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## G = C6×C54order 324 = 22·34

### Abelian group of type [6,54]

Aliases: C6×C54, SmallGroup(324,84)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C54
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27 — C3×C54 — C6×C54
 Lower central C1 — C6×C54
 Upper central 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, C3, C3, C22, C6, C9, C9, C32, C2×C6, C2×C6, C18, C3×C6, C27, C3×C9, C2×C18, C2×C18, C62, C54, C3×C18, C3×C27, C2×C54, C6×C18, C3×C54, C6×C54
Quotients: C1, C2, C3, C22, C6, C9, C32, C2×C6, C18, C3×C6, C27, C3×C9, C2×C18, C62, C54, C3×C18, C3×C27, C2×C54, C6×C18, C3×C54, C6×C54

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

324 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C3 C6 C6 C9 C9 C18 C18 C27 C54 kernel C6×C54 C3×C54 C2×C54 C6×C18 C54 C3×C18 C2×C18 C62 C18 C3×C6 C2×C6 C6 # reps 1 3 6 2 18 6 12 6 36 18 54 162

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

 46 0 0 1
,
 63 0 0 31
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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