metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C17⋊2Q16, Q8.D17, C4.4D34, C34.10D4, C68.4C22, Dic34.2C2, C17⋊3C8.1C2, (Q8×C17).1C2, C2.7(C17⋊D4), SmallGroup(272,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C17⋊Q16
G = < a,b,c | a17=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C17⋊Q16
►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)
(1 239 49 222 25 256 60 205)(2 255 50 238 26 272 61 221)(3 254 51 237 27 271 62 220)(4 253 35 236 28 270 63 219)(5 252 36 235 29 269 64 218)(6 251 37 234 30 268 65 217)(7 250 38 233 31 267 66 216)(8 249 39 232 32 266 67 215)(9 248 40 231 33 265 68 214)(10 247 41 230 34 264 52 213)(11 246 42 229 18 263 53 212)(12 245 43 228 19 262 54 211)(13 244 44 227 20 261 55 210)(14 243 45 226 21 260 56 209)(15 242 46 225 22 259 57 208)(16 241 47 224 23 258 58 207)(17 240 48 223 24 257 59 206)(69 186 129 152 97 203 113 169)(70 185 130 151 98 202 114 168)(71 184 131 150 99 201 115 167)(72 183 132 149 100 200 116 166)(73 182 133 148 101 199 117 165)(74 181 134 147 102 198 118 164)(75 180 135 146 86 197 119 163)(76 179 136 145 87 196 103 162)(77 178 120 144 88 195 104 161)(78 177 121 143 89 194 105 160)(79 176 122 142 90 193 106 159)(80 175 123 141 91 192 107 158)(81 174 124 140 92 191 108 157)(82 173 125 139 93 190 109 156)(83 172 126 138 94 189 110 155)(84 171 127 137 95 188 111 154)(85 187 128 153 96 204 112 170)
(1 95 25 84)(2 96 26 85)(3 97 27 69)(4 98 28 70)(5 99 29 71)(6 100 30 72)(7 101 31 73)(8 102 32 74)(9 86 33 75)(10 87 34 76)(11 88 18 77)(12 89 19 78)(13 90 20 79)(14 91 21 80)(15 92 22 81)(16 93 23 82)(17 94 24 83)(35 130 63 114)(36 131 64 115)(37 132 65 116)(38 133 66 117)(39 134 67 118)(40 135 68 119)(41 136 52 103)(42 120 53 104)(43 121 54 105)(44 122 55 106)(45 123 56 107)(46 124 57 108)(47 125 58 109)(48 126 59 110)(49 127 60 111)(50 128 61 112)(51 129 62 113)(137 256 154 239)(138 257 155 240)(139 258 156 241)(140 259 157 242)(141 260 158 243)(142 261 159 244)(143 262 160 245)(144 263 161 246)(145 264 162 247)(146 265 163 248)(147 266 164 249)(148 267 165 250)(149 268 166 251)(150 269 167 252)(151 270 168 253)(152 271 169 254)(153 272 170 255)(171 205 188 222)(172 206 189 223)(173 207 190 224)(174 208 191 225)(175 209 192 226)(176 210 193 227)(177 211 194 228)(178 212 195 229)(179 213 196 230)(180 214 197 231)(181 215 198 232)(182 216 199 233)(183 217 200 234)(184 218 201 235)(185 219 202 236)(186 220 203 237)(187 221 204 238)
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), (1,239,49,222,25,256,60,205)(2,255,50,238,26,272,61,221)(3,254,51,237,27,271,62,220)(4,253,35,236,28,270,63,219)(5,252,36,235,29,269,64,218)(6,251,37,234,30,268,65,217)(7,250,38,233,31,267,66,216)(8,249,39,232,32,266,67,215)(9,248,40,231,33,265,68,214)(10,247,41,230,34,264,52,213)(11,246,42,229,18,263,53,212)(12,245,43,228,19,262,54,211)(13,244,44,227,20,261,55,210)(14,243,45,226,21,260,56,209)(15,242,46,225,22,259,57,208)(16,241,47,224,23,258,58,207)(17,240,48,223,24,257,59,206)(69,186,129,152,97,203,113,169)(70,185,130,151,98,202,114,168)(71,184,131,150,99,201,115,167)(72,183,132,149,100,200,116,166)(73,182,133,148,101,199,117,165)(74,181,134,147,102,198,118,164)(75,180,135,146,86,197,119,163)(76,179,136,145,87,196,103,162)(77,178,120,144,88,195,104,161)(78,177,121,143,89,194,105,160)(79,176,122,142,90,193,106,159)(80,175,123,141,91,192,107,158)(81,174,124,140,92,191,108,157)(82,173,125,139,93,190,109,156)(83,172,126,138,94,189,110,155)(84,171,127,137,95,188,111,154)(85,187,128,153,96,204,112,170), (1,95,25,84)(2,96,26,85)(3,97,27,69)(4,98,28,70)(5,99,29,71)(6,100,30,72)(7,101,31,73)(8,102,32,74)(9,86,33,75)(10,87,34,76)(11,88,18,77)(12,89,19,78)(13,90,20,79)(14,91,21,80)(15,92,22,81)(16,93,23,82)(17,94,24,83)(35,130,63,114)(36,131,64,115)(37,132,65,116)(38,133,66,117)(39,134,67,118)(40,135,68,119)(41,136,52,103)(42,120,53,104)(43,121,54,105)(44,122,55,106)(45,123,56,107)(46,124,57,108)(47,125,58,109)(48,126,59,110)(49,127,60,111)(50,128,61,112)(51,129,62,113)(137,256,154,239)(138,257,155,240)(139,258,156,241)(140,259,157,242)(141,260,158,243)(142,261,159,244)(143,262,160,245)(144,263,161,246)(145,264,162,247)(146,265,163,248)(147,266,164,249)(148,267,165,250)(149,268,166,251)(150,269,167,252)(151,270,168,253)(152,271,169,254)(153,272,170,255)(171,205,188,222)(172,206,189,223)(173,207,190,224)(174,208,191,225)(175,209,192,226)(176,210,193,227)(177,211,194,228)(178,212,195,229)(179,213,196,230)(180,214,197,231)(181,215,198,232)(182,216,199,233)(183,217,200,234)(184,218,201,235)(185,219,202,236)(186,220,203,237)(187,221,204,238)>;
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), (1,239,49,222,25,256,60,205)(2,255,50,238,26,272,61,221)(3,254,51,237,27,271,62,220)(4,253,35,236,28,270,63,219)(5,252,36,235,29,269,64,218)(6,251,37,234,30,268,65,217)(7,250,38,233,31,267,66,216)(8,249,39,232,32,266,67,215)(9,248,40,231,33,265,68,214)(10,247,41,230,34,264,52,213)(11,246,42,229,18,263,53,212)(12,245,43,228,19,262,54,211)(13,244,44,227,20,261,55,210)(14,243,45,226,21,260,56,209)(15,242,46,225,22,259,57,208)(16,241,47,224,23,258,58,207)(17,240,48,223,24,257,59,206)(69,186,129,152,97,203,113,169)(70,185,130,151,98,202,114,168)(71,184,131,150,99,201,115,167)(72,183,132,149,100,200,116,166)(73,182,133,148,101,199,117,165)(74,181,134,147,102,198,118,164)(75,180,135,146,86,197,119,163)(76,179,136,145,87,196,103,162)(77,178,120,144,88,195,104,161)(78,177,121,143,89,194,105,160)(79,176,122,142,90,193,106,159)(80,175,123,141,91,192,107,158)(81,174,124,140,92,191,108,157)(82,173,125,139,93,190,109,156)(83,172,126,138,94,189,110,155)(84,171,127,137,95,188,111,154)(85,187,128,153,96,204,112,170), (1,95,25,84)(2,96,26,85)(3,97,27,69)(4,98,28,70)(5,99,29,71)(6,100,30,72)(7,101,31,73)(8,102,32,74)(9,86,33,75)(10,87,34,76)(11,88,18,77)(12,89,19,78)(13,90,20,79)(14,91,21,80)(15,92,22,81)(16,93,23,82)(17,94,24,83)(35,130,63,114)(36,131,64,115)(37,132,65,116)(38,133,66,117)(39,134,67,118)(40,135,68,119)(41,136,52,103)(42,120,53,104)(43,121,54,105)(44,122,55,106)(45,123,56,107)(46,124,57,108)(47,125,58,109)(48,126,59,110)(49,127,60,111)(50,128,61,112)(51,129,62,113)(137,256,154,239)(138,257,155,240)(139,258,156,241)(140,259,157,242)(141,260,158,243)(142,261,159,244)(143,262,160,245)(144,263,161,246)(145,264,162,247)(146,265,163,248)(147,266,164,249)(148,267,165,250)(149,268,166,251)(150,269,167,252)(151,270,168,253)(152,271,169,254)(153,272,170,255)(171,205,188,222)(172,206,189,223)(173,207,190,224)(174,208,191,225)(175,209,192,226)(176,210,193,227)(177,211,194,228)(178,212,195,229)(179,213,196,230)(180,214,197,231)(181,215,198,232)(182,216,199,233)(183,217,200,234)(184,218,201,235)(185,219,202,236)(186,220,203,237)(187,221,204,238) );
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)], [(1,239,49,222,25,256,60,205),(2,255,50,238,26,272,61,221),(3,254,51,237,27,271,62,220),(4,253,35,236,28,270,63,219),(5,252,36,235,29,269,64,218),(6,251,37,234,30,268,65,217),(7,250,38,233,31,267,66,216),(8,249,39,232,32,266,67,215),(9,248,40,231,33,265,68,214),(10,247,41,230,34,264,52,213),(11,246,42,229,18,263,53,212),(12,245,43,228,19,262,54,211),(13,244,44,227,20,261,55,210),(14,243,45,226,21,260,56,209),(15,242,46,225,22,259,57,208),(16,241,47,224,23,258,58,207),(17,240,48,223,24,257,59,206),(69,186,129,152,97,203,113,169),(70,185,130,151,98,202,114,168),(71,184,131,150,99,201,115,167),(72,183,132,149,100,200,116,166),(73,182,133,148,101,199,117,165),(74,181,134,147,102,198,118,164),(75,180,135,146,86,197,119,163),(76,179,136,145,87,196,103,162),(77,178,120,144,88,195,104,161),(78,177,121,143,89,194,105,160),(79,176,122,142,90,193,106,159),(80,175,123,141,91,192,107,158),(81,174,124,140,92,191,108,157),(82,173,125,139,93,190,109,156),(83,172,126,138,94,189,110,155),(84,171,127,137,95,188,111,154),(85,187,128,153,96,204,112,170)], [(1,95,25,84),(2,96,26,85),(3,97,27,69),(4,98,28,70),(5,99,29,71),(6,100,30,72),(7,101,31,73),(8,102,32,74),(9,86,33,75),(10,87,34,76),(11,88,18,77),(12,89,19,78),(13,90,20,79),(14,91,21,80),(15,92,22,81),(16,93,23,82),(17,94,24,83),(35,130,63,114),(36,131,64,115),(37,132,65,116),(38,133,66,117),(39,134,67,118),(40,135,68,119),(41,136,52,103),(42,120,53,104),(43,121,54,105),(44,122,55,106),(45,123,56,107),(46,124,57,108),(47,125,58,109),(48,126,59,110),(49,127,60,111),(50,128,61,112),(51,129,62,113),(137,256,154,239),(138,257,155,240),(139,258,156,241),(140,259,157,242),(141,260,158,243),(142,261,159,244),(143,262,160,245),(144,263,161,246),(145,264,162,247),(146,265,163,248),(147,266,164,249),(148,267,165,250),(149,268,166,251),(150,269,167,252),(151,270,168,253),(152,271,169,254),(153,272,170,255),(171,205,188,222),(172,206,189,223),(173,207,190,224),(174,208,191,225),(175,209,192,226),(176,210,193,227),(177,211,194,228),(178,212,195,229),(179,213,196,230),(180,214,197,231),(181,215,198,232),(182,216,199,233),(183,217,200,234),(184,218,201,235),(185,219,202,236),(186,220,203,237),(187,221,204,238)]])
47 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68X |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 4 | 68 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | Q16 | D17 | D34 | C17⋊D4 | C17⋊Q16 |
| kernel | C17⋊Q16 | C17⋊3C8 | Dic34 | Q8×C17 | C34 | C17 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 8 |
Matrix representation of C17⋊Q16 ►in GL4(𝔽137) generated by
| 40 | 1 | 0 | 0 |
| 77 | 91 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 67 | 133 | 0 | 0 |
| 26 | 70 | 0 | 0 |
| 0 | 0 | 24 | 128 |
| 0 | 0 | 25 | 82 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 51 | 42 |
| 0 | 0 | 62 | 86 |
G:=sub<GL(4,GF(137))| [40,77,0,0,1,91,0,0,0,0,1,0,0,0,0,1],[67,26,0,0,133,70,0,0,0,0,24,25,0,0,128,82],[1,0,0,0,0,1,0,0,0,0,51,62,0,0,42,86] >;
C17⋊Q16 in GAP, Magma, Sage, TeX
C_{17}\rtimes Q_{16} % in TeX
G:=Group("C17:Q16"); // GroupNames label
G:=SmallGroup(272,18);
// by ID
G=gap.SmallGroup(272,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,40,61,46,182,97,42,6404]);
// Polycyclic
G:=Group<a,b,c|a^17=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export