direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C17, C8.C34, Q8.C34, C136.3C2, C34.16D4, C68.19C22, C4.3(C2×C34), C2.5(D4×C17), (Q8×C17).2C2, SmallGroup(272,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C17
G = < a,b,c | a17=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q16×C17
►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)
(1 159 123 46 77 212 117 53)(2 160 124 47 78 213 118 54)(3 161 125 48 79 214 119 55)(4 162 126 49 80 215 103 56)(5 163 127 50 81 216 104 57)(6 164 128 51 82 217 105 58)(7 165 129 35 83 218 106 59)(8 166 130 36 84 219 107 60)(9 167 131 37 85 220 108 61)(10 168 132 38 69 221 109 62)(11 169 133 39 70 205 110 63)(12 170 134 40 71 206 111 64)(13 154 135 41 72 207 112 65)(14 155 136 42 73 208 113 66)(15 156 120 43 74 209 114 67)(16 157 121 44 75 210 115 68)(17 158 122 45 76 211 116 52)(18 256 90 190 174 232 146 244)(19 257 91 191 175 233 147 245)(20 258 92 192 176 234 148 246)(21 259 93 193 177 235 149 247)(22 260 94 194 178 236 150 248)(23 261 95 195 179 237 151 249)(24 262 96 196 180 238 152 250)(25 263 97 197 181 222 153 251)(26 264 98 198 182 223 137 252)(27 265 99 199 183 224 138 253)(28 266 100 200 184 225 139 254)(29 267 101 201 185 226 140 255)(30 268 102 202 186 227 141 239)(31 269 86 203 187 228 142 240)(32 270 87 204 171 229 143 241)(33 271 88 188 172 230 144 242)(34 272 89 189 173 231 145 243)
(1 232 77 256)(2 233 78 257)(3 234 79 258)(4 235 80 259)(5 236 81 260)(6 237 82 261)(7 238 83 262)(8 222 84 263)(9 223 85 264)(10 224 69 265)(11 225 70 266)(12 226 71 267)(13 227 72 268)(14 228 73 269)(15 229 74 270)(16 230 75 271)(17 231 76 272)(18 159 174 212)(19 160 175 213)(20 161 176 214)(21 162 177 215)(22 163 178 216)(23 164 179 217)(24 165 180 218)(25 166 181 219)(26 167 182 220)(27 168 183 221)(28 169 184 205)(29 170 185 206)(30 154 186 207)(31 155 187 208)(32 156 171 209)(33 157 172 210)(34 158 173 211)(35 96 59 152)(36 97 60 153)(37 98 61 137)(38 99 62 138)(39 100 63 139)(40 101 64 140)(41 102 65 141)(42 86 66 142)(43 87 67 143)(44 88 68 144)(45 89 52 145)(46 90 53 146)(47 91 54 147)(48 92 55 148)(49 93 56 149)(50 94 57 150)(51 95 58 151)(103 247 126 193)(104 248 127 194)(105 249 128 195)(106 250 129 196)(107 251 130 197)(108 252 131 198)(109 253 132 199)(110 254 133 200)(111 255 134 201)(112 239 135 202)(113 240 136 203)(114 241 120 204)(115 242 121 188)(116 243 122 189)(117 244 123 190)(118 245 124 191)(119 246 125 192)
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), (1,159,123,46,77,212,117,53)(2,160,124,47,78,213,118,54)(3,161,125,48,79,214,119,55)(4,162,126,49,80,215,103,56)(5,163,127,50,81,216,104,57)(6,164,128,51,82,217,105,58)(7,165,129,35,83,218,106,59)(8,166,130,36,84,219,107,60)(9,167,131,37,85,220,108,61)(10,168,132,38,69,221,109,62)(11,169,133,39,70,205,110,63)(12,170,134,40,71,206,111,64)(13,154,135,41,72,207,112,65)(14,155,136,42,73,208,113,66)(15,156,120,43,74,209,114,67)(16,157,121,44,75,210,115,68)(17,158,122,45,76,211,116,52)(18,256,90,190,174,232,146,244)(19,257,91,191,175,233,147,245)(20,258,92,192,176,234,148,246)(21,259,93,193,177,235,149,247)(22,260,94,194,178,236,150,248)(23,261,95,195,179,237,151,249)(24,262,96,196,180,238,152,250)(25,263,97,197,181,222,153,251)(26,264,98,198,182,223,137,252)(27,265,99,199,183,224,138,253)(28,266,100,200,184,225,139,254)(29,267,101,201,185,226,140,255)(30,268,102,202,186,227,141,239)(31,269,86,203,187,228,142,240)(32,270,87,204,171,229,143,241)(33,271,88,188,172,230,144,242)(34,272,89,189,173,231,145,243), (1,232,77,256)(2,233,78,257)(3,234,79,258)(4,235,80,259)(5,236,81,260)(6,237,82,261)(7,238,83,262)(8,222,84,263)(9,223,85,264)(10,224,69,265)(11,225,70,266)(12,226,71,267)(13,227,72,268)(14,228,73,269)(15,229,74,270)(16,230,75,271)(17,231,76,272)(18,159,174,212)(19,160,175,213)(20,161,176,214)(21,162,177,215)(22,163,178,216)(23,164,179,217)(24,165,180,218)(25,166,181,219)(26,167,182,220)(27,168,183,221)(28,169,184,205)(29,170,185,206)(30,154,186,207)(31,155,187,208)(32,156,171,209)(33,157,172,210)(34,158,173,211)(35,96,59,152)(36,97,60,153)(37,98,61,137)(38,99,62,138)(39,100,63,139)(40,101,64,140)(41,102,65,141)(42,86,66,142)(43,87,67,143)(44,88,68,144)(45,89,52,145)(46,90,53,146)(47,91,54,147)(48,92,55,148)(49,93,56,149)(50,94,57,150)(51,95,58,151)(103,247,126,193)(104,248,127,194)(105,249,128,195)(106,250,129,196)(107,251,130,197)(108,252,131,198)(109,253,132,199)(110,254,133,200)(111,255,134,201)(112,239,135,202)(113,240,136,203)(114,241,120,204)(115,242,121,188)(116,243,122,189)(117,244,123,190)(118,245,124,191)(119,246,125,192)>;
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), (1,159,123,46,77,212,117,53)(2,160,124,47,78,213,118,54)(3,161,125,48,79,214,119,55)(4,162,126,49,80,215,103,56)(5,163,127,50,81,216,104,57)(6,164,128,51,82,217,105,58)(7,165,129,35,83,218,106,59)(8,166,130,36,84,219,107,60)(9,167,131,37,85,220,108,61)(10,168,132,38,69,221,109,62)(11,169,133,39,70,205,110,63)(12,170,134,40,71,206,111,64)(13,154,135,41,72,207,112,65)(14,155,136,42,73,208,113,66)(15,156,120,43,74,209,114,67)(16,157,121,44,75,210,115,68)(17,158,122,45,76,211,116,52)(18,256,90,190,174,232,146,244)(19,257,91,191,175,233,147,245)(20,258,92,192,176,234,148,246)(21,259,93,193,177,235,149,247)(22,260,94,194,178,236,150,248)(23,261,95,195,179,237,151,249)(24,262,96,196,180,238,152,250)(25,263,97,197,181,222,153,251)(26,264,98,198,182,223,137,252)(27,265,99,199,183,224,138,253)(28,266,100,200,184,225,139,254)(29,267,101,201,185,226,140,255)(30,268,102,202,186,227,141,239)(31,269,86,203,187,228,142,240)(32,270,87,204,171,229,143,241)(33,271,88,188,172,230,144,242)(34,272,89,189,173,231,145,243), (1,232,77,256)(2,233,78,257)(3,234,79,258)(4,235,80,259)(5,236,81,260)(6,237,82,261)(7,238,83,262)(8,222,84,263)(9,223,85,264)(10,224,69,265)(11,225,70,266)(12,226,71,267)(13,227,72,268)(14,228,73,269)(15,229,74,270)(16,230,75,271)(17,231,76,272)(18,159,174,212)(19,160,175,213)(20,161,176,214)(21,162,177,215)(22,163,178,216)(23,164,179,217)(24,165,180,218)(25,166,181,219)(26,167,182,220)(27,168,183,221)(28,169,184,205)(29,170,185,206)(30,154,186,207)(31,155,187,208)(32,156,171,209)(33,157,172,210)(34,158,173,211)(35,96,59,152)(36,97,60,153)(37,98,61,137)(38,99,62,138)(39,100,63,139)(40,101,64,140)(41,102,65,141)(42,86,66,142)(43,87,67,143)(44,88,68,144)(45,89,52,145)(46,90,53,146)(47,91,54,147)(48,92,55,148)(49,93,56,149)(50,94,57,150)(51,95,58,151)(103,247,126,193)(104,248,127,194)(105,249,128,195)(106,250,129,196)(107,251,130,197)(108,252,131,198)(109,253,132,199)(110,254,133,200)(111,255,134,201)(112,239,135,202)(113,240,136,203)(114,241,120,204)(115,242,121,188)(116,243,122,189)(117,244,123,190)(118,245,124,191)(119,246,125,192) );
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)], [(1,159,123,46,77,212,117,53),(2,160,124,47,78,213,118,54),(3,161,125,48,79,214,119,55),(4,162,126,49,80,215,103,56),(5,163,127,50,81,216,104,57),(6,164,128,51,82,217,105,58),(7,165,129,35,83,218,106,59),(8,166,130,36,84,219,107,60),(9,167,131,37,85,220,108,61),(10,168,132,38,69,221,109,62),(11,169,133,39,70,205,110,63),(12,170,134,40,71,206,111,64),(13,154,135,41,72,207,112,65),(14,155,136,42,73,208,113,66),(15,156,120,43,74,209,114,67),(16,157,121,44,75,210,115,68),(17,158,122,45,76,211,116,52),(18,256,90,190,174,232,146,244),(19,257,91,191,175,233,147,245),(20,258,92,192,176,234,148,246),(21,259,93,193,177,235,149,247),(22,260,94,194,178,236,150,248),(23,261,95,195,179,237,151,249),(24,262,96,196,180,238,152,250),(25,263,97,197,181,222,153,251),(26,264,98,198,182,223,137,252),(27,265,99,199,183,224,138,253),(28,266,100,200,184,225,139,254),(29,267,101,201,185,226,140,255),(30,268,102,202,186,227,141,239),(31,269,86,203,187,228,142,240),(32,270,87,204,171,229,143,241),(33,271,88,188,172,230,144,242),(34,272,89,189,173,231,145,243)], [(1,232,77,256),(2,233,78,257),(3,234,79,258),(4,235,80,259),(5,236,81,260),(6,237,82,261),(7,238,83,262),(8,222,84,263),(9,223,85,264),(10,224,69,265),(11,225,70,266),(12,226,71,267),(13,227,72,268),(14,228,73,269),(15,229,74,270),(16,230,75,271),(17,231,76,272),(18,159,174,212),(19,160,175,213),(20,161,176,214),(21,162,177,215),(22,163,178,216),(23,164,179,217),(24,165,180,218),(25,166,181,219),(26,167,182,220),(27,168,183,221),(28,169,184,205),(29,170,185,206),(30,154,186,207),(31,155,187,208),(32,156,171,209),(33,157,172,210),(34,158,173,211),(35,96,59,152),(36,97,60,153),(37,98,61,137),(38,99,62,138),(39,100,63,139),(40,101,64,140),(41,102,65,141),(42,86,66,142),(43,87,67,143),(44,88,68,144),(45,89,52,145),(46,90,53,146),(47,91,54,147),(48,92,55,148),(49,93,56,149),(50,94,57,150),(51,95,58,151),(103,247,126,193),(104,248,127,194),(105,249,128,195),(106,250,129,196),(107,251,130,197),(108,252,131,198),(109,253,132,199),(110,254,133,200),(111,255,134,201),(112,239,135,202),(113,240,136,203),(114,241,120,204),(115,242,121,188),(116,243,122,189),(117,244,123,190),(118,245,124,191),(119,246,125,192)]])
119 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 17A | ··· | 17P | 34A | ··· | 34P | 68A | ··· | 68P | 68Q | ··· | 68AV | 136A | ··· | 136AF |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
119 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | D4 | Q16 | D4×C17 | Q16×C17 |
| kernel | Q16×C17 | C136 | Q8×C17 | Q16 | C8 | Q8 | C34 | C17 | C2 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 | 1 | 2 | 16 | 32 |
Matrix representation of Q16×C17 ►in GL3(𝔽137) generated by
| 119 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 31 |
| 0 | 53 | 31 |
| 1 | 0 | 0 |
| 0 | 91 | 8 |
| 0 | 95 | 46 |
G:=sub<GL(3,GF(137))| [119,0,0,0,1,0,0,0,1],[1,0,0,0,0,53,0,31,31],[1,0,0,0,91,95,0,8,46] >;
Q16×C17 in GAP, Magma, Sage, TeX
Q_{16}\times C_{17} % in TeX
G:=Group("Q16xC17"); // GroupNames label
G:=SmallGroup(272,27);
// by ID
G=gap.SmallGroup(272,27);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-2,680,701,686,4083,2048,58]);
// Polycyclic
G:=Group<a,b,c|a^17=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export