Copied to
clipboard

G = C3×C108order 324 = 22·34

Abelian group of type [3,108]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C108, SmallGroup(324,29)

Series: Derived Chief Lower central Upper central

C1 — C3×C108
C1C3C9C18C3×C18C3×C54 — C3×C108
C1 — C3×C108
C1 — C3×C108

Generators and relations for C3×C108
 G = < a,b | a3=b108=1, ab=ba >


Smallest permutation representation of C3×C108
Regular action on 324 points
Generators in S324
(1 121 233)(2 122 234)(3 123 235)(4 124 236)(5 125 237)(6 126 238)(7 127 239)(8 128 240)(9 129 241)(10 130 242)(11 131 243)(12 132 244)(13 133 245)(14 134 246)(15 135 247)(16 136 248)(17 137 249)(18 138 250)(19 139 251)(20 140 252)(21 141 253)(22 142 254)(23 143 255)(24 144 256)(25 145 257)(26 146 258)(27 147 259)(28 148 260)(29 149 261)(30 150 262)(31 151 263)(32 152 264)(33 153 265)(34 154 266)(35 155 267)(36 156 268)(37 157 269)(38 158 270)(39 159 271)(40 160 272)(41 161 273)(42 162 274)(43 163 275)(44 164 276)(45 165 277)(46 166 278)(47 167 279)(48 168 280)(49 169 281)(50 170 282)(51 171 283)(52 172 284)(53 173 285)(54 174 286)(55 175 287)(56 176 288)(57 177 289)(58 178 290)(59 179 291)(60 180 292)(61 181 293)(62 182 294)(63 183 295)(64 184 296)(65 185 297)(66 186 298)(67 187 299)(68 188 300)(69 189 301)(70 190 302)(71 191 303)(72 192 304)(73 193 305)(74 194 306)(75 195 307)(76 196 308)(77 197 309)(78 198 310)(79 199 311)(80 200 312)(81 201 313)(82 202 314)(83 203 315)(84 204 316)(85 205 317)(86 206 318)(87 207 319)(88 208 320)(89 209 321)(90 210 322)(91 211 323)(92 212 324)(93 213 217)(94 214 218)(95 215 219)(96 216 220)(97 109 221)(98 110 222)(99 111 223)(100 112 224)(101 113 225)(102 114 226)(103 115 227)(104 116 228)(105 117 229)(106 118 230)(107 119 231)(108 120 232)
(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)| (1,121,233)(2,122,234)(3,123,235)(4,124,236)(5,125,237)(6,126,238)(7,127,239)(8,128,240)(9,129,241)(10,130,242)(11,131,243)(12,132,244)(13,133,245)(14,134,246)(15,135,247)(16,136,248)(17,137,249)(18,138,250)(19,139,251)(20,140,252)(21,141,253)(22,142,254)(23,143,255)(24,144,256)(25,145,257)(26,146,258)(27,147,259)(28,148,260)(29,149,261)(30,150,262)(31,151,263)(32,152,264)(33,153,265)(34,154,266)(35,155,267)(36,156,268)(37,157,269)(38,158,270)(39,159,271)(40,160,272)(41,161,273)(42,162,274)(43,163,275)(44,164,276)(45,165,277)(46,166,278)(47,167,279)(48,168,280)(49,169,281)(50,170,282)(51,171,283)(52,172,284)(53,173,285)(54,174,286)(55,175,287)(56,176,288)(57,177,289)(58,178,290)(59,179,291)(60,180,292)(61,181,293)(62,182,294)(63,183,295)(64,184,296)(65,185,297)(66,186,298)(67,187,299)(68,188,300)(69,189,301)(70,190,302)(71,191,303)(72,192,304)(73,193,305)(74,194,306)(75,195,307)(76,196,308)(77,197,309)(78,198,310)(79,199,311)(80,200,312)(81,201,313)(82,202,314)(83,203,315)(84,204,316)(85,205,317)(86,206,318)(87,207,319)(88,208,320)(89,209,321)(90,210,322)(91,211,323)(92,212,324)(93,213,217)(94,214,218)(95,215,219)(96,216,220)(97,109,221)(98,110,222)(99,111,223)(100,112,224)(101,113,225)(102,114,226)(103,115,227)(104,116,228)(105,117,229)(106,118,230)(107,119,231)(108,120,232), (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( (1,121,233)(2,122,234)(3,123,235)(4,124,236)(5,125,237)(6,126,238)(7,127,239)(8,128,240)(9,129,241)(10,130,242)(11,131,243)(12,132,244)(13,133,245)(14,134,246)(15,135,247)(16,136,248)(17,137,249)(18,138,250)(19,139,251)(20,140,252)(21,141,253)(22,142,254)(23,143,255)(24,144,256)(25,145,257)(26,146,258)(27,147,259)(28,148,260)(29,149,261)(30,150,262)(31,151,263)(32,152,264)(33,153,265)(34,154,266)(35,155,267)(36,156,268)(37,157,269)(38,158,270)(39,159,271)(40,160,272)(41,161,273)(42,162,274)(43,163,275)(44,164,276)(45,165,277)(46,166,278)(47,167,279)(48,168,280)(49,169,281)(50,170,282)(51,171,283)(52,172,284)(53,173,285)(54,174,286)(55,175,287)(56,176,288)(57,177,289)(58,178,290)(59,179,291)(60,180,292)(61,181,293)(62,182,294)(63,183,295)(64,184,296)(65,185,297)(66,186,298)(67,187,299)(68,188,300)(69,189,301)(70,190,302)(71,191,303)(72,192,304)(73,193,305)(74,194,306)(75,195,307)(76,196,308)(77,197,309)(78,198,310)(79,199,311)(80,200,312)(81,201,313)(82,202,314)(83,203,315)(84,204,316)(85,205,317)(86,206,318)(87,207,319)(88,208,320)(89,209,321)(90,210,322)(91,211,323)(92,212,324)(93,213,217)(94,214,218)(95,215,219)(96,216,220)(97,109,221)(98,110,222)(99,111,223)(100,112,224)(101,113,225)(102,114,226)(103,115,227)(104,116,228)(105,117,229)(106,118,230)(107,119,231)(108,120,232), (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([(1,121,233),(2,122,234),(3,123,235),(4,124,236),(5,125,237),(6,126,238),(7,127,239),(8,128,240),(9,129,241),(10,130,242),(11,131,243),(12,132,244),(13,133,245),(14,134,246),(15,135,247),(16,136,248),(17,137,249),(18,138,250),(19,139,251),(20,140,252),(21,141,253),(22,142,254),(23,143,255),(24,144,256),(25,145,257),(26,146,258),(27,147,259),(28,148,260),(29,149,261),(30,150,262),(31,151,263),(32,152,264),(33,153,265),(34,154,266),(35,155,267),(36,156,268),(37,157,269),(38,158,270),(39,159,271),(40,160,272),(41,161,273),(42,162,274),(43,163,275),(44,164,276),(45,165,277),(46,166,278),(47,167,279),(48,168,280),(49,169,281),(50,170,282),(51,171,283),(52,172,284),(53,173,285),(54,174,286),(55,175,287),(56,176,288),(57,177,289),(58,178,290),(59,179,291),(60,180,292),(61,181,293),(62,182,294),(63,183,295),(64,184,296),(65,185,297),(66,186,298),(67,187,299),(68,188,300),(69,189,301),(70,190,302),(71,191,303),(72,192,304),(73,193,305),(74,194,306),(75,195,307),(76,196,308),(77,197,309),(78,198,310),(79,199,311),(80,200,312),(81,201,313),(82,202,314),(83,203,315),(84,204,316),(85,205,317),(86,206,318),(87,207,319),(88,208,320),(89,209,321),(90,210,322),(91,211,323),(92,212,324),(93,213,217),(94,214,218),(95,215,219),(96,216,220),(97,109,221),(98,110,222),(99,111,223),(100,112,224),(101,113,225),(102,114,226),(103,115,227),(104,116,228),(105,117,229),(106,118,230),(107,119,231),(108,120,232)], [(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···9R12A···12P18A···18R27A···27BB36A···36AJ54A···54BB108A···108DD
order123···3446···69···912···1218···1827···2736···3654···54108···108
size111···1111···11···11···11···11···11···11···11···1

324 irreducible representations

dim111111111111111111
type++
imageC1C2C3C3C4C6C6C9C9C12C12C18C18C27C36C36C54C108
kernelC3×C108C3×C54C108C3×C36C3×C27C54C3×C18C36C3×C12C27C3×C9C18C3×C6C12C9C32C6C3
# reps116226212612412654241254108

Matrix representation of C3×C108 in GL2(𝔽109) generated by

10
045
,
940
010
G:=sub<GL(2,GF(109))| [1,0,0,45],[94,0,0,10] >;

C3×C108 in GAP, Magma, Sage, TeX

C_3\times C_{108}
% in TeX

G:=Group("C3xC108");
// GroupNames label

G:=SmallGroup(324,29);
// by ID

G=gap.SmallGroup(324,29);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,223,118]);
// Polycyclic

G:=Group<a,b|a^3=b^108=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C108 in TeX

׿
×
𝔽