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G = C9×C36order 324 = 22·34

Abelian group of type [9,36]

direct product, abelian, monomial, 3-elementary

Aliases: C9×C36, SmallGroup(324,26)

Series: Derived Chief Lower central Upper central

C1 — C9×C36
C1C3C32C3×C6C3×C18C9×C18 — C9×C36
C1 — C9×C36
C1 — C9×C36

Generators and relations for C9×C36
 G = < a,b | a9=b36=1, ab=ba >

Subgroups: 69, all normal (9 characteristic)
C1, C2, C3 [×4], C4, C6 [×4], C9 [×12], C32, C12 [×4], C18 [×12], C3×C6, C3×C9 [×4], C36 [×12], C3×C12, C3×C18 [×4], C92, C3×C36 [×4], C9×C18, C9×C36
Quotients: C1, C2, C3 [×4], C4, C6 [×4], C9 [×12], C32, C12 [×4], C18 [×12], C3×C6, C3×C9 [×4], C36 [×12], C3×C12, C3×C18 [×4], C92, C3×C36 [×4], C9×C18, C9×C36

Smallest permutation representation of C9×C36
Regular action on 324 points
Generators in S324
(1 258 324 39 233 134 163 88 186)(2 259 289 40 234 135 164 89 187)(3 260 290 41 235 136 165 90 188)(4 261 291 42 236 137 166 91 189)(5 262 292 43 237 138 167 92 190)(6 263 293 44 238 139 168 93 191)(7 264 294 45 239 140 169 94 192)(8 265 295 46 240 141 170 95 193)(9 266 296 47 241 142 171 96 194)(10 267 297 48 242 143 172 97 195)(11 268 298 49 243 144 173 98 196)(12 269 299 50 244 109 174 99 197)(13 270 300 51 245 110 175 100 198)(14 271 301 52 246 111 176 101 199)(15 272 302 53 247 112 177 102 200)(16 273 303 54 248 113 178 103 201)(17 274 304 55 249 114 179 104 202)(18 275 305 56 250 115 180 105 203)(19 276 306 57 251 116 145 106 204)(20 277 307 58 252 117 146 107 205)(21 278 308 59 217 118 147 108 206)(22 279 309 60 218 119 148 73 207)(23 280 310 61 219 120 149 74 208)(24 281 311 62 220 121 150 75 209)(25 282 312 63 221 122 151 76 210)(26 283 313 64 222 123 152 77 211)(27 284 314 65 223 124 153 78 212)(28 285 315 66 224 125 154 79 213)(29 286 316 67 225 126 155 80 214)(30 287 317 68 226 127 156 81 215)(31 288 318 69 227 128 157 82 216)(32 253 319 70 228 129 158 83 181)(33 254 320 71 229 130 159 84 182)(34 255 321 72 230 131 160 85 183)(35 256 322 37 231 132 161 86 184)(36 257 323 38 232 133 162 87 185)
(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)

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

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

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

324 conjugacy classes

class 1  2 3A···3H4A4B6A···6H9A···9BT12A···12P18A···18BT36A···36EN
order123···3446···69···912···1218···1836···36
size111···1111···11···11···11···11···1

324 irreducible representations

dim111111111
type++
imageC1C2C3C4C6C9C12C18C36
kernelC9×C36C9×C18C3×C36C92C3×C18C36C3×C9C18C9
# reps11828721672144

Matrix representation of C9×C36 in GL2(𝔽37) generated by

340
09
,
30
019
G:=sub<GL(2,GF(37))| [34,0,0,9],[3,0,0,19] >;

C9×C36 in GAP, Magma, Sage, TeX

C_9\times C_{36}
% in TeX

G:=Group("C9xC36");
// GroupNames label

G:=SmallGroup(324,26);
// by ID

G=gap.SmallGroup(324,26);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,223,338]);
// Polycyclic

G:=Group<a,b|a^9=b^36=1,a*b=b*a>;
// generators/relations

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