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G = C372C9order 333 = 32·37

The semidirect product of C37 and C9 acting via C9/C3=C3

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C372C9, C111.C3, C3.(C37⋊C3), SmallGroup(333,1)

Series: Derived Chief Lower central Upper central

C1C37 — C372C9
C1C37C111 — C372C9
C37 — C372C9
C1C3

Generators and relations for C372C9
 G = < a,b | a37=b9=1, bab-1=a10 >

37C9

Smallest permutation representation of C372C9
Regular action on 333 points
Generators in S333
(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)(260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296)(297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333)
(1 308 210 105 293 150 48 225 112)(2 297 220 106 282 160 49 251 122)(3 323 193 107 271 170 50 240 132)(4 312 203 108 260 180 51 229 142)(5 301 213 109 286 153 52 255 115)(6 327 186 110 275 163 53 244 125)(7 316 196 111 264 173 54 233 135)(8 305 206 75 290 183 55 259 145)(9 331 216 76 279 156 56 248 118)(10 320 189 77 268 166 57 237 128)(11 309 199 78 294 176 58 226 138)(12 298 209 79 283 149 59 252 148)(13 324 219 80 272 159 60 241 121)(14 313 192 81 261 169 61 230 131)(15 302 202 82 287 179 62 256 141)(16 328 212 83 276 152 63 245 114)(17 317 222 84 265 162 64 234 124)(18 306 195 85 291 172 65 223 134)(19 332 205 86 280 182 66 249 144)(20 321 215 87 269 155 67 238 117)(21 310 188 88 295 165 68 227 127)(22 299 198 89 284 175 69 253 137)(23 325 208 90 273 185 70 242 147)(24 314 218 91 262 158 71 231 120)(25 303 191 92 288 168 72 257 130)(26 329 201 93 277 178 73 246 140)(27 318 211 94 266 151 74 235 113)(28 307 221 95 292 161 38 224 123)(29 333 194 96 281 171 39 250 133)(30 322 204 97 270 181 40 239 143)(31 311 214 98 296 154 41 228 116)(32 300 187 99 285 164 42 254 126)(33 326 197 100 274 174 43 243 136)(34 315 207 101 263 184 44 232 146)(35 304 217 102 289 157 45 258 119)(36 330 190 103 278 167 46 247 129)(37 319 200 104 267 177 47 236 139)

G:=sub<Sym(333)| (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)(260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333), (1,308,210,105,293,150,48,225,112)(2,297,220,106,282,160,49,251,122)(3,323,193,107,271,170,50,240,132)(4,312,203,108,260,180,51,229,142)(5,301,213,109,286,153,52,255,115)(6,327,186,110,275,163,53,244,125)(7,316,196,111,264,173,54,233,135)(8,305,206,75,290,183,55,259,145)(9,331,216,76,279,156,56,248,118)(10,320,189,77,268,166,57,237,128)(11,309,199,78,294,176,58,226,138)(12,298,209,79,283,149,59,252,148)(13,324,219,80,272,159,60,241,121)(14,313,192,81,261,169,61,230,131)(15,302,202,82,287,179,62,256,141)(16,328,212,83,276,152,63,245,114)(17,317,222,84,265,162,64,234,124)(18,306,195,85,291,172,65,223,134)(19,332,205,86,280,182,66,249,144)(20,321,215,87,269,155,67,238,117)(21,310,188,88,295,165,68,227,127)(22,299,198,89,284,175,69,253,137)(23,325,208,90,273,185,70,242,147)(24,314,218,91,262,158,71,231,120)(25,303,191,92,288,168,72,257,130)(26,329,201,93,277,178,73,246,140)(27,318,211,94,266,151,74,235,113)(28,307,221,95,292,161,38,224,123)(29,333,194,96,281,171,39,250,133)(30,322,204,97,270,181,40,239,143)(31,311,214,98,296,154,41,228,116)(32,300,187,99,285,164,42,254,126)(33,326,197,100,274,174,43,243,136)(34,315,207,101,263,184,44,232,146)(35,304,217,102,289,157,45,258,119)(36,330,190,103,278,167,46,247,129)(37,319,200,104,267,177,47,236,139)>;

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)(260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333), (1,308,210,105,293,150,48,225,112)(2,297,220,106,282,160,49,251,122)(3,323,193,107,271,170,50,240,132)(4,312,203,108,260,180,51,229,142)(5,301,213,109,286,153,52,255,115)(6,327,186,110,275,163,53,244,125)(7,316,196,111,264,173,54,233,135)(8,305,206,75,290,183,55,259,145)(9,331,216,76,279,156,56,248,118)(10,320,189,77,268,166,57,237,128)(11,309,199,78,294,176,58,226,138)(12,298,209,79,283,149,59,252,148)(13,324,219,80,272,159,60,241,121)(14,313,192,81,261,169,61,230,131)(15,302,202,82,287,179,62,256,141)(16,328,212,83,276,152,63,245,114)(17,317,222,84,265,162,64,234,124)(18,306,195,85,291,172,65,223,134)(19,332,205,86,280,182,66,249,144)(20,321,215,87,269,155,67,238,117)(21,310,188,88,295,165,68,227,127)(22,299,198,89,284,175,69,253,137)(23,325,208,90,273,185,70,242,147)(24,314,218,91,262,158,71,231,120)(25,303,191,92,288,168,72,257,130)(26,329,201,93,277,178,73,246,140)(27,318,211,94,266,151,74,235,113)(28,307,221,95,292,161,38,224,123)(29,333,194,96,281,171,39,250,133)(30,322,204,97,270,181,40,239,143)(31,311,214,98,296,154,41,228,116)(32,300,187,99,285,164,42,254,126)(33,326,197,100,274,174,43,243,136)(34,315,207,101,263,184,44,232,146)(35,304,217,102,289,157,45,258,119)(36,330,190,103,278,167,46,247,129)(37,319,200,104,267,177,47,236,139) );

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),(260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296),(297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333)], [(1,308,210,105,293,150,48,225,112),(2,297,220,106,282,160,49,251,122),(3,323,193,107,271,170,50,240,132),(4,312,203,108,260,180,51,229,142),(5,301,213,109,286,153,52,255,115),(6,327,186,110,275,163,53,244,125),(7,316,196,111,264,173,54,233,135),(8,305,206,75,290,183,55,259,145),(9,331,216,76,279,156,56,248,118),(10,320,189,77,268,166,57,237,128),(11,309,199,78,294,176,58,226,138),(12,298,209,79,283,149,59,252,148),(13,324,219,80,272,159,60,241,121),(14,313,192,81,261,169,61,230,131),(15,302,202,82,287,179,62,256,141),(16,328,212,83,276,152,63,245,114),(17,317,222,84,265,162,64,234,124),(18,306,195,85,291,172,65,223,134),(19,332,205,86,280,182,66,249,144),(20,321,215,87,269,155,67,238,117),(21,310,188,88,295,165,68,227,127),(22,299,198,89,284,175,69,253,137),(23,325,208,90,273,185,70,242,147),(24,314,218,91,262,158,71,231,120),(25,303,191,92,288,168,72,257,130),(26,329,201,93,277,178,73,246,140),(27,318,211,94,266,151,74,235,113),(28,307,221,95,292,161,38,224,123),(29,333,194,96,281,171,39,250,133),(30,322,204,97,270,181,40,239,143),(31,311,214,98,296,154,41,228,116),(32,300,187,99,285,164,42,254,126),(33,326,197,100,274,174,43,243,136),(34,315,207,101,263,184,44,232,146),(35,304,217,102,289,157,45,258,119),(36,330,190,103,278,167,46,247,129),(37,319,200,104,267,177,47,236,139)])

45 conjugacy classes

class 1 3A3B9A···9F37A···37L111A···111X
order1339···937···37111···111
size11137···373···33···3

45 irreducible representations

dim11133
type+
imageC1C3C9C37⋊C3C372C9
kernelC372C9C111C37C3C1
# reps1261224

Matrix representation of C372C9 in GL4(𝔽1999) generated by

1000
0010
0001
01915590
,
503000
08261298923
08294361020
012031432737
G:=sub<GL(4,GF(1999))| [1,0,0,0,0,0,0,1,0,1,0,915,0,0,1,590],[503,0,0,0,0,826,829,1203,0,1298,436,1432,0,923,1020,737] >;

C372C9 in GAP, Magma, Sage, TeX

C_{37}\rtimes_2C_9
% in TeX

G:=Group("C37:2C9");
// GroupNames label

G:=SmallGroup(333,1);
// by ID

G=gap.SmallGroup(333,1);
# by ID

G:=PCGroup([3,-3,-3,-37,9,2108]);
// Polycyclic

G:=Group<a,b|a^37=b^9=1,b*a*b^-1=a^10>;
// generators/relations

Export

Subgroup lattice of C372C9 in TeX

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