direct product, p-group, elementary abelian, monomial
Aliases: C172, SmallGroup(289,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C172 |
| C1 — C172 |
| C1 — C172 |
Generators and relations for C172
G = < a,b | a17=b17=1, ab=ba >
Smallest permutation representation of C172
►Regular action on 289 points
Generators in S289
(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)(273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289)
(1 114 196 76 53 40 276 142 88 215 237 164 136 182 246 26 263)(2 115 197 77 54 41 277 143 89 216 238 165 120 183 247 27 264)(3 116 198 78 55 42 278 144 90 217 222 166 121 184 248 28 265)(4 117 199 79 56 43 279 145 91 218 223 167 122 185 249 29 266)(5 118 200 80 57 44 280 146 92 219 224 168 123 186 250 30 267)(6 119 201 81 58 45 281 147 93 220 225 169 124 187 251 31 268)(7 103 202 82 59 46 282 148 94 221 226 170 125 171 252 32 269)(8 104 203 83 60 47 283 149 95 205 227 154 126 172 253 33 270)(9 105 204 84 61 48 284 150 96 206 228 155 127 173 254 34 271)(10 106 188 85 62 49 285 151 97 207 229 156 128 174 255 18 272)(11 107 189 69 63 50 286 152 98 208 230 157 129 175 239 19 256)(12 108 190 70 64 51 287 153 99 209 231 158 130 176 240 20 257)(13 109 191 71 65 35 288 137 100 210 232 159 131 177 241 21 258)(14 110 192 72 66 36 289 138 101 211 233 160 132 178 242 22 259)(15 111 193 73 67 37 273 139 102 212 234 161 133 179 243 23 260)(16 112 194 74 68 38 274 140 86 213 235 162 134 180 244 24 261)(17 113 195 75 52 39 275 141 87 214 236 163 135 181 245 25 262)
G:=sub<Sym(289)| (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)(273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289), (1,114,196,76,53,40,276,142,88,215,237,164,136,182,246,26,263)(2,115,197,77,54,41,277,143,89,216,238,165,120,183,247,27,264)(3,116,198,78,55,42,278,144,90,217,222,166,121,184,248,28,265)(4,117,199,79,56,43,279,145,91,218,223,167,122,185,249,29,266)(5,118,200,80,57,44,280,146,92,219,224,168,123,186,250,30,267)(6,119,201,81,58,45,281,147,93,220,225,169,124,187,251,31,268)(7,103,202,82,59,46,282,148,94,221,226,170,125,171,252,32,269)(8,104,203,83,60,47,283,149,95,205,227,154,126,172,253,33,270)(9,105,204,84,61,48,284,150,96,206,228,155,127,173,254,34,271)(10,106,188,85,62,49,285,151,97,207,229,156,128,174,255,18,272)(11,107,189,69,63,50,286,152,98,208,230,157,129,175,239,19,256)(12,108,190,70,64,51,287,153,99,209,231,158,130,176,240,20,257)(13,109,191,71,65,35,288,137,100,210,232,159,131,177,241,21,258)(14,110,192,72,66,36,289,138,101,211,233,160,132,178,242,22,259)(15,111,193,73,67,37,273,139,102,212,234,161,133,179,243,23,260)(16,112,194,74,68,38,274,140,86,213,235,162,134,180,244,24,261)(17,113,195,75,52,39,275,141,87,214,236,163,135,181,245,25,262)>;
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)(273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289), (1,114,196,76,53,40,276,142,88,215,237,164,136,182,246,26,263)(2,115,197,77,54,41,277,143,89,216,238,165,120,183,247,27,264)(3,116,198,78,55,42,278,144,90,217,222,166,121,184,248,28,265)(4,117,199,79,56,43,279,145,91,218,223,167,122,185,249,29,266)(5,118,200,80,57,44,280,146,92,219,224,168,123,186,250,30,267)(6,119,201,81,58,45,281,147,93,220,225,169,124,187,251,31,268)(7,103,202,82,59,46,282,148,94,221,226,170,125,171,252,32,269)(8,104,203,83,60,47,283,149,95,205,227,154,126,172,253,33,270)(9,105,204,84,61,48,284,150,96,206,228,155,127,173,254,34,271)(10,106,188,85,62,49,285,151,97,207,229,156,128,174,255,18,272)(11,107,189,69,63,50,286,152,98,208,230,157,129,175,239,19,256)(12,108,190,70,64,51,287,153,99,209,231,158,130,176,240,20,257)(13,109,191,71,65,35,288,137,100,210,232,159,131,177,241,21,258)(14,110,192,72,66,36,289,138,101,211,233,160,132,178,242,22,259)(15,111,193,73,67,37,273,139,102,212,234,161,133,179,243,23,260)(16,112,194,74,68,38,274,140,86,213,235,162,134,180,244,24,261)(17,113,195,75,52,39,275,141,87,214,236,163,135,181,245,25,262) );
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),(273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289)], [(1,114,196,76,53,40,276,142,88,215,237,164,136,182,246,26,263),(2,115,197,77,54,41,277,143,89,216,238,165,120,183,247,27,264),(3,116,198,78,55,42,278,144,90,217,222,166,121,184,248,28,265),(4,117,199,79,56,43,279,145,91,218,223,167,122,185,249,29,266),(5,118,200,80,57,44,280,146,92,219,224,168,123,186,250,30,267),(6,119,201,81,58,45,281,147,93,220,225,169,124,187,251,31,268),(7,103,202,82,59,46,282,148,94,221,226,170,125,171,252,32,269),(8,104,203,83,60,47,283,149,95,205,227,154,126,172,253,33,270),(9,105,204,84,61,48,284,150,96,206,228,155,127,173,254,34,271),(10,106,188,85,62,49,285,151,97,207,229,156,128,174,255,18,272),(11,107,189,69,63,50,286,152,98,208,230,157,129,175,239,19,256),(12,108,190,70,64,51,287,153,99,209,231,158,130,176,240,20,257),(13,109,191,71,65,35,288,137,100,210,232,159,131,177,241,21,258),(14,110,192,72,66,36,289,138,101,211,233,160,132,178,242,22,259),(15,111,193,73,67,37,273,139,102,212,234,161,133,179,243,23,260),(16,112,194,74,68,38,274,140,86,213,235,162,134,180,244,24,261),(17,113,195,75,52,39,275,141,87,214,236,163,135,181,245,25,262)]])
289 conjugacy classes
| class | 1 | 17A | ··· | 17KB |
| order | 1 | 17 | ··· | 17 |
| size | 1 | 1 | ··· | 1 |
289 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C17 |
| kernel | C172 | C17 |
| # reps | 1 | 288 |
Matrix representation of C172 ►in GL2(𝔽103) generated by
| 64 | 0 |
| 0 | 81 |
| 76 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(103))| [64,0,0,81],[76,0,0,1] >;
C172 in GAP, Magma, Sage, TeX
C_{17}^2 % in TeX
G:=Group("C17^2"); // GroupNames label
G:=SmallGroup(289,2);
// by ID
G=gap.SmallGroup(289,2);
# by ID
G:=PCGroup([2,-17,17]:ExponentLimit:=1);
// Polycyclic
G:=Group<a,b|a^17=b^17=1,a*b=b*a>;
// generators/relations
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