metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17⋊4C16, C68.5C4, C34.3C8, C8.2D17, C136.2C2, C4.2Dic17, C2.(C17⋊3C8), SmallGroup(272,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C17⋊4C16 |
Generators and relations for C17⋊4C16
G = < a,b | a17=b16=1, bab-1=a-1 >
Smallest permutation representation of C17⋊4C16
►Regular action on 272 points
Generators in S272
(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)
(1 256 126 188 55 222 99 154 28 239 109 171 50 205 70 137)(2 272 127 204 56 238 100 170 29 255 110 187 51 221 71 153)(3 271 128 203 57 237 101 169 30 254 111 186 35 220 72 152)(4 270 129 202 58 236 102 168 31 253 112 185 36 219 73 151)(5 269 130 201 59 235 86 167 32 252 113 184 37 218 74 150)(6 268 131 200 60 234 87 166 33 251 114 183 38 217 75 149)(7 267 132 199 61 233 88 165 34 250 115 182 39 216 76 148)(8 266 133 198 62 232 89 164 18 249 116 181 40 215 77 147)(9 265 134 197 63 231 90 163 19 248 117 180 41 214 78 146)(10 264 135 196 64 230 91 162 20 247 118 179 42 213 79 145)(11 263 136 195 65 229 92 161 21 246 119 178 43 212 80 144)(12 262 120 194 66 228 93 160 22 245 103 177 44 211 81 143)(13 261 121 193 67 227 94 159 23 244 104 176 45 210 82 142)(14 260 122 192 68 226 95 158 24 243 105 175 46 209 83 141)(15 259 123 191 52 225 96 157 25 242 106 174 47 208 84 140)(16 258 124 190 53 224 97 156 26 241 107 173 48 207 85 139)(17 257 125 189 54 223 98 155 27 240 108 172 49 206 69 138)
G:=sub<Sym(272)| (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), (1,256,126,188,55,222,99,154,28,239,109,171,50,205,70,137)(2,272,127,204,56,238,100,170,29,255,110,187,51,221,71,153)(3,271,128,203,57,237,101,169,30,254,111,186,35,220,72,152)(4,270,129,202,58,236,102,168,31,253,112,185,36,219,73,151)(5,269,130,201,59,235,86,167,32,252,113,184,37,218,74,150)(6,268,131,200,60,234,87,166,33,251,114,183,38,217,75,149)(7,267,132,199,61,233,88,165,34,250,115,182,39,216,76,148)(8,266,133,198,62,232,89,164,18,249,116,181,40,215,77,147)(9,265,134,197,63,231,90,163,19,248,117,180,41,214,78,146)(10,264,135,196,64,230,91,162,20,247,118,179,42,213,79,145)(11,263,136,195,65,229,92,161,21,246,119,178,43,212,80,144)(12,262,120,194,66,228,93,160,22,245,103,177,44,211,81,143)(13,261,121,193,67,227,94,159,23,244,104,176,45,210,82,142)(14,260,122,192,68,226,95,158,24,243,105,175,46,209,83,141)(15,259,123,191,52,225,96,157,25,242,106,174,47,208,84,140)(16,258,124,190,53,224,97,156,26,241,107,173,48,207,85,139)(17,257,125,189,54,223,98,155,27,240,108,172,49,206,69,138)>;
G:=Group( (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), (1,256,126,188,55,222,99,154,28,239,109,171,50,205,70,137)(2,272,127,204,56,238,100,170,29,255,110,187,51,221,71,153)(3,271,128,203,57,237,101,169,30,254,111,186,35,220,72,152)(4,270,129,202,58,236,102,168,31,253,112,185,36,219,73,151)(5,269,130,201,59,235,86,167,32,252,113,184,37,218,74,150)(6,268,131,200,60,234,87,166,33,251,114,183,38,217,75,149)(7,267,132,199,61,233,88,165,34,250,115,182,39,216,76,148)(8,266,133,198,62,232,89,164,18,249,116,181,40,215,77,147)(9,265,134,197,63,231,90,163,19,248,117,180,41,214,78,146)(10,264,135,196,64,230,91,162,20,247,118,179,42,213,79,145)(11,263,136,195,65,229,92,161,21,246,119,178,43,212,80,144)(12,262,120,194,66,228,93,160,22,245,103,177,44,211,81,143)(13,261,121,193,67,227,94,159,23,244,104,176,45,210,82,142)(14,260,122,192,68,226,95,158,24,243,105,175,46,209,83,141)(15,259,123,191,52,225,96,157,25,242,106,174,47,208,84,140)(16,258,124,190,53,224,97,156,26,241,107,173,48,207,85,139)(17,257,125,189,54,223,98,155,27,240,108,172,49,206,69,138) );
G=PermutationGroup([[(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)], [(1,256,126,188,55,222,99,154,28,239,109,171,50,205,70,137),(2,272,127,204,56,238,100,170,29,255,110,187,51,221,71,153),(3,271,128,203,57,237,101,169,30,254,111,186,35,220,72,152),(4,270,129,202,58,236,102,168,31,253,112,185,36,219,73,151),(5,269,130,201,59,235,86,167,32,252,113,184,37,218,74,150),(6,268,131,200,60,234,87,166,33,251,114,183,38,217,75,149),(7,267,132,199,61,233,88,165,34,250,115,182,39,216,76,148),(8,266,133,198,62,232,89,164,18,249,116,181,40,215,77,147),(9,265,134,197,63,231,90,163,19,248,117,180,41,214,78,146),(10,264,135,196,64,230,91,162,20,247,118,179,42,213,79,145),(11,263,136,195,65,229,92,161,21,246,119,178,43,212,80,144),(12,262,120,194,66,228,93,160,22,245,103,177,44,211,81,143),(13,261,121,193,67,227,94,159,23,244,104,176,45,210,82,142),(14,260,122,192,68,226,95,158,24,243,105,175,46,209,83,141),(15,259,123,191,52,225,96,157,25,242,106,174,47,208,84,140),(16,258,124,190,53,224,97,156,26,241,107,173,48,207,85,139),(17,257,125,189,54,223,98,155,27,240,108,172,49,206,69,138)]])
80 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P | 136A | ··· | 136AF |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D17 | Dic17 | C17⋊3C8 | C17⋊4C16 |
| kernel | C17⋊4C16 | C136 | C68 | C34 | C17 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C17⋊4C16 ►in GL2(𝔽1361) generated by
| 0 | 1 |
| 1360 | 1205 |
| 934 | 198 |
| 121 | 427 |
G:=sub<GL(2,GF(1361))| [0,1360,1,1205],[934,121,198,427] >;
C17⋊4C16 in GAP, Magma, Sage, TeX
C_{17}\rtimes_4C_{16} % in TeX
G:=Group("C17:4C16"); // GroupNames label
G:=SmallGroup(272,1);
// by ID
G=gap.SmallGroup(272,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,10,26,42,6404]);
// Polycyclic
G:=Group<a,b|a^17=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
Export