Q16).1C2, (C3×Q16).2S3, C6.18(D4⋊S3), C12.4(C3⋊D4), SmallGroup(288,36)

Series: Derived Chief Lower central Upper central

C1C72 — C9⋊Q32
C1C3C9C18C36C72Dic36 — C9⋊Q32
C9C18C36C72 — C9⋊Q32
C1C2C4C8Q16

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

4C4
36C4
2Q8
18Q8
4C12
12Dic3
9Q16
9C16
2C3×Q8
6Dic6
4Dic9
4C36
9Q32
3C3⋊C16
3Dic12
2Dic18
2Q8×C9
3C3⋊Q32

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

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

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

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

39 conjugacy classes

class 1  2  3 4A4B4C 6 8A8B9A9B9C12A12B12C16A16B16C16D18A18B18C24A24B36A36B36C36D···36I72A···72F
order12344468899912121216161616181818242436363636···3672···72
size112287222222248818181818222444448···84···4

39 irreducible representations

dim11112222222224444
type+++++++++-++-+-
imageC1C2C2C2S3D4D6D8D9C3⋊D4Q32D18C9⋊D4D4⋊S3C3⋊Q32D4⋊D9C9⋊Q32
kernelC9⋊Q32C9⋊C16Dic36C9×Q16C3×Q16C36C24C18Q16C12C9C8C4C6C3C2C1
# reps11111112324361236

Matrix representation of C9⋊Q32 in GL4(𝔽433) generated by

39735000
834700
0010
0001
,
196600
4741400
00132316
00310214
,
42140900
241200
00398177
004235
G:=sub<GL(4,GF(433))| [397,83,0,0,350,47,0,0,0,0,1,0,0,0,0,1],[19,47,0,0,66,414,0,0,0,0,132,310,0,0,316,214],[421,24,0,0,409,12,0,0,0,0,398,42,0,0,177,35] >;

C9⋊Q32 in GAP, Magma, Sage, TeX

C_9\rtimes Q_{32}
% in TeX

G:=Group("C9:Q32");
// GroupNames label

G:=SmallGroup(288,36);
// by ID

G=gap.SmallGroup(288,36);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,120,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic

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

Export

Subgroup lattice of C9⋊Q32 in TeX

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