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G = C6×C54order 324 = 22·34

Abelian group of type [6,54]

direct product, abelian, monomial

Aliases: C6×C54, SmallGroup(324,84)

Series: Derived Chief Lower central Upper central

C1 — C6×C54
C1C3C9C3×C9C3×C27C3×C54 — C6×C54
C1 — C6×C54
C1 — C6×C54

Generators and relations for C6×C54
 G = < a,b | a6=b54=1, ab=ba >

Subgroups: 70, all normal (12 characteristic)
C1, C2 [×3], C3, C3 [×3], C22, C6 [×12], C9, C9 [×2], C32, C2×C6, C2×C6 [×3], C18 [×9], C3×C6 [×3], C27 [×3], C3×C9, C2×C18, C2×C18 [×2], C62, C54 [×9], C3×C18 [×3], C3×C27, C2×C54 [×3], C6×C18, C3×C54 [×3], C6×C54
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C9 [×3], C32, C2×C6 [×4], C18 [×9], C3×C6 [×3], C27 [×3], C3×C9, C2×C18 [×3], C62, C54 [×9], C3×C18 [×3], C3×C27, C2×C54 [×3], C6×C18, C3×C54 [×3], C6×C54

Smallest permutation representation of C6×C54
Regular action on 324 points
Generators in S324
(1 255 179 316 88 141)(2 256 180 317 89 142)(3 257 181 318 90 143)(4 258 182 319 91 144)(5 259 183 320 92 145)(6 260 184 321 93 146)(7 261 185 322 94 147)(8 262 186 323 95 148)(9 263 187 324 96 149)(10 264 188 271 97 150)(11 265 189 272 98 151)(12 266 190 273 99 152)(13 267 191 274 100 153)(14 268 192 275 101 154)(15 269 193 276 102 155)(16 270 194 277 103 156)(17 217 195 278 104 157)(18 218 196 279 105 158)(19 219 197 280 106 159)(20 220 198 281 107 160)(21 221 199 282 108 161)(22 222 200 283 55 162)(23 223 201 284 56 109)(24 224 202 285 57 110)(25 225 203 286 58 111)(26 226 204 287 59 112)(27 227 205 288 60 113)(28 228 206 289 61 114)(29 229 207 290 62 115)(30 230 208 291 63 116)(31 231 209 292 64 117)(32 232 210 293 65 118)(33 233 211 294 66 119)(34 234 212 295 67 120)(35 235 213 296 68 121)(36 236 214 297 69 122)(37 237 215 298 70 123)(38 238 216 299 71 124)(39 239 163 300 72 125)(40 240 164 301 73 126)(41 241 165 302 74 127)(42 242 166 303 75 128)(43 243 167 304 76 129)(44 244 168 305 77 130)(45 245 169 306 78 131)(46 246 170 307 79 132)(47 247 171 308 80 133)(48 248 172 309 81 134)(49 249 173 310 82 135)(50 250 174 311 83 136)(51 251 175 312 84 137)(52 252 176 313 85 138)(53 253 177 314 86 139)(54 254 178 315 87 140)
(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)| (1,255,179,316,88,141)(2,256,180,317,89,142)(3,257,181,318,90,143)(4,258,182,319,91,144)(5,259,183,320,92,145)(6,260,184,321,93,146)(7,261,185,322,94,147)(8,262,186,323,95,148)(9,263,187,324,96,149)(10,264,188,271,97,150)(11,265,189,272,98,151)(12,266,190,273,99,152)(13,267,191,274,100,153)(14,268,192,275,101,154)(15,269,193,276,102,155)(16,270,194,277,103,156)(17,217,195,278,104,157)(18,218,196,279,105,158)(19,219,197,280,106,159)(20,220,198,281,107,160)(21,221,199,282,108,161)(22,222,200,283,55,162)(23,223,201,284,56,109)(24,224,202,285,57,110)(25,225,203,286,58,111)(26,226,204,287,59,112)(27,227,205,288,60,113)(28,228,206,289,61,114)(29,229,207,290,62,115)(30,230,208,291,63,116)(31,231,209,292,64,117)(32,232,210,293,65,118)(33,233,211,294,66,119)(34,234,212,295,67,120)(35,235,213,296,68,121)(36,236,214,297,69,122)(37,237,215,298,70,123)(38,238,216,299,71,124)(39,239,163,300,72,125)(40,240,164,301,73,126)(41,241,165,302,74,127)(42,242,166,303,75,128)(43,243,167,304,76,129)(44,244,168,305,77,130)(45,245,169,306,78,131)(46,246,170,307,79,132)(47,247,171,308,80,133)(48,248,172,309,81,134)(49,249,173,310,82,135)(50,250,174,311,83,136)(51,251,175,312,84,137)(52,252,176,313,85,138)(53,253,177,314,86,139)(54,254,178,315,87,140), (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( (1,255,179,316,88,141)(2,256,180,317,89,142)(3,257,181,318,90,143)(4,258,182,319,91,144)(5,259,183,320,92,145)(6,260,184,321,93,146)(7,261,185,322,94,147)(8,262,186,323,95,148)(9,263,187,324,96,149)(10,264,188,271,97,150)(11,265,189,272,98,151)(12,266,190,273,99,152)(13,267,191,274,100,153)(14,268,192,275,101,154)(15,269,193,276,102,155)(16,270,194,277,103,156)(17,217,195,278,104,157)(18,218,196,279,105,158)(19,219,197,280,106,159)(20,220,198,281,107,160)(21,221,199,282,108,161)(22,222,200,283,55,162)(23,223,201,284,56,109)(24,224,202,285,57,110)(25,225,203,286,58,111)(26,226,204,287,59,112)(27,227,205,288,60,113)(28,228,206,289,61,114)(29,229,207,290,62,115)(30,230,208,291,63,116)(31,231,209,292,64,117)(32,232,210,293,65,118)(33,233,211,294,66,119)(34,234,212,295,67,120)(35,235,213,296,68,121)(36,236,214,297,69,122)(37,237,215,298,70,123)(38,238,216,299,71,124)(39,239,163,300,72,125)(40,240,164,301,73,126)(41,241,165,302,74,127)(42,242,166,303,75,128)(43,243,167,304,76,129)(44,244,168,305,77,130)(45,245,169,306,78,131)(46,246,170,307,79,132)(47,247,171,308,80,133)(48,248,172,309,81,134)(49,249,173,310,82,135)(50,250,174,311,83,136)(51,251,175,312,84,137)(52,252,176,313,85,138)(53,253,177,314,86,139)(54,254,178,315,87,140), (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([(1,255,179,316,88,141),(2,256,180,317,89,142),(3,257,181,318,90,143),(4,258,182,319,91,144),(5,259,183,320,92,145),(6,260,184,321,93,146),(7,261,185,322,94,147),(8,262,186,323,95,148),(9,263,187,324,96,149),(10,264,188,271,97,150),(11,265,189,272,98,151),(12,266,190,273,99,152),(13,267,191,274,100,153),(14,268,192,275,101,154),(15,269,193,276,102,155),(16,270,194,277,103,156),(17,217,195,278,104,157),(18,218,196,279,105,158),(19,219,197,280,106,159),(20,220,198,281,107,160),(21,221,199,282,108,161),(22,222,200,283,55,162),(23,223,201,284,56,109),(24,224,202,285,57,110),(25,225,203,286,58,111),(26,226,204,287,59,112),(27,227,205,288,60,113),(28,228,206,289,61,114),(29,229,207,290,62,115),(30,230,208,291,63,116),(31,231,209,292,64,117),(32,232,210,293,65,118),(33,233,211,294,66,119),(34,234,212,295,67,120),(35,235,213,296,68,121),(36,236,214,297,69,122),(37,237,215,298,70,123),(38,238,216,299,71,124),(39,239,163,300,72,125),(40,240,164,301,73,126),(41,241,165,302,74,127),(42,242,166,303,75,128),(43,243,167,304,76,129),(44,244,168,305,77,130),(45,245,169,306,78,131),(46,246,170,307,79,132),(47,247,171,308,80,133),(48,248,172,309,81,134),(49,249,173,310,82,135),(50,250,174,311,83,136),(51,251,175,312,84,137),(52,252,176,313,85,138),(53,253,177,314,86,139),(54,254,178,315,87,140)], [(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 2A2B2C3A···3H6A···6X9A···9R18A···18BB27A···27BB54A···54FF
order12223···36···69···918···1827···2754···54
size11111···11···11···11···11···11···1

324 irreducible representations

dim111111111111
type++
imageC1C2C3C3C6C6C9C9C18C18C27C54
kernelC6×C54C3×C54C2×C54C6×C18C54C3×C18C2×C18C62C18C3×C6C2×C6C6
# reps1362186126361854162

Matrix representation of C6×C54 in GL2(𝔽109) generated by

460
01
,
630
031
G:=sub<GL(2,GF(109))| [46,0,0,1],[63,0,0,31] >;

C6×C54 in GAP, Magma, Sage, TeX

C_6\times C_{54}
% in TeX

G:=Group("C6xC54");
// GroupNames label

G:=SmallGroup(324,84);
// by ID

G=gap.SmallGroup(324,84);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,176,118]);
// Polycyclic

G:=Group<a,b|a^6=b^54=1,a*b=b*a>;
// generators/relations

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