direct product, abelian, monomial, 2-elementary
Aliases: C4×C68, SmallGroup(272,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C68 |
| C1 — C4×C68 |
| C1 — C4×C68 |
Generators and relations for C4×C68
G = < a,b | a4=b68=1, ab=ba >
Smallest permutation representation of C4×C68
►Regular action on 272 points
Generators in S272
(1 103 261 151)(2 104 262 152)(3 105 263 153)(4 106 264 154)(5 107 265 155)(6 108 266 156)(7 109 267 157)(8 110 268 158)(9 111 269 159)(10 112 270 160)(11 113 271 161)(12 114 272 162)(13 115 205 163)(14 116 206 164)(15 117 207 165)(16 118 208 166)(17 119 209 167)(18 120 210 168)(19 121 211 169)(20 122 212 170)(21 123 213 171)(22 124 214 172)(23 125 215 173)(24 126 216 174)(25 127 217 175)(26 128 218 176)(27 129 219 177)(28 130 220 178)(29 131 221 179)(30 132 222 180)(31 133 223 181)(32 134 224 182)(33 135 225 183)(34 136 226 184)(35 69 227 185)(36 70 228 186)(37 71 229 187)(38 72 230 188)(39 73 231 189)(40 74 232 190)(41 75 233 191)(42 76 234 192)(43 77 235 193)(44 78 236 194)(45 79 237 195)(46 80 238 196)(47 81 239 197)(48 82 240 198)(49 83 241 199)(50 84 242 200)(51 85 243 201)(52 86 244 202)(53 87 245 203)(54 88 246 204)(55 89 247 137)(56 90 248 138)(57 91 249 139)(58 92 250 140)(59 93 251 141)(60 94 252 142)(61 95 253 143)(62 96 254 144)(63 97 255 145)(64 98 256 146)(65 99 257 147)(66 100 258 148)(67 101 259 149)(68 102 260 150)
(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)
G:=sub<Sym(272)| (1,103,261,151)(2,104,262,152)(3,105,263,153)(4,106,264,154)(5,107,265,155)(6,108,266,156)(7,109,267,157)(8,110,268,158)(9,111,269,159)(10,112,270,160)(11,113,271,161)(12,114,272,162)(13,115,205,163)(14,116,206,164)(15,117,207,165)(16,118,208,166)(17,119,209,167)(18,120,210,168)(19,121,211,169)(20,122,212,170)(21,123,213,171)(22,124,214,172)(23,125,215,173)(24,126,216,174)(25,127,217,175)(26,128,218,176)(27,129,219,177)(28,130,220,178)(29,131,221,179)(30,132,222,180)(31,133,223,181)(32,134,224,182)(33,135,225,183)(34,136,226,184)(35,69,227,185)(36,70,228,186)(37,71,229,187)(38,72,230,188)(39,73,231,189)(40,74,232,190)(41,75,233,191)(42,76,234,192)(43,77,235,193)(44,78,236,194)(45,79,237,195)(46,80,238,196)(47,81,239,197)(48,82,240,198)(49,83,241,199)(50,84,242,200)(51,85,243,201)(52,86,244,202)(53,87,245,203)(54,88,246,204)(55,89,247,137)(56,90,248,138)(57,91,249,139)(58,92,250,140)(59,93,251,141)(60,94,252,142)(61,95,253,143)(62,96,254,144)(63,97,255,145)(64,98,256,146)(65,99,257,147)(66,100,258,148)(67,101,259,149)(68,102,260,150), (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)>;
G:=Group( (1,103,261,151)(2,104,262,152)(3,105,263,153)(4,106,264,154)(5,107,265,155)(6,108,266,156)(7,109,267,157)(8,110,268,158)(9,111,269,159)(10,112,270,160)(11,113,271,161)(12,114,272,162)(13,115,205,163)(14,116,206,164)(15,117,207,165)(16,118,208,166)(17,119,209,167)(18,120,210,168)(19,121,211,169)(20,122,212,170)(21,123,213,171)(22,124,214,172)(23,125,215,173)(24,126,216,174)(25,127,217,175)(26,128,218,176)(27,129,219,177)(28,130,220,178)(29,131,221,179)(30,132,222,180)(31,133,223,181)(32,134,224,182)(33,135,225,183)(34,136,226,184)(35,69,227,185)(36,70,228,186)(37,71,229,187)(38,72,230,188)(39,73,231,189)(40,74,232,190)(41,75,233,191)(42,76,234,192)(43,77,235,193)(44,78,236,194)(45,79,237,195)(46,80,238,196)(47,81,239,197)(48,82,240,198)(49,83,241,199)(50,84,242,200)(51,85,243,201)(52,86,244,202)(53,87,245,203)(54,88,246,204)(55,89,247,137)(56,90,248,138)(57,91,249,139)(58,92,250,140)(59,93,251,141)(60,94,252,142)(61,95,253,143)(62,96,254,144)(63,97,255,145)(64,98,256,146)(65,99,257,147)(66,100,258,148)(67,101,259,149)(68,102,260,150), (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) );
G=PermutationGroup([[(1,103,261,151),(2,104,262,152),(3,105,263,153),(4,106,264,154),(5,107,265,155),(6,108,266,156),(7,109,267,157),(8,110,268,158),(9,111,269,159),(10,112,270,160),(11,113,271,161),(12,114,272,162),(13,115,205,163),(14,116,206,164),(15,117,207,165),(16,118,208,166),(17,119,209,167),(18,120,210,168),(19,121,211,169),(20,122,212,170),(21,123,213,171),(22,124,214,172),(23,125,215,173),(24,126,216,174),(25,127,217,175),(26,128,218,176),(27,129,219,177),(28,130,220,178),(29,131,221,179),(30,132,222,180),(31,133,223,181),(32,134,224,182),(33,135,225,183),(34,136,226,184),(35,69,227,185),(36,70,228,186),(37,71,229,187),(38,72,230,188),(39,73,231,189),(40,74,232,190),(41,75,233,191),(42,76,234,192),(43,77,235,193),(44,78,236,194),(45,79,237,195),(46,80,238,196),(47,81,239,197),(48,82,240,198),(49,83,241,199),(50,84,242,200),(51,85,243,201),(52,86,244,202),(53,87,245,203),(54,88,246,204),(55,89,247,137),(56,90,248,138),(57,91,249,139),(58,92,250,140),(59,93,251,141),(60,94,252,142),(61,95,253,143),(62,96,254,144),(63,97,255,145),(64,98,256,146),(65,99,257,147),(66,100,258,148),(67,101,259,149),(68,102,260,150)], [(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)]])
272 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 17A | ··· | 17P | 34A | ··· | 34AV | 68A | ··· | 68GJ |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
272 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C17 | C34 | C68 |
| kernel | C4×C68 | C2×C68 | C68 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 16 | 48 | 192 |
Matrix representation of C4×C68 ►in GL2(𝔽137) generated by
| 37 | 0 |
| 0 | 100 |
| 126 | 0 |
| 0 | 133 |
G:=sub<GL(2,GF(137))| [37,0,0,100],[126,0,0,133] >;
C4×C68 in GAP, Magma, Sage, TeX
C_4\times C_{68} % in TeX
G:=Group("C4xC68"); // GroupNames label
G:=SmallGroup(272,20);
// by ID
G=gap.SmallGroup(272,20);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-2,340,686]);
// Polycyclic
G:=Group<a,b|a^4=b^68=1,a*b=b*a>;
// generators/relations
Export