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G = C7×C42order 294 = 2·3·72

Abelian group of type [7,42]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C42, SmallGroup(294,23)

Series: Derived Chief Lower central Upper central

C1 — C7×C42
C1C7C72C7×C21 — C7×C42
C1 — C7×C42
C1 — C7×C42

Generators and relations for C7×C42
 G = < a,b | a7=b42=1, ab=ba >


Smallest permutation representation of C7×C42
Regular action on 294 points
Generators in S294
(1 169 244 81 104 253 154)(2 170 245 82 105 254 155)(3 171 246 83 106 255 156)(4 172 247 84 107 256 157)(5 173 248 43 108 257 158)(6 174 249 44 109 258 159)(7 175 250 45 110 259 160)(8 176 251 46 111 260 161)(9 177 252 47 112 261 162)(10 178 211 48 113 262 163)(11 179 212 49 114 263 164)(12 180 213 50 115 264 165)(13 181 214 51 116 265 166)(14 182 215 52 117 266 167)(15 183 216 53 118 267 168)(16 184 217 54 119 268 127)(17 185 218 55 120 269 128)(18 186 219 56 121 270 129)(19 187 220 57 122 271 130)(20 188 221 58 123 272 131)(21 189 222 59 124 273 132)(22 190 223 60 125 274 133)(23 191 224 61 126 275 134)(24 192 225 62 85 276 135)(25 193 226 63 86 277 136)(26 194 227 64 87 278 137)(27 195 228 65 88 279 138)(28 196 229 66 89 280 139)(29 197 230 67 90 281 140)(30 198 231 68 91 282 141)(31 199 232 69 92 283 142)(32 200 233 70 93 284 143)(33 201 234 71 94 285 144)(34 202 235 72 95 286 145)(35 203 236 73 96 287 146)(36 204 237 74 97 288 147)(37 205 238 75 98 289 148)(38 206 239 76 99 290 149)(39 207 240 77 100 291 150)(40 208 241 78 101 292 151)(41 209 242 79 102 293 152)(42 210 243 80 103 294 153)
(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)

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

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

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

294 conjugacy classes

class 1  2 3A3B6A6B7A···7AV14A···14AV21A···21CR42A···42CR
order1233667···714···1421···2142···42
size1111111···11···11···11···1

294 irreducible representations

dim11111111
type++
imageC1C2C3C6C7C14C21C42
kernelC7×C42C7×C21C7×C14C72C42C21C14C7
# reps112248489696

Matrix representation of C7×C42 in GL2(𝔽43) generated by

160
04
,
70
02
G:=sub<GL(2,GF(43))| [16,0,0,4],[7,0,0,2] >;

C7×C42 in GAP, Magma, Sage, TeX

C_7\times C_{42}
% in TeX

G:=Group("C7xC42");
// GroupNames label

G:=SmallGroup(294,23);
// by ID

G=gap.SmallGroup(294,23);
# by ID

G:=PCGroup([4,-2,-3,-7,-7]);
// Polycyclic

G:=Group<a,b|a^7=b^42=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C7×C42 in TeX

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