metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17⋊3C16, C68.2C4, C34.2C8, C2.(C17⋊2C8), C4.2(C17⋊C4), C17⋊3C8.2C2, SmallGroup(272,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C17⋊3C16 |
Generators and relations for C17⋊3C16
G = < a,b | a17=b16=1, bab-1=a4 >
Smallest permutation representation of C17⋊3C16
►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)
(1 257 120 201 64 228 86 164 22 242 103 183 47 219 69 149)(2 270 136 188 65 224 102 168 23 255 119 187 48 215 85 153)(3 266 135 192 66 237 101 155 24 251 118 174 49 211 84 140)(4 262 134 196 67 233 100 159 25 247 117 178 50 207 83 144)(5 258 133 200 68 229 99 163 26 243 116 182 51 220 82 148)(6 271 132 204 52 225 98 167 27 239 115 186 35 216 81 152)(7 267 131 191 53 238 97 154 28 252 114 173 36 212 80 139)(8 263 130 195 54 234 96 158 29 248 113 177 37 208 79 143)(9 259 129 199 55 230 95 162 30 244 112 181 38 221 78 147)(10 272 128 203 56 226 94 166 31 240 111 185 39 217 77 151)(11 268 127 190 57 222 93 170 32 253 110 172 40 213 76 138)(12 264 126 194 58 235 92 157 33 249 109 176 41 209 75 142)(13 260 125 198 59 231 91 161 34 245 108 180 42 205 74 146)(14 256 124 202 60 227 90 165 18 241 107 184 43 218 73 150)(15 269 123 189 61 223 89 169 19 254 106 171 44 214 72 137)(16 265 122 193 62 236 88 156 20 250 105 175 45 210 71 141)(17 261 121 197 63 232 87 160 21 246 104 179 46 206 70 145)
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), (1,257,120,201,64,228,86,164,22,242,103,183,47,219,69,149)(2,270,136,188,65,224,102,168,23,255,119,187,48,215,85,153)(3,266,135,192,66,237,101,155,24,251,118,174,49,211,84,140)(4,262,134,196,67,233,100,159,25,247,117,178,50,207,83,144)(5,258,133,200,68,229,99,163,26,243,116,182,51,220,82,148)(6,271,132,204,52,225,98,167,27,239,115,186,35,216,81,152)(7,267,131,191,53,238,97,154,28,252,114,173,36,212,80,139)(8,263,130,195,54,234,96,158,29,248,113,177,37,208,79,143)(9,259,129,199,55,230,95,162,30,244,112,181,38,221,78,147)(10,272,128,203,56,226,94,166,31,240,111,185,39,217,77,151)(11,268,127,190,57,222,93,170,32,253,110,172,40,213,76,138)(12,264,126,194,58,235,92,157,33,249,109,176,41,209,75,142)(13,260,125,198,59,231,91,161,34,245,108,180,42,205,74,146)(14,256,124,202,60,227,90,165,18,241,107,184,43,218,73,150)(15,269,123,189,61,223,89,169,19,254,106,171,44,214,72,137)(16,265,122,193,62,236,88,156,20,250,105,175,45,210,71,141)(17,261,121,197,63,232,87,160,21,246,104,179,46,206,70,145)>;
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), (1,257,120,201,64,228,86,164,22,242,103,183,47,219,69,149)(2,270,136,188,65,224,102,168,23,255,119,187,48,215,85,153)(3,266,135,192,66,237,101,155,24,251,118,174,49,211,84,140)(4,262,134,196,67,233,100,159,25,247,117,178,50,207,83,144)(5,258,133,200,68,229,99,163,26,243,116,182,51,220,82,148)(6,271,132,204,52,225,98,167,27,239,115,186,35,216,81,152)(7,267,131,191,53,238,97,154,28,252,114,173,36,212,80,139)(8,263,130,195,54,234,96,158,29,248,113,177,37,208,79,143)(9,259,129,199,55,230,95,162,30,244,112,181,38,221,78,147)(10,272,128,203,56,226,94,166,31,240,111,185,39,217,77,151)(11,268,127,190,57,222,93,170,32,253,110,172,40,213,76,138)(12,264,126,194,58,235,92,157,33,249,109,176,41,209,75,142)(13,260,125,198,59,231,91,161,34,245,108,180,42,205,74,146)(14,256,124,202,60,227,90,165,18,241,107,184,43,218,73,150)(15,269,123,189,61,223,89,169,19,254,106,171,44,214,72,137)(16,265,122,193,62,236,88,156,20,250,105,175,45,210,71,141)(17,261,121,197,63,232,87,160,21,246,104,179,46,206,70,145) );
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)], [(1,257,120,201,64,228,86,164,22,242,103,183,47,219,69,149),(2,270,136,188,65,224,102,168,23,255,119,187,48,215,85,153),(3,266,135,192,66,237,101,155,24,251,118,174,49,211,84,140),(4,262,134,196,67,233,100,159,25,247,117,178,50,207,83,144),(5,258,133,200,68,229,99,163,26,243,116,182,51,220,82,148),(6,271,132,204,52,225,98,167,27,239,115,186,35,216,81,152),(7,267,131,191,53,238,97,154,28,252,114,173,36,212,80,139),(8,263,130,195,54,234,96,158,29,248,113,177,37,208,79,143),(9,259,129,199,55,230,95,162,30,244,112,181,38,221,78,147),(10,272,128,203,56,226,94,166,31,240,111,185,39,217,77,151),(11,268,127,190,57,222,93,170,32,253,110,172,40,213,76,138),(12,264,126,194,58,235,92,157,33,249,109,176,41,209,75,142),(13,260,125,198,59,231,91,161,34,245,108,180,42,205,74,146),(14,256,124,202,60,227,90,165,18,241,107,184,43,218,73,150),(15,269,123,189,61,223,89,169,19,254,106,171,44,214,72,137),(16,265,122,193,62,236,88,156,20,250,105,175,45,210,71,141),(17,261,121,197,63,232,87,160,21,246,104,179,46,206,70,145)]])
32 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 68A | ··· | 68H |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 17 | 17 | 17 | 17 | 34 | 34 | 34 | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 17 | ··· | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C4 | C8 | C16 | C17⋊C4 | C17⋊2C8 | C17⋊3C16 |
| kernel | C17⋊3C16 | C17⋊3C8 | C68 | C34 | C17 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of C17⋊3C16 ►in GL5(𝔽1361)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1360 | 1 | 0 | 0 |
| 0 | 1360 | 0 | 1 | 0 |
| 0 | 1360 | 0 | 0 | 1 |
| 0 | 480 | 19 | 1342 | 880 |
| 787 | 0 | 0 | 0 | 0 |
| 0 | 145 | 861 | 974 | 1303 |
| 0 | 1126 | 54 | 502 | 848 |
| 0 | 891 | 916 | 361 | 1080 |
| 0 | 51 | 66 | 1001 | 801 |
G:=sub<GL(5,GF(1361))| [1,0,0,0,0,0,1360,1360,1360,480,0,1,0,0,19,0,0,1,0,1342,0,0,0,1,880],[787,0,0,0,0,0,145,1126,891,51,0,861,54,916,66,0,974,502,361,1001,0,1303,848,1080,801] >;
C17⋊3C16 in GAP, Magma, Sage, TeX
C_{17}\rtimes_3C_{16} % in TeX
G:=Group("C17:3C16"); // GroupNames label
G:=SmallGroup(272,3);
// by ID
G=gap.SmallGroup(272,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,10,26,42,5204,3209]);
// Polycyclic
G:=Group<a,b|a^17=b^16=1,b*a*b^-1=a^4>;
// generators/relations
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