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G = C172order 289 = 172

Elementary abelian group of type [17,17]

direct product, p-group, elementary abelian, monomial

Aliases: C172, SmallGroup(289,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C172
C1C17 — C172
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 36 220 85 133 117 192 247 258 139 173 63 223 285 97 30 163)(2 37 221 69 134 118 193 248 259 140 174 64 224 286 98 31 164)(3 38 205 70 135 119 194 249 260 141 175 65 225 287 99 32 165)(4 39 206 71 136 103 195 250 261 142 176 66 226 288 100 33 166)(5 40 207 72 120 104 196 251 262 143 177 67 227 289 101 34 167)(6 41 208 73 121 105 197 252 263 144 178 68 228 273 102 18 168)(7 42 209 74 122 106 198 253 264 145 179 52 229 274 86 19 169)(8 43 210 75 123 107 199 254 265 146 180 53 230 275 87 20 170)(9 44 211 76 124 108 200 255 266 147 181 54 231 276 88 21 154)(10 45 212 77 125 109 201 239 267 148 182 55 232 277 89 22 155)(11 46 213 78 126 110 202 240 268 149 183 56 233 278 90 23 156)(12 47 214 79 127 111 203 241 269 150 184 57 234 279 91 24 157)(13 48 215 80 128 112 204 242 270 151 185 58 235 280 92 25 158)(14 49 216 81 129 113 188 243 271 152 186 59 236 281 93 26 159)(15 50 217 82 130 114 189 244 272 153 187 60 237 282 94 27 160)(16 51 218 83 131 115 190 245 256 137 171 61 238 283 95 28 161)(17 35 219 84 132 116 191 246 257 138 172 62 222 284 96 29 162)

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), (1,36,220,85,133,117,192,247,258,139,173,63,223,285,97,30,163)(2,37,221,69,134,118,193,248,259,140,174,64,224,286,98,31,164)(3,38,205,70,135,119,194,249,260,141,175,65,225,287,99,32,165)(4,39,206,71,136,103,195,250,261,142,176,66,226,288,100,33,166)(5,40,207,72,120,104,196,251,262,143,177,67,227,289,101,34,167)(6,41,208,73,121,105,197,252,263,144,178,68,228,273,102,18,168)(7,42,209,74,122,106,198,253,264,145,179,52,229,274,86,19,169)(8,43,210,75,123,107,199,254,265,146,180,53,230,275,87,20,170)(9,44,211,76,124,108,200,255,266,147,181,54,231,276,88,21,154)(10,45,212,77,125,109,201,239,267,148,182,55,232,277,89,22,155)(11,46,213,78,126,110,202,240,268,149,183,56,233,278,90,23,156)(12,47,214,79,127,111,203,241,269,150,184,57,234,279,91,24,157)(13,48,215,80,128,112,204,242,270,151,185,58,235,280,92,25,158)(14,49,216,81,129,113,188,243,271,152,186,59,236,281,93,26,159)(15,50,217,82,130,114,189,244,272,153,187,60,237,282,94,27,160)(16,51,218,83,131,115,190,245,256,137,171,61,238,283,95,28,161)(17,35,219,84,132,116,191,246,257,138,172,62,222,284,96,29,162)>;

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), (1,36,220,85,133,117,192,247,258,139,173,63,223,285,97,30,163)(2,37,221,69,134,118,193,248,259,140,174,64,224,286,98,31,164)(3,38,205,70,135,119,194,249,260,141,175,65,225,287,99,32,165)(4,39,206,71,136,103,195,250,261,142,176,66,226,288,100,33,166)(5,40,207,72,120,104,196,251,262,143,177,67,227,289,101,34,167)(6,41,208,73,121,105,197,252,263,144,178,68,228,273,102,18,168)(7,42,209,74,122,106,198,253,264,145,179,52,229,274,86,19,169)(8,43,210,75,123,107,199,254,265,146,180,53,230,275,87,20,170)(9,44,211,76,124,108,200,255,266,147,181,54,231,276,88,21,154)(10,45,212,77,125,109,201,239,267,148,182,55,232,277,89,22,155)(11,46,213,78,126,110,202,240,268,149,183,56,233,278,90,23,156)(12,47,214,79,127,111,203,241,269,150,184,57,234,279,91,24,157)(13,48,215,80,128,112,204,242,270,151,185,58,235,280,92,25,158)(14,49,216,81,129,113,188,243,271,152,186,59,236,281,93,26,159)(15,50,217,82,130,114,189,244,272,153,187,60,237,282,94,27,160)(16,51,218,83,131,115,190,245,256,137,171,61,238,283,95,28,161)(17,35,219,84,132,116,191,246,257,138,172,62,222,284,96,29,162) );

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)], [(1,36,220,85,133,117,192,247,258,139,173,63,223,285,97,30,163),(2,37,221,69,134,118,193,248,259,140,174,64,224,286,98,31,164),(3,38,205,70,135,119,194,249,260,141,175,65,225,287,99,32,165),(4,39,206,71,136,103,195,250,261,142,176,66,226,288,100,33,166),(5,40,207,72,120,104,196,251,262,143,177,67,227,289,101,34,167),(6,41,208,73,121,105,197,252,263,144,178,68,228,273,102,18,168),(7,42,209,74,122,106,198,253,264,145,179,52,229,274,86,19,169),(8,43,210,75,123,107,199,254,265,146,180,53,230,275,87,20,170),(9,44,211,76,124,108,200,255,266,147,181,54,231,276,88,21,154),(10,45,212,77,125,109,201,239,267,148,182,55,232,277,89,22,155),(11,46,213,78,126,110,202,240,268,149,183,56,233,278,90,23,156),(12,47,214,79,127,111,203,241,269,150,184,57,234,279,91,24,157),(13,48,215,80,128,112,204,242,270,151,185,58,235,280,92,25,158),(14,49,216,81,129,113,188,243,271,152,186,59,236,281,93,26,159),(15,50,217,82,130,114,189,244,272,153,187,60,237,282,94,27,160),(16,51,218,83,131,115,190,245,256,137,171,61,238,283,95,28,161),(17,35,219,84,132,116,191,246,257,138,172,62,222,284,96,29,162)])

289 conjugacy classes

class 1 17A···17KB
order117···17
size11···1

289 irreducible representations

dim11
type+
imageC1C17
kernelC172C17
# reps1288

Matrix representation of C172 in GL2(𝔽103) generated by

640
081
,
760
01
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

Export

Subgroup lattice of C172 in TeX

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